Interval Arithmetic (IA) Fundamentals
What is IA? Variations Interval Arithmetic: Underlying Rationale Fundamentals, History, and Semantics Successes Famous Proofs Engineering Verifications History Ralph Baker Kearfott Early Moore Karlsruhe Russian Department of Mathematics
Logical Pitfalls University of Louisiana at Lafayette Constraints Simplex representations Existence Verification BIRS Casa Oaxaca Seminar, November 13, 2016 The IEEE Standard
1/31
1 / 31 Outline
What is IA? Interval Arithmetic (IA) Fundamentals Variations
What is IA? Underlying Rationale Variations Successes Underlying Rationale Famous Proofs
Successes Engineering Verifications Famous Proofs Engineering Verifications History History Early Early Moore Moore Karlsruhe Russian Karlsruhe Logical Pitfalls Russian Constraints Simplex representations Existence Verification Logical Pitfalls The IEEE Constraints Standard Simplex representations Existence Verification 2/31 The IEEE Standard
2 / 31 I The result of the operation is defined logically for ∈ {+, −, ×, ÷} as x y = {x y | x ∈ x and y ∈ y}.
I The logical definition leads to operational definitions: x + y = [x + y, x + y], x − y = [x − y, x − y], x × y = [min{xy, xy, xy, xy}, max{xy, xy, xy, xy}] 1 1 1 = [ , ] if x > 0 or x < 0 x x x 1 x ÷ y = x × y (There are alternatives for × and ÷ more efficient for certain architectures.)
Classical Interval Arithmetic Definition
Interval Arithmetic (IA) I Operations are defined over the set of closed and Fundamentals bounded intervals x = [x, x]. What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
3/31
3 / 31 I The logical definition leads to operational definitions: x + y = [x + y, x + y], x − y = [x − y, x − y], x × y = [min{xy, xy, xy, xy}, max{xy, xy, xy, xy}] 1 1 1 = [ , ] if x > 0 or x < 0 x x x 1 x ÷ y = x × y (There are alternatives for × and ÷ more efficient for certain architectures.)
Classical Interval Arithmetic Definition
Interval Arithmetic (IA) I Operations are defined over the set of closed and Fundamentals bounded intervals x = [x, x]. What is IA? I The result of the operation is defined logically for Variations ∈ {+, −, ×, ÷} as x y = {x y | x ∈ x and y ∈ y}. Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
3/31
3 / 31 Classical Interval Arithmetic Definition
Interval Arithmetic (IA) I Operations are defined over the set of closed and Fundamentals bounded intervals x = [x, x]. What is IA? I The result of the operation is defined logically for Variations ∈ {+, −, ×, ÷} as x y = {x y | x ∈ x and y ∈ y}. Underlying Rationale I The logical definition leads to operational definitions: Successes Famous Proofs x + y = [x + y, x + y], Engineering Verifications
History x − y = [x − y, x − y], Early Moore Karlsruhe x × y = [min{xy, xy, xy, xy}, max{xy, xy, xy, xy}] Russian 1 1 1 Logical Pitfalls = [ , ] > < Constraints if x 0 or x 0 Simplex representations x x x Existence Verification 1 The IEEE x ÷ y = x × Standard y (There are alternatives for × and ÷ more efficient for certain
architectures.) 3/31
3 / 31 I Evaluating an expression in interval arithmetic does not give an exact range of the expression, but does give bounds on the range of the expression.
I Example (interval dependence) If f (x) = (x + 1)(x − 1), then
f ([−2, 2]) = [−2, 2] + 1 [−2, 2] − 1 = [−1, 3][−3, 1] = [−9, 3],
whereas the exact range is [−1, 3].
I The interval [−9, 3] represents the exact range of ˜f (x, y) = (x + 1)(y − 1) over the rectangle x ∈ [−2, 2], y ∈ [−2, 2] (when x and y vary independently).
Classical Interval Arithmetic What does this definition do?
Interval Arithmetic (IA) I In exact arithmetic, the operational definitions give the Fundamentals exact ranges of the elementary operations.
What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
4/31
4 / 31 I Example (interval dependence) If f (x) = (x + 1)(x − 1), then
f ([−2, 2]) = [−2, 2] + 1 [−2, 2] − 1 = [−1, 3][−3, 1] = [−9, 3],
whereas the exact range is [−1, 3].
I The interval [−9, 3] represents the exact range of ˜f (x, y) = (x + 1)(y − 1) over the rectangle x ∈ [−2, 2], y ∈ [−2, 2] (when x and y vary independently).
Classical Interval Arithmetic What does this definition do?
Interval Arithmetic (IA) I In exact arithmetic, the operational definitions give the Fundamentals exact ranges of the elementary operations.
What is IA? I Evaluating an expression in interval arithmetic does not Variations give an exact range of the expression, but does give Underlying Rationale bounds on the range of the expression.
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
4/31
4 / 31 I The interval [−9, 3] represents the exact range of ˜f (x, y) = (x + 1)(y − 1) over the rectangle x ∈ [−2, 2], y ∈ [−2, 2] (when x and y vary independently).
Classical Interval Arithmetic What does this definition do?
Interval Arithmetic (IA) I In exact arithmetic, the operational definitions give the Fundamentals exact ranges of the elementary operations.
What is IA? I Evaluating an expression in interval arithmetic does not Variations give an exact range of the expression, but does give Underlying Rationale bounds on the range of the expression.
Successes Famous Proofs I Example (interval dependence) Engineering Verifications
History If f (x) = (x + 1)(x − 1), then Early Moore Karlsruhe f ([−2, 2]) = [−2, 2] + 1 [−2, 2] − 1 Russian Logical Pitfalls = [−1, 3][−3, 1] = [−9, 3], Constraints Simplex representations Existence Verification whereas the exact range is [−1, 3]. The IEEE Standard
4/31
4 / 31 Classical Interval Arithmetic What does this definition do?
Interval Arithmetic (IA) I In exact arithmetic, the operational definitions give the Fundamentals exact ranges of the elementary operations.
What is IA? I Evaluating an expression in interval arithmetic does not Variations give an exact range of the expression, but does give Underlying Rationale bounds on the range of the expression.
Successes Famous Proofs I Example (interval dependence) Engineering Verifications
History If f (x) = (x + 1)(x − 1), then Early Moore Karlsruhe f ([−2, 2]) = [−2, 2] + 1 [−2, 2] − 1 Russian Logical Pitfalls = [−1, 3][−3, 1] = [−9, 3], Constraints Simplex representations Existence Verification whereas the exact range is [−1, 3]. The IEEE Standard I The interval [−9, 3] represents the exact range of ˜f (x, y) = (x + 1)(y − 1) over the rectangle x ∈ [−2, 2], y ∈ [−2, 2] (when x and y vary independently). 4/31
4 / 31 I Modern computational environments (such as IEEE 754-compliant ones) allow rounding down to the nearest machine number less than the exact result and rounding up to the nearest machine number greater than the exact result.
I If we use downward rounding to compute the lower end point and upward rounding to compute the upper end point, the result of each elementary operation contains the exact range of that operation.
I Hence, an interval evaluation of an expression on a machine mathematically rigorously contains the range of the expression.
Classical Interval Arithmetic Why can this be mathematically rigorous with approximate arithmetic?
Interval Arithmetic (IA) Fundamentals I The operational definitions give approximate end points.
What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
5/31
5 / 31 I If we use downward rounding to compute the lower end point and upward rounding to compute the upper end point, the result of each elementary operation contains the exact range of that operation.
I Hence, an interval evaluation of an expression on a machine mathematically rigorously contains the range of the expression.
Classical Interval Arithmetic Why can this be mathematically rigorous with approximate arithmetic?
Interval Arithmetic (IA) Fundamentals I The operational definitions give approximate end points.
What is IA? I Modern computational environments (such as IEEE Variations 754-compliant ones) allow rounding down to the nearest Underlying Rationale machine number less than the exact result and rounding Successes up to the nearest machine number greater than the Famous Proofs Engineering Verifications exact result. History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
5/31
5 / 31 I Hence, an interval evaluation of an expression on a machine mathematically rigorously contains the range of the expression.
Classical Interval Arithmetic Why can this be mathematically rigorous with approximate arithmetic?
Interval Arithmetic (IA) Fundamentals I The operational definitions give approximate end points.
What is IA? I Modern computational environments (such as IEEE Variations 754-compliant ones) allow rounding down to the nearest Underlying Rationale machine number less than the exact result and rounding Successes up to the nearest machine number greater than the Famous Proofs Engineering Verifications exact result.
History I If we use downward rounding to compute the lower end Early Moore point and upward rounding to compute the upper end Karlsruhe Russian point, the result of each elementary operation contains Logical Pitfalls Constraints the exact range of that operation. Simplex representations Existence Verification
The IEEE Standard
5/31
5 / 31 Classical Interval Arithmetic Why can this be mathematically rigorous with approximate arithmetic?
Interval Arithmetic (IA) Fundamentals I The operational definitions give approximate end points.
What is IA? I Modern computational environments (such as IEEE Variations 754-compliant ones) allow rounding down to the nearest Underlying Rationale machine number less than the exact result and rounding Successes up to the nearest machine number greater than the Famous Proofs Engineering Verifications exact result.
History I If we use downward rounding to compute the lower end Early Moore point and upward rounding to compute the upper end Karlsruhe Russian point, the result of each elementary operation contains Logical Pitfalls Constraints the exact range of that operation. Simplex representations Existence Verification I Hence, an interval evaluation of an expression on a The IEEE machine mathematically rigorously contains the range of Standard the expression.
5/31
5 / 31 I There are no additive and multiplicative inverses. [1, 2] − [1, 2] = [−1, 1] I For example: 1 [1, 2] / [1, 2] = 2 , 2 I Interval arithmetic is only subdistributive: a(b + c) ⊆ ab + bc.
I For example, [−1, 1] [−3, −2] + [2, 3] = [−1, 1][−1, 1] =[ −1, 1], while [−1, 1][−3, −2]+[−1, 1][2, 3] = [−3, 3]+[−3, 3] =[ −6, 6].
I Theorem (Single Use Expressions — SUE) In an algebraic expression evaluated in exact interval arithmetic, the result is the exact range if each variable occurs only once in the expression. • Note: The converse is not true.
Algebraic Properties (or lack thereof)
Interval Arithmetic (IA) I Interval arithmetic is commutative and associative. Fundamentals
What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
6/31
6 / 31 [1, 2] − [1, 2] = [−1, 1] I For example: 1 [1, 2] / [1, 2] = 2 , 2 I Interval arithmetic is only subdistributive: a(b + c) ⊆ ab + bc.
I For example, [−1, 1] [−3, −2] + [2, 3] = [−1, 1][−1, 1] =[ −1, 1], while [−1, 1][−3, −2]+[−1, 1][2, 3] = [−3, 3]+[−3, 3] =[ −6, 6].
I Theorem (Single Use Expressions — SUE) In an algebraic expression evaluated in exact interval arithmetic, the result is the exact range if each variable occurs only once in the expression. • Note: The converse is not true.
Algebraic Properties (or lack thereof)
Interval Arithmetic (IA) I Interval arithmetic is commutative and associative. Fundamentals I There are no additive and multiplicative inverses. What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
6/31
6 / 31 I Interval arithmetic is only subdistributive: a(b + c) ⊆ ab + bc.
I For example, [−1, 1] [−3, −2] + [2, 3] = [−1, 1][−1, 1] =[ −1, 1], while [−1, 1][−3, −2]+[−1, 1][2, 3] = [−3, 3]+[−3, 3] =[ −6, 6].
I Theorem (Single Use Expressions — SUE) In an algebraic expression evaluated in exact interval arithmetic, the result is the exact range if each variable occurs only once in the expression. • Note: The converse is not true.
Algebraic Properties (or lack thereof)
Interval Arithmetic (IA) I Interval arithmetic is commutative and associative. Fundamentals I There are no additive and multiplicative inverses. What is IA? [1, 2] − [1, 2] = [−1, 1] Variations I For example: 1 Underlying [1, 2] / [1, 2] = 2 , 2 Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
6/31
6 / 31 I For example, [−1, 1] [−3, −2] + [2, 3] = [−1, 1][−1, 1] =[ −1, 1], while [−1, 1][−3, −2]+[−1, 1][2, 3] = [−3, 3]+[−3, 3] =[ −6, 6].
I Theorem (Single Use Expressions — SUE) In an algebraic expression evaluated in exact interval arithmetic, the result is the exact range if each variable occurs only once in the expression. • Note: The converse is not true.
Algebraic Properties (or lack thereof)
Interval Arithmetic (IA) I Interval arithmetic is commutative and associative. Fundamentals I There are no additive and multiplicative inverses. What is IA? [1, 2] − [1, 2] = [−1, 1] Variations I For example: 1 Underlying [1, 2] / [1, 2] = 2 , 2 Rationale I Interval arithmetic is only subdistributive: Successes Famous Proofs a(b + c) ⊆ ab + bc. Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
6/31
6 / 31 I Theorem (Single Use Expressions — SUE) In an algebraic expression evaluated in exact interval arithmetic, the result is the exact range if each variable occurs only once in the expression. • Note: The converse is not true.
Algebraic Properties (or lack thereof)
Interval Arithmetic (IA) I Interval arithmetic is commutative and associative. Fundamentals I There are no additive and multiplicative inverses. What is IA? [1, 2] − [1, 2] = [−1, 1] Variations I For example: 1 Underlying [1, 2] / [1, 2] = 2 , 2 Rationale I Interval arithmetic is only subdistributive: Successes Famous Proofs a(b + c) ⊆ ab + bc. Engineering Verifications
History I For example, Early Moore [−1, 1] [−3, −2] + [2, 3] = [−1, 1][−1, 1] =[ −1, 1], while Karlsruhe Russian [−1, 1][−3, −2]+[−1, 1][2, 3] = [−3, 3]+[−3, 3] =[ −6, 6].
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
6/31
6 / 31 • Note: The converse is not true.
Algebraic Properties (or lack thereof)
Interval Arithmetic (IA) I Interval arithmetic is commutative and associative. Fundamentals I There are no additive and multiplicative inverses. What is IA? [1, 2] − [1, 2] = [−1, 1] Variations I For example: 1 Underlying [1, 2] / [1, 2] = 2 , 2 Rationale I Interval arithmetic is only subdistributive: Successes Famous Proofs a(b + c) ⊆ ab + bc. Engineering Verifications
History I For example, Early Moore [−1, 1] [−3, −2] + [2, 3] = [−1, 1][−1, 1] =[ −1, 1], while Karlsruhe Russian [−1, 1][−3, −2]+[−1, 1][2, 3] = [−3, 3]+[−3, 3] =[ −6, 6].
Logical Pitfalls Constraints I Theorem (Single Use Expressions — SUE) Simplex representations Existence Verification In an algebraic expression evaluated in exact interval The IEEE Standard arithmetic, the result is the exact range if each variable occurs only once in the expression.
6/31
6 / 31 Algebraic Properties (or lack thereof)
Interval Arithmetic (IA) I Interval arithmetic is commutative and associative. Fundamentals I There are no additive and multiplicative inverses. What is IA? [1, 2] − [1, 2] = [−1, 1] Variations I For example: 1 Underlying [1, 2] / [1, 2] = 2 , 2 Rationale I Interval arithmetic is only subdistributive: Successes Famous Proofs a(b + c) ⊆ ab + bc. Engineering Verifications
History I For example, Early Moore [−1, 1] [−3, −2] + [2, 3] = [−1, 1][−1, 1] =[ −1, 1], while Karlsruhe Russian [−1, 1][−3, −2]+[−1, 1][2, 3] = [−3, 3]+[−3, 3] =[ −6, 6].
Logical Pitfalls Constraints I Theorem (Single Use Expressions — SUE) Simplex representations Existence Verification In an algebraic expression evaluated in exact interval The IEEE Standard arithmetic, the result is the exact range if each variable occurs only once in the expression.
• Note: The converse is not true. 6/31
6 / 31 Circular arithmetic: Representation as midpoint-radius, but with the midpoint in the complex plane. Elementary operations are not exact, but are mere enclosures. Rectangular arithmetic: An alternative complex interval arithmetic. Addition is exact, but multiplication just gives an enclosure. Kaucher arithmetic, modal arithmetic etc.: Algebraically completes interval arithmetic with additive inverses. It has uses, but interpretation of the results is more complicated, sometimes depending on monotonicity properties.
Alternative “Interval” Systems (Different representations or different semantics)
Interval Midpoint-radius arithmetic: Intervals represented in terms of Arithmetic (IA) Fundamentals midpoint and error; addition gives exact range
What is IA? but multiplication just gives an enclosure for the
Variations range.
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
7/31
7 / 31 Rectangular arithmetic: An alternative complex interval arithmetic. Addition is exact, but multiplication just gives an enclosure. Kaucher arithmetic, modal arithmetic etc.: Algebraically completes interval arithmetic with additive inverses. It has uses, but interpretation of the results is more complicated, sometimes depending on monotonicity properties.
Alternative “Interval” Systems (Different representations or different semantics)
Interval Midpoint-radius arithmetic: Intervals represented in terms of Arithmetic (IA) Fundamentals midpoint and error; addition gives exact range
What is IA? but multiplication just gives an enclosure for the
Variations range. Underlying Circular arithmetic: Representation as midpoint-radius, but Rationale
Successes with the midpoint in the complex plane. Famous Proofs Elementary operations are not exact, but are Engineering Verifications
History mere enclosures. Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
7/31
7 / 31 Kaucher arithmetic, modal arithmetic etc.: Algebraically completes interval arithmetic with additive inverses. It has uses, but interpretation of the results is more complicated, sometimes depending on monotonicity properties.
Alternative “Interval” Systems (Different representations or different semantics)
Interval Midpoint-radius arithmetic: Intervals represented in terms of Arithmetic (IA) Fundamentals midpoint and error; addition gives exact range
What is IA? but multiplication just gives an enclosure for the
Variations range. Underlying Circular arithmetic: Representation as midpoint-radius, but Rationale
Successes with the midpoint in the complex plane. Famous Proofs Elementary operations are not exact, but are Engineering Verifications
History mere enclosures. Early Moore Rectangular arithmetic: An alternative complex interval Karlsruhe Russian arithmetic. Addition is exact, but multiplication Logical Pitfalls just gives an enclosure. Constraints Simplex representations Existence Verification
The IEEE Standard
7/31
7 / 31 Alternative “Interval” Systems (Different representations or different semantics)
Interval Midpoint-radius arithmetic: Intervals represented in terms of Arithmetic (IA) Fundamentals midpoint and error; addition gives exact range
What is IA? but multiplication just gives an enclosure for the
Variations range. Underlying Circular arithmetic: Representation as midpoint-radius, but Rationale
Successes with the midpoint in the complex plane. Famous Proofs Elementary operations are not exact, but are Engineering Verifications
History mere enclosures. Early Moore Rectangular arithmetic: An alternative complex interval Karlsruhe Russian arithmetic. Addition is exact, but multiplication Logical Pitfalls just gives an enclosure. Constraints Simplex representations Existence Verification Kaucher arithmetic, modal arithmetic etc.: Algebraically
The IEEE completes interval arithmetic with additive Standard inverses. It has uses, but interpretation of the results is more complicated, sometimes depending on monotonicity properties. 7/31
7 / 31 1 1 1 I In our operational definition, = , − ??? y 4 3 a I The arguments contain undefined quantities for 0 a ∈ [1, 2], but ...
I The range of the operation over defined quantities is 1 S 1 − ∞, − 3 4 , ∞ . I Different definitions for the operation’s result and different interpretations are appropriate in different contexts. (More to be said later.)
I This has been carefully considered and defined in an exception-tracking framework in the IEEE 1788-2015 standard for interval arithmetic.
Extensions What do we do with this?
Interval x Arithmetic (IA) Consider = [1, 2]/[−3, 4]. Fundamentals y
What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
8/31
8 / 31 a I The arguments contain undefined quantities for 0 a ∈ [1, 2], but ...
I The range of the operation over defined quantities is 1 S 1 − ∞, − 3 4 , ∞ . I Different definitions for the operation’s result and different interpretations are appropriate in different contexts. (More to be said later.)
I This has been carefully considered and defined in an exception-tracking framework in the IEEE 1788-2015 standard for interval arithmetic.
Extensions What do we do with this?
Interval x Arithmetic (IA) Consider = [1, 2]/[−3, 4]. Fundamentals y What is IA? 1 1 1 I In our operational definition, = , − ??? Variations y 4 3 Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
8/31
8 / 31 I The range of the operation over defined quantities is 1 S 1 − ∞, − 3 4 , ∞ . I Different definitions for the operation’s result and different interpretations are appropriate in different contexts. (More to be said later.)
I This has been carefully considered and defined in an exception-tracking framework in the IEEE 1788-2015 standard for interval arithmetic.
Extensions What do we do with this?
Interval x Arithmetic (IA) Consider = [1, 2]/[−3, 4]. Fundamentals y What is IA? 1 1 1 I In our operational definition, = , − ??? Variations y 4 3 Underlying a Rationale I The arguments contain undefined quantities for Successes 0 Famous Proofs a ∈ [1, 2], but ... Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
8/31
8 / 31 I Different definitions for the operation’s result and different interpretations are appropriate in different contexts. (More to be said later.)
I This has been carefully considered and defined in an exception-tracking framework in the IEEE 1788-2015 standard for interval arithmetic.
Extensions What do we do with this?
Interval x Arithmetic (IA) Consider = [1, 2]/[−3, 4]. Fundamentals y What is IA? 1 1 1 I In our operational definition, = , − ??? Variations y 4 3 Underlying a Rationale I The arguments contain undefined quantities for Successes 0 Famous Proofs a ∈ [1, 2], but ... Engineering Verifications
History I The range of the operation over defined quantities is Early − ∞, − 1 S 1 , ∞. Moore 3 4 Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
8/31
8 / 31 I This has been carefully considered and defined in an exception-tracking framework in the IEEE 1788-2015 standard for interval arithmetic.
Extensions What do we do with this?
Interval x Arithmetic (IA) Consider = [1, 2]/[−3, 4]. Fundamentals y What is IA? 1 1 1 I In our operational definition, = , − ??? Variations y 4 3 Underlying a Rationale I The arguments contain undefined quantities for Successes 0 Famous Proofs a ∈ [1, 2], but ... Engineering Verifications
History I The range of the operation over defined quantities is Early − ∞, − 1 S 1 , ∞. Moore 3 4 Karlsruhe Russian I Different definitions for the operation’s result and Logical Pitfalls different interpretations are appropriate in different Constraints Simplex representations contexts. (More to be said later.) Existence Verification
The IEEE Standard
8/31
8 / 31 Extensions What do we do with this?
Interval x Arithmetic (IA) Consider = [1, 2]/[−3, 4]. Fundamentals y What is IA? 1 1 1 I In our operational definition, = , − ??? Variations y 4 3 Underlying a Rationale I The arguments contain undefined quantities for Successes 0 Famous Proofs a ∈ [1, 2], but ... Engineering Verifications
History I The range of the operation over defined quantities is Early − ∞, − 1 S 1 , ∞. Moore 3 4 Karlsruhe Russian I Different definitions for the operation’s result and Logical Pitfalls different interpretations are appropriate in different Constraints Simplex representations contexts. (More to be said later.) Existence Verification
The IEEE I This has been carefully considered and defined in an Standard exception-tracking framework in the IEEE 1788-2015 standard for interval arithmetic. 8/31
8 / 31 • proving the hypotheses of fixed point theorems, • bounding the objective function and proving or disproving feasibility in global optimization algorithms, • proving collision avoidance in robotics, navigation systems, celestial mechanics, • etc.
I Interval widths start out small, on the order of the machine precision, but ...
I overestimation can make results meaningless, and obtaining meaningful results is often tricky. Bounding function ranges over large domains I provides a polynomial-time computation that often gives helpful bounds, for ...
Reasons for Interval Arithmetic (general uses)
Interval Arithmetic (IA) Rigorously bounding roundoff error in floating point Fundamentals computations. What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
9/31
9 / 31 • proving the hypotheses of fixed point theorems, • bounding the objective function and proving or disproving feasibility in global optimization algorithms, • proving collision avoidance in robotics, navigation systems, celestial mechanics, • etc.
I overestimation can make results meaningless, and obtaining meaningful results is often tricky. Bounding function ranges over large domains I provides a polynomial-time computation that often gives helpful bounds, for ...
Reasons for Interval Arithmetic (general uses)
Interval Arithmetic (IA) Rigorously bounding roundoff error in floating point Fundamentals computations. What is IA? I Interval widths start out small, on the order of the Variations machine precision, but ... Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
9/31
9 / 31 • proving the hypotheses of fixed point theorems, • bounding the objective function and proving or disproving feasibility in global optimization algorithms, • proving collision avoidance in robotics, navigation systems, celestial mechanics, • etc.
Bounding function ranges over large domains I provides a polynomial-time computation that often gives helpful bounds, for ...
Reasons for Interval Arithmetic (general uses)
Interval Arithmetic (IA) Rigorously bounding roundoff error in floating point Fundamentals computations. What is IA? I Interval widths start out small, on the order of the Variations machine precision, but ... Underlying Rationale I overestimation can make results meaningless, and Successes obtaining meaningful results is often tricky. Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
9/31
9 / 31 • proving the hypotheses of fixed point theorems, • bounding the objective function and proving or disproving feasibility in global optimization algorithms, • proving collision avoidance in robotics, navigation systems, celestial mechanics, • etc.
I provides a polynomial-time computation that often gives helpful bounds, for ...
Reasons for Interval Arithmetic (general uses)
Interval Arithmetic (IA) Rigorously bounding roundoff error in floating point Fundamentals computations. What is IA? I Interval widths start out small, on the order of the Variations machine precision, but ... Underlying Rationale I overestimation can make results meaningless, and Successes obtaining meaningful results is often tricky. Famous Proofs Engineering Verifications Bounding function ranges over large domains History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
9/31
9 / 31 • proving the hypotheses of fixed point theorems, • bounding the objective function and proving or disproving feasibility in global optimization algorithms, • proving collision avoidance in robotics, navigation systems, celestial mechanics, • etc.
Reasons for Interval Arithmetic (general uses)
Interval Arithmetic (IA) Rigorously bounding roundoff error in floating point Fundamentals computations. What is IA? I Interval widths start out small, on the order of the Variations machine precision, but ... Underlying Rationale I overestimation can make results meaningless, and Successes obtaining meaningful results is often tricky. Famous Proofs Engineering Verifications Bounding function ranges over large domains History Early I provides a polynomial-time computation that often gives Moore ... Karlsruhe helpful bounds, for Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
9/31
9 / 31 • bounding the objective function and proving or disproving feasibility in global optimization algorithms, • proving collision avoidance in robotics, navigation systems, celestial mechanics, • etc.
Reasons for Interval Arithmetic (general uses)
Interval Arithmetic (IA) Rigorously bounding roundoff error in floating point Fundamentals computations. What is IA? I Interval widths start out small, on the order of the Variations machine precision, but ... Underlying Rationale I overestimation can make results meaningless, and Successes obtaining meaningful results is often tricky. Famous Proofs Engineering Verifications Bounding function ranges over large domains History Early I provides a polynomial-time computation that often gives Moore ... Karlsruhe helpful bounds, for Russian • proving the hypotheses of fixed point theorems, Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
9/31
9 / 31 • proving collision avoidance in robotics, navigation systems, celestial mechanics, • etc.
Reasons for Interval Arithmetic (general uses)
Interval Arithmetic (IA) Rigorously bounding roundoff error in floating point Fundamentals computations. What is IA? I Interval widths start out small, on the order of the Variations machine precision, but ... Underlying Rationale I overestimation can make results meaningless, and Successes obtaining meaningful results is often tricky. Famous Proofs Engineering Verifications Bounding function ranges over large domains History Early I provides a polynomial-time computation that often gives Moore ... Karlsruhe helpful bounds, for Russian • proving the hypotheses of fixed point theorems, Logical Pitfalls • bounding the objective function and proving or disproving Constraints Simplex representations feasibility in global optimization algorithms, Existence Verification
The IEEE Standard
9/31
9 / 31 • etc.
Reasons for Interval Arithmetic (general uses)
Interval Arithmetic (IA) Rigorously bounding roundoff error in floating point Fundamentals computations. What is IA? I Interval widths start out small, on the order of the Variations machine precision, but ... Underlying Rationale I overestimation can make results meaningless, and Successes obtaining meaningful results is often tricky. Famous Proofs Engineering Verifications Bounding function ranges over large domains History Early I provides a polynomial-time computation that often gives Moore ... Karlsruhe helpful bounds, for Russian • proving the hypotheses of fixed point theorems, Logical Pitfalls • bounding the objective function and proving or disproving Constraints Simplex representations feasibility in global optimization algorithms, Existence Verification • proving collision avoidance in robotics, navigation The IEEE Standard systems, celestial mechanics,
9/31
9 / 31 Reasons for Interval Arithmetic (general uses)
Interval Arithmetic (IA) Rigorously bounding roundoff error in floating point Fundamentals computations. What is IA? I Interval widths start out small, on the order of the Variations machine precision, but ... Underlying Rationale I overestimation can make results meaningless, and Successes obtaining meaningful results is often tricky. Famous Proofs Engineering Verifications Bounding function ranges over large domains History Early I provides a polynomial-time computation that often gives Moore ... Karlsruhe helpful bounds, for Russian • proving the hypotheses of fixed point theorems, Logical Pitfalls • bounding the objective function and proving or disproving Constraints Simplex representations feasibility in global optimization algorithms, Existence Verification • proving collision avoidance in robotics, navigation The IEEE Standard systems, celestial mechanics, • etc.
9/31
9 / 31 I The Kepler Conjecture (made by Johannes Kepler in 1611) states that the densest packing of spheres in 3-dimensional space does not exceed that of the face-centered cubic packing.
I Thomas Hales used a blueprint proposed by Toth in 1957, for exhaustive enumeration.
I He computed lower bounds on over 5000 cases using linear programming (1998).
I The bounds were verified with interval arithmetic.
I A formal proof is proceeding with the Isabelle and HOL proof systems.
I See https://arxiv.org/abs/1501.02155v1 and https: //en.wikipedia.org/wiki/Kepler_conjecture.
Proof of Important Conjectures Proof of the Kepler Conjecture (Thomas Hales)
Interval Arithmetic (IA) Fundamentals
What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
10/31
10 / 31 I Thomas Hales used a blueprint proposed by Toth in 1957, for exhaustive enumeration.
I He computed lower bounds on over 5000 cases using linear programming (1998).
I The bounds were verified with interval arithmetic.
I A formal proof is proceeding with the Isabelle and HOL proof systems.
I See https://arxiv.org/abs/1501.02155v1 and https: //en.wikipedia.org/wiki/Kepler_conjecture.
Proof of Important Conjectures Proof of the Kepler Conjecture (Thomas Hales)
Interval Arithmetic (IA) Fundamentals I The Kepler Conjecture (made by Johannes Kepler in 1611) states that the densest packing of spheres in What is IA? 3-dimensional space does not exceed that of the Variations face-centered cubic packing. Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
10/31
10 / 31 I He computed lower bounds on over 5000 cases using linear programming (1998).
I The bounds were verified with interval arithmetic.
I A formal proof is proceeding with the Isabelle and HOL proof systems.
I See https://arxiv.org/abs/1501.02155v1 and https: //en.wikipedia.org/wiki/Kepler_conjecture.
Proof of Important Conjectures Proof of the Kepler Conjecture (Thomas Hales)
Interval Arithmetic (IA) Fundamentals I The Kepler Conjecture (made by Johannes Kepler in 1611) states that the densest packing of spheres in What is IA? 3-dimensional space does not exceed that of the Variations face-centered cubic packing. Underlying Rationale I Thomas Hales used a blueprint proposed by Toth in Successes Famous Proofs 1957, for exhaustive enumeration. Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
10/31
10 / 31 I The bounds were verified with interval arithmetic.
I A formal proof is proceeding with the Isabelle and HOL proof systems.
I See https://arxiv.org/abs/1501.02155v1 and https: //en.wikipedia.org/wiki/Kepler_conjecture.
Proof of Important Conjectures Proof of the Kepler Conjecture (Thomas Hales)
Interval Arithmetic (IA) Fundamentals I The Kepler Conjecture (made by Johannes Kepler in 1611) states that the densest packing of spheres in What is IA? 3-dimensional space does not exceed that of the Variations face-centered cubic packing. Underlying Rationale I Thomas Hales used a blueprint proposed by Toth in Successes Famous Proofs 1957, for exhaustive enumeration. Engineering Verifications
History I He computed lower bounds on over 5000 cases using Early linear programming (1998). Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
10/31
10 / 31 I A formal proof is proceeding with the Isabelle and HOL proof systems.
I See https://arxiv.org/abs/1501.02155v1 and https: //en.wikipedia.org/wiki/Kepler_conjecture.
Proof of Important Conjectures Proof of the Kepler Conjecture (Thomas Hales)
Interval Arithmetic (IA) Fundamentals I The Kepler Conjecture (made by Johannes Kepler in 1611) states that the densest packing of spheres in What is IA? 3-dimensional space does not exceed that of the Variations face-centered cubic packing. Underlying Rationale I Thomas Hales used a blueprint proposed by Toth in Successes Famous Proofs 1957, for exhaustive enumeration. Engineering Verifications
History I He computed lower bounds on over 5000 cases using Early linear programming (1998). Moore Karlsruhe Russian I The bounds were verified with interval arithmetic.
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
10/31
10 / 31 I See https://arxiv.org/abs/1501.02155v1 and https: //en.wikipedia.org/wiki/Kepler_conjecture.
Proof of Important Conjectures Proof of the Kepler Conjecture (Thomas Hales)
Interval Arithmetic (IA) Fundamentals I The Kepler Conjecture (made by Johannes Kepler in 1611) states that the densest packing of spheres in What is IA? 3-dimensional space does not exceed that of the Variations face-centered cubic packing. Underlying Rationale I Thomas Hales used a blueprint proposed by Toth in Successes Famous Proofs 1957, for exhaustive enumeration. Engineering Verifications
History I He computed lower bounds on over 5000 cases using Early linear programming (1998). Moore Karlsruhe Russian I The bounds were verified with interval arithmetic.
Logical Pitfalls I A formal proof is proceeding with the Isabelle and HOL Constraints Simplex representations proof systems. Existence Verification
The IEEE Standard
10/31
10 / 31 Proof of Important Conjectures Proof of the Kepler Conjecture (Thomas Hales)
Interval Arithmetic (IA) Fundamentals I The Kepler Conjecture (made by Johannes Kepler in 1611) states that the densest packing of spheres in What is IA? 3-dimensional space does not exceed that of the Variations face-centered cubic packing. Underlying Rationale I Thomas Hales used a blueprint proposed by Toth in Successes Famous Proofs 1957, for exhaustive enumeration. Engineering Verifications
History I He computed lower bounds on over 5000 cases using Early linear programming (1998). Moore Karlsruhe Russian I The bounds were verified with interval arithmetic.
Logical Pitfalls I A formal proof is proceeding with the Isabelle and HOL Constraints Simplex representations proof systems. Existence Verification The IEEE I See https://arxiv.org/abs/1501.02155v1 and Standard https: //en.wikipedia.org/wiki/Kepler_conjecture. 10/31
10 / 31 1994 Hassard, Zhang, Hastings, and Troy use a mathematically rigorous interval-arithmetic-based ODE integrator to prove existence of chaotic solutions in the Lorenz equations. 1998 (and earlier) Mischaikov and Mrozek use Conley index theory and interval arithmetic to prove chaotic solutions in the Lorenz equations for an explicit parameter value. 2001 Warwick Tucker (in dissertation work) used normal form theory and interval arithmetic to solve Stephen Smale’s 14-th problem, namely, that the Lorenz equations have a strange attractor that persists under perturbations of the coefficients in the differential equations.
Proof of Important Conjectures Chaos and attractors for the Lorenz equations (various researchers – 1994 to 2001)
Interval Arithmetic (IA) Fundamentals (The Lorenz equations are a simplified model of weather prediction.) What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
11/31
11 / 31 1998 (and earlier) Mischaikov and Mrozek use Conley index theory and interval arithmetic to prove chaotic solutions in the Lorenz equations for an explicit parameter value. 2001 Warwick Tucker (in dissertation work) used normal form theory and interval arithmetic to solve Stephen Smale’s 14-th problem, namely, that the Lorenz equations have a strange attractor that persists under perturbations of the coefficients in the differential equations.
Proof of Important Conjectures Chaos and attractors for the Lorenz equations (various researchers – 1994 to 2001)
Interval Arithmetic (IA) Fundamentals (The Lorenz equations are a simplified model of weather prediction.) What is IA?
Variations 1994 Hassard, Zhang, Hastings, and Troy use a
Underlying mathematically rigorous interval-arithmetic-based ODE Rationale integrator to prove existence of chaotic solutions in the Successes Famous Proofs Lorenz equations. Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
11/31
11 / 31 2001 Warwick Tucker (in dissertation work) used normal form theory and interval arithmetic to solve Stephen Smale’s 14-th problem, namely, that the Lorenz equations have a strange attractor that persists under perturbations of the coefficients in the differential equations.
Proof of Important Conjectures Chaos and attractors for the Lorenz equations (various researchers – 1994 to 2001)
Interval Arithmetic (IA) Fundamentals (The Lorenz equations are a simplified model of weather prediction.) What is IA?
Variations 1994 Hassard, Zhang, Hastings, and Troy use a
Underlying mathematically rigorous interval-arithmetic-based ODE Rationale integrator to prove existence of chaotic solutions in the Successes Famous Proofs Lorenz equations. Engineering Verifications 1998 (and earlier) Mischaikov and Mrozek use Conley index History Early theory and interval arithmetic to prove chaotic solutions Moore Karlsruhe in the Lorenz equations for an explicit parameter value. Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
11/31
11 / 31 Proof of Important Conjectures Chaos and attractors for the Lorenz equations (various researchers – 1994 to 2001)
Interval Arithmetic (IA) Fundamentals (The Lorenz equations are a simplified model of weather prediction.) What is IA?
Variations 1994 Hassard, Zhang, Hastings, and Troy use a
Underlying mathematically rigorous interval-arithmetic-based ODE Rationale integrator to prove existence of chaotic solutions in the Successes Famous Proofs Lorenz equations. Engineering Verifications 1998 (and earlier) Mischaikov and Mrozek use Conley index History Early theory and interval arithmetic to prove chaotic solutions Moore Karlsruhe in the Lorenz equations for an explicit parameter value. Russian Logical Pitfalls 2001 Warwick Tucker (in dissertation work) used normal form Constraints Simplex representations theory and interval arithmetic to solve Stephen Smale’s Existence Verification 14-th problem, namely, that the Lorenz equations have a The IEEE Standard strange attractor that persists under perturbations of the coefficients in the differential equations.
11/31
11 / 31 2014 Kenta Kobayashi for Computer-Assisted Uniqueness Proof for Stokes’ Wave of Extreme Form, and 2016 Banhelyi, Csendes, Krisztin, and Neumaier for Global attractivity of the zero solution for Wright’s equation (a model in population genetics)
Proof of Important Conjectures Additional work
Interval Arithmetic (IA) Fundamentals
What is IA? The R. E. Moore Prize for application of interval arithmetic Variations has been awarded to various researchers for proving certain Underlying Rationale mathematical conjectures. See http:
Successes //www.cs.utep.edu/interval-comp/honors.html. Famous Proofs Engineering Verifications Among these are:
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
12/31
12 / 31 2016 Banhelyi, Csendes, Krisztin, and Neumaier for Global attractivity of the zero solution for Wright’s equation (a model in population genetics)
Proof of Important Conjectures Additional work
Interval Arithmetic (IA) Fundamentals
What is IA? The R. E. Moore Prize for application of interval arithmetic Variations has been awarded to various researchers for proving certain Underlying Rationale mathematical conjectures. See http:
Successes //www.cs.utep.edu/interval-comp/honors.html. Famous Proofs Engineering Verifications Among these are: History 2014 Kenta Kobayashi for Computer-Assisted Uniqueness Early Moore Proof for Stokes’ Wave of Extreme Form, and Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
12/31
12 / 31 Proof of Important Conjectures Additional work
Interval Arithmetic (IA) Fundamentals
What is IA? The R. E. Moore Prize for application of interval arithmetic Variations has been awarded to various researchers for proving certain Underlying Rationale mathematical conjectures. See http:
Successes //www.cs.utep.edu/interval-comp/honors.html. Famous Proofs Engineering Verifications Among these are: History 2014 Kenta Kobayashi for Computer-Assisted Uniqueness Early Moore Proof for Stokes’ Wave of Extreme Form, and Karlsruhe Russian 2016 Banhelyi, Csendes, Krisztin, and Neumaier for Global Logical Pitfalls Constraints attractivity of the zero solution for Wright’s equation (a Simplex representations Existence Verification model in population genetics)
The IEEE Standard
12/31
12 / 31 1. Simple use of range bounds; 2. Incorporation of range bounds in exhaustive domain searches (branch and bound algorithms) to enclose a global optimum of a minimization problem; 3. Incorporation of range bounds to rigorously enclose solution sets to differential equations in sophisticated mathematically rigorous ODE integrators.
2. Stadtherr et al Correction of major errors in widely used tables of vapor-liquid equilibria. 3. Berz et al Proof of stability of the beam, given assumed tolerances on the geometry and magnets, of the once-proposed superconducting supercollider (and the software continues to be used for other cyclotrons).
Engineering Questions Rigorously Resolved Physics and chemical engineering
Interval Arithmetic (IA) These include: Fundamentals
What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
13/31
13 / 31 2. Incorporation of range bounds in exhaustive domain searches (branch and bound algorithms) to enclose a global optimum of a minimization problem; 3. Incorporation of range bounds to rigorously enclose solution sets to differential equations in sophisticated mathematically rigorous ODE integrators.
2. Stadtherr et al Correction of major errors in widely used tables of vapor-liquid equilibria. 3. Berz et al Proof of stability of the beam, given assumed tolerances on the geometry and magnets, of the once-proposed superconducting supercollider (and the software continues to be used for other cyclotrons).
Engineering Questions Rigorously Resolved Physics and chemical engineering
Interval Arithmetic (IA) These include: Fundamentals 1. Simple use of range bounds; What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
13/31
13 / 31 3. Incorporation of range bounds to rigorously enclose solution sets to differential equations in sophisticated mathematically rigorous ODE integrators.
3. Berz et al Proof of stability of the beam, given assumed tolerances on the geometry and magnets, of the once-proposed superconducting supercollider (and the software continues to be used for other cyclotrons).
Engineering Questions Rigorously Resolved Physics and chemical engineering
Interval Arithmetic (IA) These include: Fundamentals 1. Simple use of range bounds; What is IA? 2. Incorporation of range bounds in exhaustive domain Variations searches (branch and bound algorithms) to enclose a Underlying Rationale global optimum of a minimization problem; Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian 2. Stadtherr et al Correction of major errors in widely used Logical Pitfalls tables of vapor-liquid equilibria. Constraints Simplex representations Existence Verification
The IEEE Standard
13/31
13 / 31 Engineering Questions Rigorously Resolved Physics and chemical engineering
Interval Arithmetic (IA) These include: Fundamentals 1. Simple use of range bounds; What is IA? 2. Incorporation of range bounds in exhaustive domain Variations searches (branch and bound algorithms) to enclose a Underlying Rationale global optimum of a minimization problem; Successes Famous Proofs 3. Incorporation of range bounds to rigorously enclose Engineering Verifications solution sets to differential equations in sophisticated History Early mathematically rigorous ODE integrators. Moore Karlsruhe Russian 2. Stadtherr et al Correction of major errors in widely used Logical Pitfalls tables of vapor-liquid equilibria. Constraints Simplex representations Existence Verification 3. Berz et al Proof of stability of the beam, given assumed The IEEE tolerances on the geometry and magnets, of the Standard once-proposed superconducting supercollider (and the software continues to be used for other cyclotrons). 13/31
13 / 31 I Luc Jaulin et al have used interval constraint propagation to increase both reliability and efficiency of underwater robot control and data analysis in generating maps. (Luc is the 2012 Moore Prize recipient.)
I (Earlier work continuing to the present) The forward manipulator problem (computation of joint angles for a particular robot hand location) is easily solved with exhaustive search (branch and bound) to the corresponding systems of nonlinear equations.
I Interval arithmetic can be used in collision avoidance. In early work (1988) yours truly used Fortran-77-based software to show the set of published solutions to a manipulator problem posed by Alexander Morgan at General Motors was incorrect. This led to discovery of an incorrectly-given coefficient in the paper and to improvement in the software in use at General Motors.
Engineering Questions Rigorously Resolved Robotics
Interval Arithmetic (IA) Fundamentals
What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
14/31
14 / 31 I (Earlier work continuing to the present) The forward manipulator problem (computation of joint angles for a particular robot hand location) is easily solved with exhaustive search (branch and bound) to the corresponding systems of nonlinear equations.
I Interval arithmetic can be used in collision avoidance. In early work (1988) yours truly used Fortran-77-based software to show the set of published solutions to a manipulator problem posed by Alexander Morgan at General Motors was incorrect. This led to discovery of an incorrectly-given coefficient in the paper and to improvement in the software in use at General Motors.
Engineering Questions Rigorously Resolved Robotics
Interval I Luc Jaulin et al have used interval constraint Arithmetic (IA) Fundamentals propagation to increase both reliability and efficiency of
What is IA? underwater robot control and data analysis in generating
Variations maps. (Luc is the 2012 Moore Prize recipient.)
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
14/31
14 / 31 I Interval arithmetic can be used in collision avoidance. In early work (1988) yours truly used Fortran-77-based software to show the set of published solutions to a manipulator problem posed by Alexander Morgan at General Motors was incorrect. This led to discovery of an incorrectly-given coefficient in the paper and to improvement in the software in use at General Motors.
Engineering Questions Rigorously Resolved Robotics
Interval I Luc Jaulin et al have used interval constraint Arithmetic (IA) Fundamentals propagation to increase both reliability and efficiency of
What is IA? underwater robot control and data analysis in generating
Variations maps. (Luc is the 2012 Moore Prize recipient.)
Underlying I (Earlier work continuing to the present) The forward Rationale
Successes manipulator problem (computation of joint angles for a Famous Proofs particular robot hand location) is easily solved with Engineering Verifications
History exhaustive search (branch and bound) to the Early Moore corresponding systems of nonlinear equations. Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
14/31
14 / 31 In early work (1988) yours truly used Fortran-77-based software to show the set of published solutions to a manipulator problem posed by Alexander Morgan at General Motors was incorrect. This led to discovery of an incorrectly-given coefficient in the paper and to improvement in the software in use at General Motors.
Engineering Questions Rigorously Resolved Robotics
Interval I Luc Jaulin et al have used interval constraint Arithmetic (IA) Fundamentals propagation to increase both reliability and efficiency of
What is IA? underwater robot control and data analysis in generating
Variations maps. (Luc is the 2012 Moore Prize recipient.)
Underlying I (Earlier work continuing to the present) The forward Rationale
Successes manipulator problem (computation of joint angles for a Famous Proofs particular robot hand location) is easily solved with Engineering Verifications
History exhaustive search (branch and bound) to the Early Moore corresponding systems of nonlinear equations. Karlsruhe Russian I Interval arithmetic can be used in collision avoidance. Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
14/31
14 / 31 Engineering Questions Rigorously Resolved Robotics
Interval I Luc Jaulin et al have used interval constraint Arithmetic (IA) Fundamentals propagation to increase both reliability and efficiency of
What is IA? underwater robot control and data analysis in generating
Variations maps. (Luc is the 2012 Moore Prize recipient.)
Underlying I (Earlier work continuing to the present) The forward Rationale
Successes manipulator problem (computation of joint angles for a Famous Proofs particular robot hand location) is easily solved with Engineering Verifications
History exhaustive search (branch and bound) to the Early Moore corresponding systems of nonlinear equations. Karlsruhe Russian I Interval arithmetic can be used in collision avoidance. Logical Pitfalls Constraints In early work (1988) yours truly used Fortran-77-based Simplex representations Existence Verification software to show the set of published solutions to a
The IEEE manipulator problem posed by Alexander Morgan at Standard General Motors was incorrect. This led to discovery of an incorrectly-given coefficient in the paper and to improvement in the software in use at General Motors.14/31
14 / 31 Rosaline Cecily Young( Mathematische Annalen, 1931,) prior to digital computers) “The Algebra of Many-Valued Quantities.” The focus is on an arithmetic on limits, where lim inf f (x) and lim sup f (x) are distinct x→x0 x→x0 (such as in in generalized gradients of nonsmooth functions). Ranges and roundoff error do not seem to have been the primary motivation. Paul S. Dwyer (Chapter in Linear Computations, 1951) “Computation with Approximate Numbers.” Interval computations are introduced as an integral part of roundoff error analysis. Mieczyslaw Warmus( Calculus of Approximations, 1956) The motivation is apparently to provide a sound theoretical backing to numerical computation.
History Early work
Interval The same basic interval operations described in all of the Arithmetic (IA) Fundamentals early work, although it was apparently done independently.
What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
15/31
15 / 31 prior to digital computers) “The Algebra of Many-Valued Quantities.” The focus is on an arithmetic on limits, where lim inf f (x) and lim sup f (x) are distinct x→x0 x→x0 (such as in in generalized gradients of nonsmooth functions). Ranges and roundoff error do not seem to have been the primary motivation. Paul S. Dwyer (Chapter in Linear Computations, 1951) “Computation with Approximate Numbers.” Interval computations are introduced as an integral part of roundoff error analysis. Mieczyslaw Warmus( Calculus of Approximations, 1956) The motivation is apparently to provide a sound theoretical backing to numerical computation.
History Early work
Interval The same basic interval operations described in all of the Arithmetic (IA) Fundamentals early work, although it was apparently done independently. Rosaline Cecily Young( Mathematische Annalen, 1931,) What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
15/31
15 / 31 Paul S. Dwyer (Chapter in Linear Computations, 1951) “Computation with Approximate Numbers.” Interval computations are introduced as an integral part of roundoff error analysis. Mieczyslaw Warmus( Calculus of Approximations, 1956) The motivation is apparently to provide a sound theoretical backing to numerical computation.
History Early work
Interval The same basic interval operations described in all of the Arithmetic (IA) Fundamentals early work, although it was apparently done independently. Rosaline Cecily Young( Mathematische Annalen, 1931,) What is IA? prior to digital computers) “The Algebra of Many-Valued Variations
Underlying Quantities.” The focus is on an arithmetic on limits, Rationale where lim inf f (x) and lim sup f (x) are distinct x→x0 x→x0 Successes (such as in in generalized gradients of nonsmooth Famous Proofs Engineering Verifications functions). Ranges and roundoff error do not seem to History Early have been the primary motivation. Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
15/31
15 / 31 “Computation with Approximate Numbers.” Interval computations are introduced as an integral part of roundoff error analysis. Mieczyslaw Warmus( Calculus of Approximations, 1956) The motivation is apparently to provide a sound theoretical backing to numerical computation.
History Early work
Interval The same basic interval operations described in all of the Arithmetic (IA) Fundamentals early work, although it was apparently done independently. Rosaline Cecily Young( Mathematische Annalen, 1931,) What is IA? prior to digital computers) “The Algebra of Many-Valued Variations
Underlying Quantities.” The focus is on an arithmetic on limits, Rationale where lim inf f (x) and lim sup f (x) are distinct x→x0 x→x0 Successes (such as in in generalized gradients of nonsmooth Famous Proofs Engineering Verifications functions). Ranges and roundoff error do not seem to History Early have been the primary motivation. Moore Karlsruhe Paul S. Dwyer (Chapter in Linear Computations, 1951) Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
15/31
15 / 31 Mieczyslaw Warmus( Calculus of Approximations, 1956) The motivation is apparently to provide a sound theoretical backing to numerical computation.
History Early work
Interval The same basic interval operations described in all of the Arithmetic (IA) Fundamentals early work, although it was apparently done independently. Rosaline Cecily Young( Mathematische Annalen, 1931,) What is IA? prior to digital computers) “The Algebra of Many-Valued Variations
Underlying Quantities.” The focus is on an arithmetic on limits, Rationale where lim inf f (x) and lim sup f (x) are distinct x→x0 x→x0 Successes (such as in in generalized gradients of nonsmooth Famous Proofs Engineering Verifications functions). Ranges and roundoff error do not seem to History Early have been the primary motivation. Moore Karlsruhe Paul S. Dwyer (Chapter in Linear Computations, 1951) Russian “Computation with Approximate Numbers.” Interval Logical Pitfalls Constraints computations are introduced as an integral part of Simplex representations Existence Verification roundoff error analysis. The IEEE Standard
15/31
15 / 31 The motivation is apparently to provide a sound theoretical backing to numerical computation.
History Early work
Interval The same basic interval operations described in all of the Arithmetic (IA) Fundamentals early work, although it was apparently done independently. Rosaline Cecily Young( Mathematische Annalen, 1931,) What is IA? prior to digital computers) “The Algebra of Many-Valued Variations
Underlying Quantities.” The focus is on an arithmetic on limits, Rationale where lim inf f (x) and lim sup f (x) are distinct x→x0 x→x0 Successes (such as in in generalized gradients of nonsmooth Famous Proofs Engineering Verifications functions). Ranges and roundoff error do not seem to History Early have been the primary motivation. Moore Karlsruhe Paul S. Dwyer (Chapter in Linear Computations, 1951) Russian “Computation with Approximate Numbers.” Interval Logical Pitfalls Constraints computations are introduced as an integral part of Simplex representations Existence Verification roundoff error analysis. The IEEE Standard Mieczyslaw Warmus( Calculus of Approximations, 1956)
15/31
15 / 31 History Early work
Interval The same basic interval operations described in all of the Arithmetic (IA) Fundamentals early work, although it was apparently done independently. Rosaline Cecily Young( Mathematische Annalen, 1931,) What is IA? prior to digital computers) “The Algebra of Many-Valued Variations
Underlying Quantities.” The focus is on an arithmetic on limits, Rationale where lim inf f (x) and lim sup f (x) are distinct x→x0 x→x0 Successes (such as in in generalized gradients of nonsmooth Famous Proofs Engineering Verifications functions). Ranges and roundoff error do not seem to History Early have been the primary motivation. Moore Karlsruhe Paul S. Dwyer (Chapter in Linear Computations, 1951) Russian “Computation with Approximate Numbers.” Interval Logical Pitfalls Constraints computations are introduced as an integral part of Simplex representations Existence Verification roundoff error analysis. The IEEE Standard Mieczyslaw Warmus( Calculus of Approximations, 1956) The motivation is apparently to provide a sound
theoretical backing to numerical computation. 15/31
15 / 31 I Archimedes’ 2-sided approximation of the circumference of a circle;
I a 1900 book Lectures on Numerical Computing (in German) with error bounds for +, −, ·, / and innacurate input data;
I an 1896 article “On computing with inexact numbers” (in German) in the Journal for Junior Highschool Studies, giving the impression interval computations were standard fare in middle schools;
I 1887, 1879, and 1854 French work where explicit formulas for the elementary operations and rigorous error bounds were given;
I An 1809 work by Gauß in Latin where explicit computation of error bounds, including rounding errors, appears.
Really Early Work (from a talk on the Origin of Intervals by Siegfried Rump) Rump mentions
Interval Arithmetic (IA) Fundamentals
What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
16/31
16 / 31 I a 1900 book Lectures on Numerical Computing (in German) with error bounds for +, −, ·, / and innacurate input data;
I an 1896 article “On computing with inexact numbers” (in German) in the Journal for Junior Highschool Studies, giving the impression interval computations were standard fare in middle schools;
I 1887, 1879, and 1854 French work where explicit formulas for the elementary operations and rigorous error bounds were given;
I An 1809 work by Gauß in Latin where explicit computation of error bounds, including rounding errors, appears.
Really Early Work (from a talk on the Origin of Intervals by Siegfried Rump) Rump mentions
Interval I Archimedes’ 2-sided approximation of the circumference Arithmetic (IA) Fundamentals of a circle;
What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
16/31
16 / 31 I an 1896 article “On computing with inexact numbers” (in German) in the Journal for Junior Highschool Studies, giving the impression interval computations were standard fare in middle schools;
I 1887, 1879, and 1854 French work where explicit formulas for the elementary operations and rigorous error bounds were given;
I An 1809 work by Gauß in Latin where explicit computation of error bounds, including rounding errors, appears.
Really Early Work (from a talk on the Origin of Intervals by Siegfried Rump) Rump mentions
Interval I Archimedes’ 2-sided approximation of the circumference Arithmetic (IA) Fundamentals of a circle;
What is IA? I a 1900 book Lectures on Numerical Computing (in Variations German) with error bounds for +, −, ·, / and innacurate Underlying input data; Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
16/31
16 / 31 I 1887, 1879, and 1854 French work where explicit formulas for the elementary operations and rigorous error bounds were given;
I An 1809 work by Gauß in Latin where explicit computation of error bounds, including rounding errors, appears.
Really Early Work (from a talk on the Origin of Intervals by Siegfried Rump) Rump mentions
Interval I Archimedes’ 2-sided approximation of the circumference Arithmetic (IA) Fundamentals of a circle;
What is IA? I a 1900 book Lectures on Numerical Computing (in Variations German) with error bounds for +, −, ·, / and innacurate Underlying input data; Rationale Successes I an 1896 article “On computing with inexact numbers” (in Famous Proofs Engineering Verifications German) in the Journal for Junior Highschool Studies, History giving the impression interval computations were Early Moore standard fare in middle schools; Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
16/31
16 / 31 I An 1809 work by Gauß in Latin where explicit computation of error bounds, including rounding errors, appears.
Really Early Work (from a talk on the Origin of Intervals by Siegfried Rump) Rump mentions
Interval I Archimedes’ 2-sided approximation of the circumference Arithmetic (IA) Fundamentals of a circle;
What is IA? I a 1900 book Lectures on Numerical Computing (in Variations German) with error bounds for +, −, ·, / and innacurate Underlying input data; Rationale Successes I an 1896 article “On computing with inexact numbers” (in Famous Proofs Engineering Verifications German) in the Journal for Junior Highschool Studies, History giving the impression interval computations were Early Moore standard fare in middle schools; Karlsruhe Russian I 1887, 1879, and 1854 French work where explicit Logical Pitfalls Constraints formulas for the elementary operations and rigorous Simplex representations Existence Verification error bounds were given;
The IEEE Standard
16/31
16 / 31 Really Early Work (from a talk on the Origin of Intervals by Siegfried Rump) Rump mentions
Interval I Archimedes’ 2-sided approximation of the circumference Arithmetic (IA) Fundamentals of a circle;
What is IA? I a 1900 book Lectures on Numerical Computing (in Variations German) with error bounds for +, −, ·, / and innacurate Underlying input data; Rationale Successes I an 1896 article “On computing with inexact numbers” (in Famous Proofs Engineering Verifications German) in the Journal for Junior Highschool Studies, History giving the impression interval computations were Early Moore standard fare in middle schools; Karlsruhe Russian I 1887, 1879, and 1854 French work where explicit Logical Pitfalls Constraints formulas for the elementary operations and rigorous Simplex representations Existence Verification error bounds were given; The IEEE I An 1809 work by Gauß in Latin where explicit Standard computation of error bounds, including rounding errors, appears. 16/31
16 / 31 Teruro Sunaga( RAAG Memoirs, 1958) “Theory of an Interval Algebra and its Application to Numerical Analysis.” The emphasis is on mathematical theory, but the motivation appears to be automatically accounting for uncertainty and error in measurement and computation. Ray Moore (Lockheed Technical Report, 1959) “Automatic Error Analysis in Digital Computation.” The basic operations are given in this monograph. • Numerical solution of ODEs, numerical integration, etc. based on intervals appear in Moore’s 1962 dissertation. • It is made clear that interval computations promise rigorous bounds on the exact result, even when finite (rounded) computer arithmetic is used.
History of Interval Arithmetic It takes off.
Interval Arithmetic (IA) Fundamentals
What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
17/31
17 / 31 “Theory of an Interval Algebra and its Application to Numerical Analysis.” The emphasis is on mathematical theory, but the motivation appears to be automatically accounting for uncertainty and error in measurement and computation. Ray Moore (Lockheed Technical Report, 1959) “Automatic Error Analysis in Digital Computation.” The basic operations are given in this monograph. • Numerical solution of ODEs, numerical integration, etc. based on intervals appear in Moore’s 1962 dissertation. • It is made clear that interval computations promise rigorous bounds on the exact result, even when finite (rounded) computer arithmetic is used.
History of Interval Arithmetic It takes off.
Interval Arithmetic (IA) Fundamentals Teruro Sunaga( RAAG Memoirs, 1958)
What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
17/31
17 / 31 Ray Moore (Lockheed Technical Report, 1959) “Automatic Error Analysis in Digital Computation.” The basic operations are given in this monograph. • Numerical solution of ODEs, numerical integration, etc. based on intervals appear in Moore’s 1962 dissertation. • It is made clear that interval computations promise rigorous bounds on the exact result, even when finite (rounded) computer arithmetic is used.
History of Interval Arithmetic It takes off.
Interval Arithmetic (IA) Fundamentals Teruro Sunaga( RAAG Memoirs, 1958) “Theory of an Interval Algebra and its Application to What is IA? Numerical Analysis.” The emphasis is on mathematical Variations theory, but the motivation appears to be automatically Underlying Rationale accounting for uncertainty and error in measurement Successes and computation. Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
17/31
17 / 31 “Automatic Error Analysis in Digital Computation.” The basic operations are given in this monograph. • Numerical solution of ODEs, numerical integration, etc. based on intervals appear in Moore’s 1962 dissertation. • It is made clear that interval computations promise rigorous bounds on the exact result, even when finite (rounded) computer arithmetic is used.
History of Interval Arithmetic It takes off.
Interval Arithmetic (IA) Fundamentals Teruro Sunaga( RAAG Memoirs, 1958) “Theory of an Interval Algebra and its Application to What is IA? Numerical Analysis.” The emphasis is on mathematical Variations theory, but the motivation appears to be automatically Underlying Rationale accounting for uncertainty and error in measurement Successes and computation. Famous Proofs Engineering Verifications Ray Moore (Lockheed Technical Report, 1959) History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
17/31
17 / 31 • Numerical solution of ODEs, numerical integration, etc. based on intervals appear in Moore’s 1962 dissertation. • It is made clear that interval computations promise rigorous bounds on the exact result, even when finite (rounded) computer arithmetic is used.
History of Interval Arithmetic It takes off.
Interval Arithmetic (IA) Fundamentals Teruro Sunaga( RAAG Memoirs, 1958) “Theory of an Interval Algebra and its Application to What is IA? Numerical Analysis.” The emphasis is on mathematical Variations theory, but the motivation appears to be automatically Underlying Rationale accounting for uncertainty and error in measurement Successes and computation. Famous Proofs Engineering Verifications Ray Moore (Lockheed Technical Report, 1959) History Early “Automatic Error Analysis in Digital Computation.” The Moore Karlsruhe basic operations are given in this monograph. Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
17/31
17 / 31 • It is made clear that interval computations promise rigorous bounds on the exact result, even when finite (rounded) computer arithmetic is used.
History of Interval Arithmetic It takes off.
Interval Arithmetic (IA) Fundamentals Teruro Sunaga( RAAG Memoirs, 1958) “Theory of an Interval Algebra and its Application to What is IA? Numerical Analysis.” The emphasis is on mathematical Variations theory, but the motivation appears to be automatically Underlying Rationale accounting for uncertainty and error in measurement Successes and computation. Famous Proofs Engineering Verifications Ray Moore (Lockheed Technical Report, 1959) History Early “Automatic Error Analysis in Digital Computation.” The Moore Karlsruhe basic operations are given in this monograph. Russian
Logical Pitfalls • Numerical solution of ODEs, numerical integration, etc. Constraints Simplex representations based on intervals appear in Moore’s 1962 dissertation. Existence Verification
The IEEE Standard
17/31
17 / 31 History of Interval Arithmetic It takes off.
Interval Arithmetic (IA) Fundamentals Teruro Sunaga( RAAG Memoirs, 1958) “Theory of an Interval Algebra and its Application to What is IA? Numerical Analysis.” The emphasis is on mathematical Variations theory, but the motivation appears to be automatically Underlying Rationale accounting for uncertainty and error in measurement Successes and computation. Famous Proofs Engineering Verifications Ray Moore (Lockheed Technical Report, 1959) History Early “Automatic Error Analysis in Digital Computation.” The Moore Karlsruhe basic operations are given in this monograph. Russian
Logical Pitfalls • Numerical solution of ODEs, numerical integration, etc. Constraints Simplex representations based on intervals appear in Moore’s 1962 dissertation. Existence Verification • It is made clear that interval computations promise The IEEE Standard rigorous bounds on the exact result, even when finite (rounded) computer arithmetic is used. 17/31
17 / 31 - saw to directed roundings in IEEE 754 (whether or not just for IA).
generated controversy with his IA patents.
Eldon Hansen worked on interval global optimization, with • early collaboration with Ray Moore. • 1992 Global Optimization with Interval Analysis book, • with a second edition in 2003 with William Walster. William Kahan, retired from U.C. Berkeley, • Proposed extended interval arithmetic in the 1960’s, • Chaired the IEEE 754 floating point arithmetic standard working group, and
• mentored several currently prominent students. G. William (Bill) Walster, a long-time interval advocate, • Promoted intervals at Sun Microsystems in the 1990’s. • Oversaw a Sun IA project, but ...
History of Interval Arithmetic (other Americans)
Interval Arithmetic (IA) Fundamentals
What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
18/31
18 / 31 - saw to directed roundings in IEEE 754 (whether or not just for IA).
generated controversy with his IA patents.
• early collaboration with Ray Moore. • 1992 Global Optimization with Interval Analysis book, • with a second edition in 2003 with William Walster. William Kahan, retired from U.C. Berkeley, • Proposed extended interval arithmetic in the 1960’s, • Chaired the IEEE 754 floating point arithmetic standard working group, and
• mentored several currently prominent students. G. William (Bill) Walster, a long-time interval advocate, • Promoted intervals at Sun Microsystems in the 1990’s. • Oversaw a Sun IA project, but ...
History of Interval Arithmetic (other Americans)
Interval Eldon Hansen worked on interval global optimization, with Arithmetic (IA) Fundamentals
What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
18/31
18 / 31 - saw to directed roundings in IEEE 754 (whether or not just for IA).
generated controversy with his IA patents.
• 1992 Global Optimization with Interval Analysis book, • with a second edition in 2003 with William Walster. William Kahan, retired from U.C. Berkeley, • Proposed extended interval arithmetic in the 1960’s, • Chaired the IEEE 754 floating point arithmetic standard working group, and
• mentored several currently prominent students. G. William (Bill) Walster, a long-time interval advocate, • Promoted intervals at Sun Microsystems in the 1990’s. • Oversaw a Sun IA project, but ...
History of Interval Arithmetic (other Americans)
Interval Eldon Hansen worked on interval global optimization, with Arithmetic (IA) Fundamentals • early collaboration with Ray Moore.
What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
18/31
18 / 31 - saw to directed roundings in IEEE 754 (whether or not just for IA).
generated controversy with his IA patents.
• with a second edition in 2003 with William Walster. William Kahan, retired from U.C. Berkeley, • Proposed extended interval arithmetic in the 1960’s, • Chaired the IEEE 754 floating point arithmetic standard working group, and
• mentored several currently prominent students. G. William (Bill) Walster, a long-time interval advocate, • Promoted intervals at Sun Microsystems in the 1990’s. • Oversaw a Sun IA project, but ...
History of Interval Arithmetic (other Americans)
Interval Eldon Hansen worked on interval global optimization, with Arithmetic (IA) Fundamentals • early collaboration with Ray Moore. What is IA? • 1992 Global Optimization with Interval Analysis book, Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
18/31
18 / 31 - saw to directed roundings in IEEE 754 (whether or not just for IA).
generated controversy with his IA patents.
William Kahan, retired from U.C. Berkeley, • Proposed extended interval arithmetic in the 1960’s, • Chaired the IEEE 754 floating point arithmetic standard working group, and
• mentored several currently prominent students. G. William (Bill) Walster, a long-time interval advocate, • Promoted intervals at Sun Microsystems in the 1990’s. • Oversaw a Sun IA project, but ...
History of Interval Arithmetic (other Americans)
Interval Eldon Hansen worked on interval global optimization, with Arithmetic (IA) Fundamentals • early collaboration with Ray Moore. What is IA? • 1992 Global Optimization with Interval Analysis book, Variations • with a second edition in 2003 with William Walster. Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
18/31
18 / 31 - saw to directed roundings in IEEE 754 (whether or not just for IA).
generated controversy with his IA patents.
• Proposed extended interval arithmetic in the 1960’s, • Chaired the IEEE 754 floating point arithmetic standard working group, and
• mentored several currently prominent students. G. William (Bill) Walster, a long-time interval advocate, • Promoted intervals at Sun Microsystems in the 1990’s. • Oversaw a Sun IA project, but ...
History of Interval Arithmetic (other Americans)
Interval Eldon Hansen worked on interval global optimization, with Arithmetic (IA) Fundamentals • early collaboration with Ray Moore. What is IA? • 1992 Global Optimization with Interval Analysis book, Variations • with a second edition in 2003 with William Walster. Underlying Rationale William Kahan, retired from U.C. Berkeley, Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
18/31
18 / 31 - saw to directed roundings in IEEE 754 (whether or not just for IA).
generated controversy with his IA patents.
• Chaired the IEEE 754 floating point arithmetic standard working group, and
• mentored several currently prominent students. G. William (Bill) Walster, a long-time interval advocate, • Promoted intervals at Sun Microsystems in the 1990’s. • Oversaw a Sun IA project, but ...
History of Interval Arithmetic (other Americans)
Interval Eldon Hansen worked on interval global optimization, with Arithmetic (IA) Fundamentals • early collaboration with Ray Moore. What is IA? • 1992 Global Optimization with Interval Analysis book, Variations • with a second edition in 2003 with William Walster. Underlying Rationale William Kahan, retired from U.C. Berkeley, Successes Famous Proofs • Proposed extended interval arithmetic in the 1960’s, Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
18/31
18 / 31 generated controversy with his IA patents.
- saw to directed roundings in IEEE 754 (whether or not just for IA). • mentored several currently prominent students. G. William (Bill) Walster, a long-time interval advocate, • Promoted intervals at Sun Microsystems in the 1990’s. • Oversaw a Sun IA project, but ...
History of Interval Arithmetic (other Americans)
Interval Eldon Hansen worked on interval global optimization, with Arithmetic (IA) Fundamentals • early collaboration with Ray Moore. What is IA? • 1992 Global Optimization with Interval Analysis book, Variations • with a second edition in 2003 with William Walster. Underlying Rationale William Kahan, retired from U.C. Berkeley, Successes Famous Proofs • Proposed extended interval arithmetic in the 1960’s, Engineering Verifications • Chaired the IEEE 754 floating point arithmetic standard History Early working group, and Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
18/31
18 / 31 generated controversy with his IA patents.
• mentored several currently prominent students. G. William (Bill) Walster, a long-time interval advocate, • Promoted intervals at Sun Microsystems in the 1990’s. • Oversaw a Sun IA project, but ...
History of Interval Arithmetic (other Americans)
Interval Eldon Hansen worked on interval global optimization, with Arithmetic (IA) Fundamentals • early collaboration with Ray Moore. What is IA? • 1992 Global Optimization with Interval Analysis book, Variations • with a second edition in 2003 with William Walster. Underlying Rationale William Kahan, retired from U.C. Berkeley, Successes Famous Proofs • Proposed extended interval arithmetic in the 1960’s, Engineering Verifications • Chaired the IEEE 754 floating point arithmetic standard History Early working group, and Moore Karlsruhe - saw to directed roundings in IEEE 754 (whether or not Russian just for IA). Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
18/31
18 / 31 generated controversy with his IA patents.
G. William (Bill) Walster, a long-time interval advocate, • Promoted intervals at Sun Microsystems in the 1990’s. • Oversaw a Sun IA project, but ...
History of Interval Arithmetic (other Americans)
Interval Eldon Hansen worked on interval global optimization, with Arithmetic (IA) Fundamentals • early collaboration with Ray Moore. What is IA? • 1992 Global Optimization with Interval Analysis book, Variations • with a second edition in 2003 with William Walster. Underlying Rationale William Kahan, retired from U.C. Berkeley, Successes Famous Proofs • Proposed extended interval arithmetic in the 1960’s, Engineering Verifications • Chaired the IEEE 754 floating point arithmetic standard History Early working group, and Moore Karlsruhe - saw to directed roundings in IEEE 754 (whether or not Russian just for IA). Logical Pitfalls Constraints • mentored several currently prominent students. Simplex representations Existence Verification
The IEEE Standard
18/31
18 / 31 generated controversy with his IA patents.
• Promoted intervals at Sun Microsystems in the 1990’s. • Oversaw a Sun IA project, but ...
History of Interval Arithmetic (other Americans)
Interval Eldon Hansen worked on interval global optimization, with Arithmetic (IA) Fundamentals • early collaboration with Ray Moore. What is IA? • 1992 Global Optimization with Interval Analysis book, Variations • with a second edition in 2003 with William Walster. Underlying Rationale William Kahan, retired from U.C. Berkeley, Successes Famous Proofs • Proposed extended interval arithmetic in the 1960’s, Engineering Verifications • Chaired the IEEE 754 floating point arithmetic standard History Early working group, and Moore Karlsruhe - saw to directed roundings in IEEE 754 (whether or not Russian just for IA). Logical Pitfalls Constraints • mentored several currently prominent students. Simplex representations Existence Verification G. William (Bill) Walster, a long-time interval advocate, The IEEE Standard
18/31
18 / 31 generated controversy with his IA patents.
• Oversaw a Sun IA project, but ...
History of Interval Arithmetic (other Americans)
Interval Eldon Hansen worked on interval global optimization, with Arithmetic (IA) Fundamentals • early collaboration with Ray Moore. What is IA? • 1992 Global Optimization with Interval Analysis book, Variations • with a second edition in 2003 with William Walster. Underlying Rationale William Kahan, retired from U.C. Berkeley, Successes Famous Proofs • Proposed extended interval arithmetic in the 1960’s, Engineering Verifications • Chaired the IEEE 754 floating point arithmetic standard History Early working group, and Moore Karlsruhe - saw to directed roundings in IEEE 754 (whether or not Russian just for IA). Logical Pitfalls Constraints • mentored several currently prominent students. Simplex representations Existence Verification G. William (Bill) Walster, a long-time interval advocate, The IEEE Standard • Promoted intervals at Sun Microsystems in the 1990’s.
18/31
18 / 31 generated controversy with his IA patents.
History of Interval Arithmetic (other Americans)
Interval Eldon Hansen worked on interval global optimization, with Arithmetic (IA) Fundamentals • early collaboration with Ray Moore. What is IA? • 1992 Global Optimization with Interval Analysis book, Variations • with a second edition in 2003 with William Walster. Underlying Rationale William Kahan, retired from U.C. Berkeley, Successes Famous Proofs • Proposed extended interval arithmetic in the 1960’s, Engineering Verifications • Chaired the IEEE 754 floating point arithmetic standard History Early working group, and Moore Karlsruhe - saw to directed roundings in IEEE 754 (whether or not Russian just for IA). Logical Pitfalls Constraints • mentored several currently prominent students. Simplex representations Existence Verification G. William (Bill) Walster, a long-time interval advocate, The IEEE Standard • Promoted intervals at Sun Microsystems in the 1990’s. • Oversaw a Sun IA project, but ...
18/31
18 / 31 History of Interval Arithmetic (other Americans)
Interval Eldon Hansen worked on interval global optimization, with Arithmetic (IA) Fundamentals • early collaboration with Ray Moore. What is IA? • 1992 Global Optimization with Interval Analysis book, Variations • with a second edition in 2003 with William Walster. Underlying Rationale William Kahan, retired from U.C. Berkeley, Successes Famous Proofs • Proposed extended interval arithmetic in the 1960’s, Engineering Verifications • Chaired the IEEE 754 floating point arithmetic standard History Early working group, and Moore Karlsruhe - saw to directed roundings in IEEE 754 (whether or not Russian just for IA). Logical Pitfalls Constraints • mentored several currently prominent students. Simplex representations Existence Verification G. William (Bill) Walster, a long-time interval advocate, The IEEE Standard • Promoted intervals at Sun Microsystems in the 1990’s. • Oversaw a Sun IA project, but ... generated controversy with his IA patents. 18/31
18 / 31 - Some of his students have recently proposed alternative algorithms to implement it, and his original proposed implementation is controversial.
Rudolf Krawczyk published his famous Krawczyk method for existence / uniqueness proofs (“Newton-Algorithmen zur Bestimming von Nullstellen mit Fehlerschranken,” 1969) Ulrich Kulisch at Karlsruhe mentored many students. • He is perhaps the most influential figure in the strong German interval analysis school. • (1980’s to the present): His group produced the “SC” languages, with an interval data type. • He pointed out the importance of an accurate dot product, although ...
History of Interval Arithmetic Karlsruhe
Interval Arithmetic (IA) Fundamentals
What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
19/31
19 / 31 - Some of his students have recently proposed alternative algorithms to implement it, and his original proposed implementation is controversial.
Ulrich Kulisch at Karlsruhe mentored many students. • He is perhaps the most influential figure in the strong German interval analysis school. • (1980’s to the present): His group produced the “SC” languages, with an interval data type. • He pointed out the importance of an accurate dot product, although ...
History of Interval Arithmetic Karlsruhe
Interval Arithmetic (IA) Fundamentals Rudolf Krawczyk published his famous Krawczyk method for
What is IA? existence / uniqueness proofs
Variations (“Newton-Algorithmen zur Bestimming von
Underlying Nullstellen mit Fehlerschranken,” 1969) Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
19/31
19 / 31 - Some of his students have recently proposed alternative algorithms to implement it, and his original proposed implementation is controversial.
• He is perhaps the most influential figure in the strong German interval analysis school. • (1980’s to the present): His group produced the “SC” languages, with an interval data type. • He pointed out the importance of an accurate dot product, although ...
History of Interval Arithmetic Karlsruhe
Interval Arithmetic (IA) Fundamentals Rudolf Krawczyk published his famous Krawczyk method for
What is IA? existence / uniqueness proofs
Variations (“Newton-Algorithmen zur Bestimming von
Underlying Nullstellen mit Fehlerschranken,” 1969) Rationale Ulrich Kulisch at Karlsruhe mentored many students. Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
19/31
19 / 31 - Some of his students have recently proposed alternative algorithms to implement it, and his original proposed implementation is controversial.
• (1980’s to the present): His group produced the “SC” languages, with an interval data type. • He pointed out the importance of an accurate dot product, although ...
History of Interval Arithmetic Karlsruhe
Interval Arithmetic (IA) Fundamentals Rudolf Krawczyk published his famous Krawczyk method for
What is IA? existence / uniqueness proofs
Variations (“Newton-Algorithmen zur Bestimming von
Underlying Nullstellen mit Fehlerschranken,” 1969) Rationale Ulrich Kulisch at Karlsruhe mentored many students. Successes Famous Proofs • He is perhaps the most influential figure in the strong Engineering Verifications
History German interval analysis school. Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
19/31
19 / 31 - Some of his students have recently proposed alternative algorithms to implement it, and his original proposed implementation is controversial.
• He pointed out the importance of an accurate dot product, although ...
History of Interval Arithmetic Karlsruhe
Interval Arithmetic (IA) Fundamentals Rudolf Krawczyk published his famous Krawczyk method for
What is IA? existence / uniqueness proofs
Variations (“Newton-Algorithmen zur Bestimming von
Underlying Nullstellen mit Fehlerschranken,” 1969) Rationale Ulrich Kulisch at Karlsruhe mentored many students. Successes Famous Proofs • He is perhaps the most influential figure in the strong Engineering Verifications
History German interval analysis school. Early Moore • (1980’s to the present): His group produced the “SC” Karlsruhe Russian languages, with an interval data type. Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
19/31
19 / 31 - Some of his students have recently proposed alternative algorithms to implement it, and his original proposed implementation is controversial.
History of Interval Arithmetic Karlsruhe
Interval Arithmetic (IA) Fundamentals Rudolf Krawczyk published his famous Krawczyk method for
What is IA? existence / uniqueness proofs
Variations (“Newton-Algorithmen zur Bestimming von
Underlying Nullstellen mit Fehlerschranken,” 1969) Rationale Ulrich Kulisch at Karlsruhe mentored many students. Successes Famous Proofs • He is perhaps the most influential figure in the strong Engineering Verifications
History German interval analysis school. Early Moore • (1980’s to the present): His group produced the “SC” Karlsruhe Russian languages, with an interval data type. Logical Pitfalls • He pointed out the importance of an accurate dot Constraints Simplex representations product, although ... Existence Verification
The IEEE Standard
19/31
19 / 31 History of Interval Arithmetic Karlsruhe
Interval Arithmetic (IA) Fundamentals Rudolf Krawczyk published his famous Krawczyk method for
What is IA? existence / uniqueness proofs
Variations (“Newton-Algorithmen zur Bestimming von
Underlying Nullstellen mit Fehlerschranken,” 1969) Rationale Ulrich Kulisch at Karlsruhe mentored many students. Successes Famous Proofs • He is perhaps the most influential figure in the strong Engineering Verifications
History German interval analysis school. Early Moore • (1980’s to the present): His group produced the “SC” Karlsruhe Russian languages, with an interval data type. Logical Pitfalls • He pointed out the importance of an accurate dot Constraints Simplex representations product, although ... Existence Verification - Some of his students have recently proposed alternative The IEEE Standard algorithms to implement it, and his original proposed implementation is controversial.
19/31
19 / 31 - perhaps the most widely used and cited IA package today.
Götz Alefeld, also at Karlsruhe, • Wrote EInführung in die Intervallrechnung with Jürgen Herzberger in 1974, appearing in English in 1983 as Introduction to Interval Computations. • Did much editorial work, and presided over GAMM (the German society for applied mathematics) Siegfried Rump, a student of Kulisch, • Developed Fortran-SC in the 1980’s, a Matlab-like language with an interval data type, accessing the ACRITH interval package. • Developed INTLAB, a Matlab toolbox for IA,
• Founded the Institute for Reliable Computing at Hamburg, educating students and developing software.
History More Karlsruhe, and the Institute for Reliable Computing
Interval Arithmetic (IA) Fundamentals
What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
20/31
20 / 31 - perhaps the most widely used and cited IA package today.
• Wrote EInführung in die Intervallrechnung with Jürgen Herzberger in 1974, appearing in English in 1983 as Introduction to Interval Computations. • Did much editorial work, and presided over GAMM (the German society for applied mathematics) Siegfried Rump, a student of Kulisch, • Developed Fortran-SC in the 1980’s, a Matlab-like language with an interval data type, accessing the ACRITH interval package. • Developed INTLAB, a Matlab toolbox for IA,
• Founded the Institute for Reliable Computing at Hamburg, educating students and developing software.
History More Karlsruhe, and the Institute for Reliable Computing
Interval Arithmetic (IA) Götz Alefeld, also at Karlsruhe, Fundamentals
What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
20/31
20 / 31 - perhaps the most widely used and cited IA package today.
• Did much editorial work, and presided over GAMM (the German society for applied mathematics) Siegfried Rump, a student of Kulisch, • Developed Fortran-SC in the 1980’s, a Matlab-like language with an interval data type, accessing the ACRITH interval package. • Developed INTLAB, a Matlab toolbox for IA,
• Founded the Institute for Reliable Computing at Hamburg, educating students and developing software.
History More Karlsruhe, and the Institute for Reliable Computing
Interval Arithmetic (IA) Götz Alefeld, also at Karlsruhe, Fundamentals • Wrote EInführung in die Intervallrechnung with Jürgen
What is IA? Herzberger in 1974, appearing in English in 1983 as Variations Introduction to Interval Computations. Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
20/31
20 / 31 - perhaps the most widely used and cited IA package today.
Siegfried Rump, a student of Kulisch, • Developed Fortran-SC in the 1980’s, a Matlab-like language with an interval data type, accessing the ACRITH interval package. • Developed INTLAB, a Matlab toolbox for IA,
• Founded the Institute for Reliable Computing at Hamburg, educating students and developing software.
History More Karlsruhe, and the Institute for Reliable Computing
Interval Arithmetic (IA) Götz Alefeld, also at Karlsruhe, Fundamentals • Wrote EInführung in die Intervallrechnung with Jürgen
What is IA? Herzberger in 1974, appearing in English in 1983 as Variations Introduction to Interval Computations. Underlying Rationale • Did much editorial work, and presided over GAMM (the
Successes German society for applied mathematics) Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
20/31
20 / 31 • Developed Fortran-SC in the 1980’s, a Matlab-like language with an interval data type, accessing the ACRITH interval package. • Developed INTLAB, a Matlab toolbox for IA, - perhaps the most widely used and cited IA package today. • Founded the Institute for Reliable Computing at Hamburg, educating students and developing software.
History More Karlsruhe, and the Institute for Reliable Computing
Interval Arithmetic (IA) Götz Alefeld, also at Karlsruhe, Fundamentals • Wrote EInführung in die Intervallrechnung with Jürgen
What is IA? Herzberger in 1974, appearing in English in 1983 as Variations Introduction to Interval Computations. Underlying Rationale • Did much editorial work, and presided over GAMM (the
Successes German society for applied mathematics) Famous Proofs Engineering Verifications Siegfried Rump, a student of Kulisch, History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
20/31
20 / 31 • Developed INTLAB, a Matlab toolbox for IA, - perhaps the most widely used and cited IA package today. • Founded the Institute for Reliable Computing at Hamburg, educating students and developing software.
History More Karlsruhe, and the Institute for Reliable Computing
Interval Arithmetic (IA) Götz Alefeld, also at Karlsruhe, Fundamentals • Wrote EInführung in die Intervallrechnung with Jürgen
What is IA? Herzberger in 1974, appearing in English in 1983 as Variations Introduction to Interval Computations. Underlying Rationale • Did much editorial work, and presided over GAMM (the
Successes German society for applied mathematics) Famous Proofs Engineering Verifications Siegfried Rump, a student of Kulisch, History • Developed Fortran-SC in the 1980’s, a Matlab-like Early Moore language with an interval data type, accessing the Karlsruhe Russian ACRITH interval package. Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
20/31
20 / 31 • Founded the Institute for Reliable Computing at Hamburg, educating students and developing software.
History More Karlsruhe, and the Institute for Reliable Computing
Interval Arithmetic (IA) Götz Alefeld, also at Karlsruhe, Fundamentals • Wrote EInführung in die Intervallrechnung with Jürgen
What is IA? Herzberger in 1974, appearing in English in 1983 as Variations Introduction to Interval Computations. Underlying Rationale • Did much editorial work, and presided over GAMM (the
Successes German society for applied mathematics) Famous Proofs Engineering Verifications Siegfried Rump, a student of Kulisch, History • Developed Fortran-SC in the 1980’s, a Matlab-like Early Moore language with an interval data type, accessing the Karlsruhe Russian ACRITH interval package. Logical Pitfalls Constraints • Developed INTLAB, a Matlab toolbox for IA, Simplex representations - perhaps the most widely used and cited IA package Existence Verification
The IEEE today. Standard
20/31
20 / 31 History More Karlsruhe, and the Institute for Reliable Computing
Interval Arithmetic (IA) Götz Alefeld, also at Karlsruhe, Fundamentals • Wrote EInführung in die Intervallrechnung with Jürgen
What is IA? Herzberger in 1974, appearing in English in 1983 as Variations Introduction to Interval Computations. Underlying Rationale • Did much editorial work, and presided over GAMM (the
Successes German society for applied mathematics) Famous Proofs Engineering Verifications Siegfried Rump, a student of Kulisch, History • Developed Fortran-SC in the 1980’s, a Matlab-like Early Moore language with an interval data type, accessing the Karlsruhe Russian ACRITH interval package. Logical Pitfalls Constraints • Developed INTLAB, a Matlab toolbox for IA, Simplex representations - perhaps the most widely used and cited IA package Existence Verification
The IEEE today. Standard • Founded the Institute for Reliable Computing at Hamburg, educating students and developing software. 20/31
20 / 31 Peter Henrici at ETH Zürich, • Developed circular complex arithmetic (1972). Karl Nickel at Universität Freiburg, • Advocated IA from the mid-1960’s. • Published the Freiburger Intervallberichte preprint series from 1978 to 1987. (Jürgen Garloff has a complete set; they will be scanned.) • Mentored many successful students and researchers. Arnold Neumaier at Universität Vienna, published • Interval Methods for Systems of Equations (1990). • Leads an exceptional research group at Vienna. • A leader in Global optimization, maintains a global optimization website at http://www.mat.univie.ac.at/~neum/glopt.html
History (Zürich, Freiburg, Vienna)
Interval Arithmetic (IA) Fundamentals
What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
21/31
21 / 31 • Developed circular complex arithmetic (1972). Karl Nickel at Universität Freiburg, • Advocated IA from the mid-1960’s. • Published the Freiburger Intervallberichte preprint series from 1978 to 1987. (Jürgen Garloff has a complete set; they will be scanned.) • Mentored many successful students and researchers. Arnold Neumaier at Universität Vienna, published • Interval Methods for Systems of Equations (1990). • Leads an exceptional research group at Vienna. • A leader in Global optimization, maintains a global optimization website at http://www.mat.univie.ac.at/~neum/glopt.html
History (Zürich, Freiburg, Vienna)
Interval Arithmetic (IA) Peter Henrici at ETH Zürich, Fundamentals
What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
21/31
21 / 31 Karl Nickel at Universität Freiburg, • Advocated IA from the mid-1960’s. • Published the Freiburger Intervallberichte preprint series from 1978 to 1987. (Jürgen Garloff has a complete set; they will be scanned.) • Mentored many successful students and researchers. Arnold Neumaier at Universität Vienna, published • Interval Methods for Systems of Equations (1990). • Leads an exceptional research group at Vienna. • A leader in Global optimization, maintains a global optimization website at http://www.mat.univie.ac.at/~neum/glopt.html
History (Zürich, Freiburg, Vienna)
Interval Arithmetic (IA) Peter Henrici at ETH Zürich, Fundamentals • Developed circular complex arithmetic (1972). What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
21/31
21 / 31 • Advocated IA from the mid-1960’s. • Published the Freiburger Intervallberichte preprint series from 1978 to 1987. (Jürgen Garloff has a complete set; they will be scanned.) • Mentored many successful students and researchers. Arnold Neumaier at Universität Vienna, published • Interval Methods for Systems of Equations (1990). • Leads an exceptional research group at Vienna. • A leader in Global optimization, maintains a global optimization website at http://www.mat.univie.ac.at/~neum/glopt.html
History (Zürich, Freiburg, Vienna)
Interval Arithmetic (IA) Peter Henrici at ETH Zürich, Fundamentals • Developed circular complex arithmetic (1972). What is IA? Karl Nickel at Universität Freiburg, Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
21/31
21 / 31 • Published the Freiburger Intervallberichte preprint series from 1978 to 1987. (Jürgen Garloff has a complete set; they will be scanned.) • Mentored many successful students and researchers. Arnold Neumaier at Universität Vienna, published • Interval Methods for Systems of Equations (1990). • Leads an exceptional research group at Vienna. • A leader in Global optimization, maintains a global optimization website at http://www.mat.univie.ac.at/~neum/glopt.html
History (Zürich, Freiburg, Vienna)
Interval Arithmetic (IA) Peter Henrici at ETH Zürich, Fundamentals • Developed circular complex arithmetic (1972). What is IA? Karl Nickel at Universität Freiburg, Variations
Underlying • Advocated IA from the mid-1960’s. Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
21/31
21 / 31 • Mentored many successful students and researchers. Arnold Neumaier at Universität Vienna, published • Interval Methods for Systems of Equations (1990). • Leads an exceptional research group at Vienna. • A leader in Global optimization, maintains a global optimization website at http://www.mat.univie.ac.at/~neum/glopt.html
History (Zürich, Freiburg, Vienna)
Interval Arithmetic (IA) Peter Henrici at ETH Zürich, Fundamentals • Developed circular complex arithmetic (1972). What is IA? Karl Nickel at Universität Freiburg, Variations
Underlying • Advocated IA from the mid-1960’s. Rationale • Published the Freiburger Intervallberichte preprint series Successes Famous Proofs from 1978 to 1987. (Jürgen Garloff has a complete set; they Engineering Verifications will be scanned.) History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
21/31
21 / 31 Arnold Neumaier at Universität Vienna, published • Interval Methods for Systems of Equations (1990). • Leads an exceptional research group at Vienna. • A leader in Global optimization, maintains a global optimization website at http://www.mat.univie.ac.at/~neum/glopt.html
History (Zürich, Freiburg, Vienna)
Interval Arithmetic (IA) Peter Henrici at ETH Zürich, Fundamentals • Developed circular complex arithmetic (1972). What is IA? Karl Nickel at Universität Freiburg, Variations
Underlying • Advocated IA from the mid-1960’s. Rationale • Published the Freiburger Intervallberichte preprint series Successes Famous Proofs from 1978 to 1987. (Jürgen Garloff has a complete set; they Engineering Verifications will be scanned.) History Early Moore • Mentored many successful students and researchers. Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
21/31
21 / 31 • Interval Methods for Systems of Equations (1990). • Leads an exceptional research group at Vienna. • A leader in Global optimization, maintains a global optimization website at http://www.mat.univie.ac.at/~neum/glopt.html
History (Zürich, Freiburg, Vienna)
Interval Arithmetic (IA) Peter Henrici at ETH Zürich, Fundamentals • Developed circular complex arithmetic (1972). What is IA? Karl Nickel at Universität Freiburg, Variations
Underlying • Advocated IA from the mid-1960’s. Rationale • Published the Freiburger Intervallberichte preprint series Successes Famous Proofs from 1978 to 1987. (Jürgen Garloff has a complete set; they Engineering Verifications will be scanned.) History Early Moore • Mentored many successful students and researchers. Karlsruhe Russian Arnold Neumaier at Universität Vienna, published Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
21/31
21 / 31 • Leads an exceptional research group at Vienna. • A leader in Global optimization, maintains a global optimization website at http://www.mat.univie.ac.at/~neum/glopt.html
History (Zürich, Freiburg, Vienna)
Interval Arithmetic (IA) Peter Henrici at ETH Zürich, Fundamentals • Developed circular complex arithmetic (1972). What is IA? Karl Nickel at Universität Freiburg, Variations
Underlying • Advocated IA from the mid-1960’s. Rationale • Published the Freiburger Intervallberichte preprint series Successes Famous Proofs from 1978 to 1987. (Jürgen Garloff has a complete set; they Engineering Verifications will be scanned.) History Early Moore • Mentored many successful students and researchers. Karlsruhe Russian Arnold Neumaier at Universität Vienna, published Logical Pitfalls • Interval Methods for Systems of Equations (1990). Constraints Simplex representations Existence Verification
The IEEE Standard
21/31
21 / 31 • A leader in Global optimization, maintains a global optimization website at http://www.mat.univie.ac.at/~neum/glopt.html
History (Zürich, Freiburg, Vienna)
Interval Arithmetic (IA) Peter Henrici at ETH Zürich, Fundamentals • Developed circular complex arithmetic (1972). What is IA? Karl Nickel at Universität Freiburg, Variations
Underlying • Advocated IA from the mid-1960’s. Rationale • Published the Freiburger Intervallberichte preprint series Successes Famous Proofs from 1978 to 1987. (Jürgen Garloff has a complete set; they Engineering Verifications will be scanned.) History Early Moore • Mentored many successful students and researchers. Karlsruhe Russian Arnold Neumaier at Universität Vienna, published Logical Pitfalls • Interval Methods for Systems of Equations (1990). Constraints Simplex representations Existence Verification • Leads an exceptional research group at Vienna. The IEEE Standard
21/31
21 / 31 History (Zürich, Freiburg, Vienna)
Interval Arithmetic (IA) Peter Henrici at ETH Zürich, Fundamentals • Developed circular complex arithmetic (1972). What is IA? Karl Nickel at Universität Freiburg, Variations
Underlying • Advocated IA from the mid-1960’s. Rationale • Published the Freiburger Intervallberichte preprint series Successes Famous Proofs from 1978 to 1987. (Jürgen Garloff has a complete set; they Engineering Verifications will be scanned.) History Early Moore • Mentored many successful students and researchers. Karlsruhe Russian Arnold Neumaier at Universität Vienna, published Logical Pitfalls • Interval Methods for Systems of Equations (1990). Constraints Simplex representations Existence Verification • Leads an exceptional research group at Vienna. The IEEE Standard • A leader in Global optimization, maintains a global optimization website at http://www.mat.univie.ac.at/~neum/glopt.html 21/31
21 / 31 now published by the University of Louisiana at Lafayette; see: http://interval.louisiana.edu/ reliable-computing-journal/RC.html
I Vyacheslav Nesterov, Alexander Yakovlev, Eldar Musaev, and others founded the Reliable Computing Journal in 1991,
I A many other salient Russian IA researchers and teachers are Boris Dobronets, Sergey Shary, Irina Dugarova, Nikolaj Glazunov, Grigory Menshikov, ... .
History The Russian school
Interval Arithmetic (IA) Fundamentals I There are extensive publications in all aspects of IA, from 1972, with Yuri Shokin; see What is IA? http://interval.louisiana.edu/ Variations reliable-computing-journal/Supplementum-1/ Underlying for Rationale a list of over 400 such publications. Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
22/31
22 / 31 now published by the University of Louisiana at Lafayette; see: http://interval.louisiana.edu/ reliable-computing-journal/RC.html
I A many other salient Russian IA researchers and teachers are Boris Dobronets, Sergey Shary, Irina Dugarova, Nikolaj Glazunov, Grigory Menshikov, ... .
History The Russian school
Interval Arithmetic (IA) Fundamentals I There are extensive publications in all aspects of IA, from 1972, with Yuri Shokin; see What is IA? http://interval.louisiana.edu/ Variations reliable-computing-journal/Supplementum-1/ Underlying for Rationale a list of over 400 such publications. Successes Famous Proofs I Vyacheslav Nesterov, Alexander Yakovlev, Eldar Engineering Verifications Musaev, and others founded the Reliable Computing History Journal in 1991, Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
22/31
22 / 31 I A many other salient Russian IA researchers and teachers are Boris Dobronets, Sergey Shary, Irina Dugarova, Nikolaj Glazunov, Grigory Menshikov, ... .
History The Russian school
Interval Arithmetic (IA) Fundamentals I There are extensive publications in all aspects of IA, from 1972, with Yuri Shokin; see What is IA? http://interval.louisiana.edu/ Variations reliable-computing-journal/Supplementum-1/ Underlying for Rationale a list of over 400 such publications. Successes Famous Proofs I Vyacheslav Nesterov, Alexander Yakovlev, Eldar Engineering Verifications Musaev, and others founded the Reliable Computing History Journal in 1991, Early Moore now published by the University of Louisiana at Karlsruhe Russian Lafayette; see: Logical Pitfalls http://interval.louisiana.edu/ Constraints Simplex representations reliable-computing-journal/RC.html Existence Verification
The IEEE Standard
22/31
22 / 31 History The Russian school
Interval Arithmetic (IA) Fundamentals I There are extensive publications in all aspects of IA, from 1972, with Yuri Shokin; see What is IA? http://interval.louisiana.edu/ Variations reliable-computing-journal/Supplementum-1/ Underlying for Rationale a list of over 400 such publications. Successes Famous Proofs I Vyacheslav Nesterov, Alexander Yakovlev, Eldar Engineering Verifications Musaev, and others founded the Reliable Computing History Journal in 1991, Early Moore now published by the University of Louisiana at Karlsruhe Russian Lafayette; see: Logical Pitfalls http://interval.louisiana.edu/ Constraints Simplex representations reliable-computing-journal/RC.html Existence Verification
The IEEE I A many other salient Russian IA researchers and Standard teachers are Boris Dobronets, Sergey Shary, Irina Dugarova, Nikolaj Glazunov, Grigory Menshikov, ... . 22/31
22 / 31 • has been active in conferences and educating students, from 1999; • has successfully applied IA to robotics applications; • has authored several books, including Applied Interval Analysis (2000). Jiri Rohn, at the Czech Academy of Sciences, Prague (Charles University), • has advanced the state of the art in interval linear systems and interval linear programming, from 1975. • has made available the VERSOFT Matlab package for interval linear algebra (see http://uivtx.cs.cas.cz/~rohn/matlab/); • has mentored Milan Hladik, active in IA in optimization, and others.
History Others, among many
Interval Arithmetic (IA) Luc Jaulin, at ENSTA-Bretagne, France, Fundamentals
What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
23/31
23 / 31 • has successfully applied IA to robotics applications; • has authored several books, including Applied Interval Analysis (2000). Jiri Rohn, at the Czech Academy of Sciences, Prague (Charles University), • has advanced the state of the art in interval linear systems and interval linear programming, from 1975. • has made available the VERSOFT Matlab package for interval linear algebra (see http://uivtx.cs.cas.cz/~rohn/matlab/); • has mentored Milan Hladik, active in IA in optimization, and others.
History Others, among many
Interval Arithmetic (IA) Luc Jaulin, at ENSTA-Bretagne, France, Fundamentals • has been active in conferences and educating students, What is IA? from 1999; Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
23/31
23 / 31 • has authored several books, including Applied Interval Analysis (2000). Jiri Rohn, at the Czech Academy of Sciences, Prague (Charles University), • has advanced the state of the art in interval linear systems and interval linear programming, from 1975. • has made available the VERSOFT Matlab package for interval linear algebra (see http://uivtx.cs.cas.cz/~rohn/matlab/); • has mentored Milan Hladik, active in IA in optimization, and others.
History Others, among many
Interval Arithmetic (IA) Luc Jaulin, at ENSTA-Bretagne, France, Fundamentals • has been active in conferences and educating students, What is IA? from 1999; Variations • has successfully applied IA to robotics applications; Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
23/31
23 / 31 Jiri Rohn, at the Czech Academy of Sciences, Prague (Charles University), • has advanced the state of the art in interval linear systems and interval linear programming, from 1975. • has made available the VERSOFT Matlab package for interval linear algebra (see http://uivtx.cs.cas.cz/~rohn/matlab/); • has mentored Milan Hladik, active in IA in optimization, and others.
History Others, among many
Interval Arithmetic (IA) Luc Jaulin, at ENSTA-Bretagne, France, Fundamentals • has been active in conferences and educating students, What is IA? from 1999; Variations • has successfully applied IA to robotics applications; Underlying Rationale • has authored several books, including Applied Interval Successes Famous Proofs Analysis (2000). Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
23/31
23 / 31 • has advanced the state of the art in interval linear systems and interval linear programming, from 1975. • has made available the VERSOFT Matlab package for interval linear algebra (see http://uivtx.cs.cas.cz/~rohn/matlab/); • has mentored Milan Hladik, active in IA in optimization, and others.
History Others, among many
Interval Arithmetic (IA) Luc Jaulin, at ENSTA-Bretagne, France, Fundamentals • has been active in conferences and educating students, What is IA? from 1999; Variations • has successfully applied IA to robotics applications; Underlying Rationale • has authored several books, including Applied Interval Successes Famous Proofs Analysis (2000). Engineering Verifications Jiri Rohn, at the Czech Academy of Sciences, Prague History Early (Charles University), Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
23/31
23 / 31 • has made available the VERSOFT Matlab package for interval linear algebra (see http://uivtx.cs.cas.cz/~rohn/matlab/); • has mentored Milan Hladik, active in IA in optimization, and others.
History Others, among many
Interval Arithmetic (IA) Luc Jaulin, at ENSTA-Bretagne, France, Fundamentals • has been active in conferences and educating students, What is IA? from 1999; Variations • has successfully applied IA to robotics applications; Underlying Rationale • has authored several books, including Applied Interval Successes Famous Proofs Analysis (2000). Engineering Verifications Jiri Rohn, at the Czech Academy of Sciences, Prague History Early (Charles University), Moore Karlsruhe • has advanced the state of the art in interval linear Russian
Logical Pitfalls systems and interval linear programming, from 1975. Constraints Simplex representations Existence Verification
The IEEE Standard
23/31
23 / 31 • has mentored Milan Hladik, active in IA in optimization, and others.
History Others, among many
Interval Arithmetic (IA) Luc Jaulin, at ENSTA-Bretagne, France, Fundamentals • has been active in conferences and educating students, What is IA? from 1999; Variations • has successfully applied IA to robotics applications; Underlying Rationale • has authored several books, including Applied Interval Successes Famous Proofs Analysis (2000). Engineering Verifications Jiri Rohn, at the Czech Academy of Sciences, Prague History Early (Charles University), Moore Karlsruhe • has advanced the state of the art in interval linear Russian
Logical Pitfalls systems and interval linear programming, from 1975. Constraints Simplex representations • has made available the VERSOFT Matlab package for Existence Verification interval linear algebra (see The IEEE Standard http://uivtx.cs.cas.cz/~rohn/matlab/);
23/31
23 / 31 History Others, among many
Interval Arithmetic (IA) Luc Jaulin, at ENSTA-Bretagne, France, Fundamentals • has been active in conferences and educating students, What is IA? from 1999; Variations • has successfully applied IA to robotics applications; Underlying Rationale • has authored several books, including Applied Interval Successes Famous Proofs Analysis (2000). Engineering Verifications Jiri Rohn, at the Czech Academy of Sciences, Prague History Early (Charles University), Moore Karlsruhe • has advanced the state of the art in interval linear Russian
Logical Pitfalls systems and interval linear programming, from 1975. Constraints Simplex representations • has made available the VERSOFT Matlab package for Existence Verification interval linear algebra (see The IEEE Standard http://uivtx.cs.cas.cz/~rohn/matlab/); • has mentored Milan Hladik, active in IA in optimization, and others. 23/31
23 / 31 p • in this context, one may interpret ± [−3, 0] as √ evaluation over the portion of the domain where · is defined (partial or loose evaluation). • We obtain x2 ∈ [0, 0], showing that (1, 0) is the only feasible point within the search box. p • Note that ± [−3, 0] represents the set of all x2 with x1 ∈ [1, 2] satisfying the constraint; no problem here.
I If we are searching in the box ([1, 2], [−0.1, 1]), 2 2 I We may solve for, say, x2 in x1 + x2 = 1 to obtain q 2 p p x2 = ± 1 − [1, 2] = ± 1 − [1, 4] = ± [−3, 0]. √ I · is only defined over part of the argument [−3, 0]. However:
Logical Pitfalls Constraint propagation: Interpretation in equality constraints
Interval Arithmetic (IA) Consider minimization of some objective subject to the Fundamentals 2 2 equality constraint x1 + x2 = 1. What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
24/31
24 / 31 p • in this context, one may interpret ± [−3, 0] as √ evaluation over the portion of the domain where · is defined (partial or loose evaluation). • We obtain x2 ∈ [0, 0], showing that (1, 0) is the only feasible point within the search box. p • Note that ± [−3, 0] represents the set of all x2 with x1 ∈ [1, 2] satisfying the constraint; no problem here.
2 2 I We may solve for, say, x2 in x1 + x2 = 1 to obtain q 2 p p x2 = ± 1 − [1, 2] = ± 1 − [1, 4] = ± [−3, 0]. √ I · is only defined over part of the argument [−3, 0]. However:
Logical Pitfalls Constraint propagation: Interpretation in equality constraints
Interval Arithmetic (IA) Consider minimization of some objective subject to the Fundamentals 2 2 equality constraint x1 + x2 = 1. What is IA? I If we are searching in the box ([1, 2], [−0.1, 1]), Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
24/31
24 / 31 p • in this context, one may interpret ± [−3, 0] as √ evaluation over the portion of the domain where · is defined (partial or loose evaluation). • We obtain x2 ∈ [0, 0], showing that (1, 0) is the only feasible point within the search box. p • Note that ± [−3, 0] represents the set of all x2 with x1 ∈ [1, 2] satisfying the constraint; no problem here.
√ I · is only defined over part of the argument [−3, 0]. However:
Logical Pitfalls Constraint propagation: Interpretation in equality constraints
Interval Arithmetic (IA) Consider minimization of some objective subject to the Fundamentals 2 2 equality constraint x1 + x2 = 1. What is IA? I If we are searching in the box ([1, 2], [−0.1, 1]), Variations 2 2 I We may solve for, say, x2 in x + x = 1 to obtain Underlying 1 2 Rationale q Successes 2 p p Famous Proofs x2 = ± 1 − [1, 2] = ± 1 − [1, 4] = ± [−3, 0]. Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
24/31
24 / 31 p • in this context, one may interpret ± [−3, 0] as √ evaluation over the portion of the domain where · is defined (partial or loose evaluation). • We obtain x2 ∈ [0, 0], showing that (1, 0) is the only feasible point within the search box. p • Note that ± [−3, 0] represents the set of all x2 with x1 ∈ [1, 2] satisfying the constraint; no problem here.
Logical Pitfalls Constraint propagation: Interpretation in equality constraints
Interval Arithmetic (IA) Consider minimization of some objective subject to the Fundamentals 2 2 equality constraint x1 + x2 = 1. What is IA? I If we are searching in the box ([1, 2], [−0.1, 1]), Variations 2 2 I We may solve for, say, x2 in x + x = 1 to obtain Underlying 1 2 Rationale q Successes 2 p p Famous Proofs x2 = ± 1 − [1, 2] = ± 1 − [1, 4] = ± [−3, 0]. Engineering Verifications
History √ Early I · is only defined over part of the argument [−3, 0]. Moore However: Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
24/31
24 / 31 • We obtain x2 ∈ [0, 0], showing that (1, 0) is the only feasible point within the search box. p • Note that ± [−3, 0] represents the set of all x2 with x1 ∈ [1, 2] satisfying the constraint; no problem here.
Logical Pitfalls Constraint propagation: Interpretation in equality constraints
Interval Arithmetic (IA) Consider minimization of some objective subject to the Fundamentals 2 2 equality constraint x1 + x2 = 1. What is IA? I If we are searching in the box ([1, 2], [−0.1, 1]), Variations 2 2 I We may solve for, say, x2 in x + x = 1 to obtain Underlying 1 2 Rationale q Successes 2 p p Famous Proofs x2 = ± 1 − [1, 2] = ± 1 − [1, 4] = ± [−3, 0]. Engineering Verifications
History √ Early I · is only defined over part of the argument [−3, 0]. Moore However: Karlsruhe Russian p • in this context, one may interpret ± [−3, 0] as √ Logical Pitfalls evaluation over the portion of the domain where · is Constraints Simplex representations defined (partial or loose evaluation). Existence Verification
The IEEE Standard
24/31
24 / 31 p • Note that ± [−3, 0] represents the set of all x2 with x1 ∈ [1, 2] satisfying the constraint; no problem here.
Logical Pitfalls Constraint propagation: Interpretation in equality constraints
Interval Arithmetic (IA) Consider minimization of some objective subject to the Fundamentals 2 2 equality constraint x1 + x2 = 1. What is IA? I If we are searching in the box ([1, 2], [−0.1, 1]), Variations 2 2 I We may solve for, say, x2 in x + x = 1 to obtain Underlying 1 2 Rationale q Successes 2 p p Famous Proofs x2 = ± 1 − [1, 2] = ± 1 − [1, 4] = ± [−3, 0]. Engineering Verifications
History √ Early I · is only defined over part of the argument [−3, 0]. Moore However: Karlsruhe Russian p • in this context, one may interpret ± [−3, 0] as √ Logical Pitfalls evaluation over the portion of the domain where · is Constraints Simplex representations defined (partial or loose evaluation). Existence Verification • We obtain x2 ∈ [0, 0], showing that (1, 0) is the only The IEEE Standard feasible point within the search box.
24/31
24 / 31 Logical Pitfalls Constraint propagation: Interpretation in equality constraints
Interval Arithmetic (IA) Consider minimization of some objective subject to the Fundamentals 2 2 equality constraint x1 + x2 = 1. What is IA? I If we are searching in the box ([1, 2], [−0.1, 1]), Variations 2 2 I We may solve for, say, x2 in x + x = 1 to obtain Underlying 1 2 Rationale q Successes 2 p p Famous Proofs x2 = ± 1 − [1, 2] = ± 1 − [1, 4] = ± [−3, 0]. Engineering Verifications
History √ Early I · is only defined over part of the argument [−3, 0]. Moore However: Karlsruhe Russian p • in this context, one may interpret ± [−3, 0] as √ Logical Pitfalls evaluation over the portion of the domain where · is Constraints Simplex representations defined (partial or loose evaluation). Existence Verification • We obtain x2 ∈ [0, 0], showing that (1, 0) is the only The IEEE Standard feasible point within the search box. p • Note that ± [−3, 0] represents the set of all x2 with x ∈ [1, 2] satisfying the constraint; no problem here. 1 24/31
24 / 31 • If we solve for x1, we obtain √ √ x1 ≤ [1, 2] orx 1 ≤ [− 2, −1]. √ √ • Here, our conclusion is that x1 ∈ [− 2, −1] ∪ [1, 2], and the computation and logic are straightforward.
• solving√ for x1, we obtain x1 ∈ (−∞, −1] ∪ [1, ∞). • [1, 2] must be replaced√ by [1, ∞); this depends on ≥ and monotonicity of ·. • The interpretation of the interval arithmetic result is different for ≥ than for ≤.
2 2 I Consider an inequality constraint x1 − x2 ≤ 1 within the box ([−3, 3], [−0.1, 1]).
2 2 I If the equality instead had been reversed, x1 − x2 ≥ 1,
Logical Pitfalls Constraint propagation: Interpretation in inequality constraints Which bounds to use and the sense can be confusing.
Interval Arithmetic (IA) Fundamentals
What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
25/31
25 / 31 • solving√ for x1, we obtain x1 ∈ (−∞, −1] ∪ [1, ∞). • [1, 2] must be replaced√ by [1, ∞); this depends on ≥ and monotonicity of ·. • The interpretation of the interval arithmetic result is different for ≥ than for ≤.
• If we solve for x1, we obtain √ √ x1 ≤ [1, 2] orx 1 ≤ [− 2, −1]. √ √ • Here, our conclusion is that x1 ∈ [− 2, −1] ∪ [1, 2], and the computation and logic are straightforward.
2 2 I If the equality instead had been reversed, x1 − x2 ≥ 1,
Logical Pitfalls Constraint propagation: Interpretation in inequality constraints Which bounds to use and the sense can be confusing.
Interval Arithmetic (IA) Fundamentals 2 2 I Consider an inequality constraint x1 − x2 ≤ 1 within the What is IA? box ([−3, 3], [−0.1, 1]).
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
25/31
25 / 31 • solving√ for x1, we obtain x1 ∈ (−∞, −1] ∪ [1, ∞). • [1, 2] must be replaced√ by [1, ∞); this depends on ≥ and monotonicity of ·. • The interpretation of the interval arithmetic result is different for ≥ than for ≤.
√ √ • Here, our conclusion is that x1 ∈ [− 2, −1] ∪ [1, 2], and the computation and logic are straightforward.
2 2 I If the equality instead had been reversed, x1 − x2 ≥ 1,
Logical Pitfalls Constraint propagation: Interpretation in inequality constraints Which bounds to use and the sense can be confusing.
Interval Arithmetic (IA) Fundamentals 2 2 I Consider an inequality constraint x1 − x2 ≤ 1 within the What is IA? box ([−3, 3], [−0.1, 1]). Variations • If we solve for x1, we obtain Underlying √ √ Rationale x1 ≤ [1, 2] orx 1 ≤ [− 2, −1]. Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
25/31
25 / 31 • solving√ for x1, we obtain x1 ∈ (−∞, −1] ∪ [1, ∞). • [1, 2] must be replaced√ by [1, ∞); this depends on ≥ and monotonicity of ·. • The interpretation of the interval arithmetic result is different for ≥ than for ≤.
2 2 I If the equality instead had been reversed, x1 − x2 ≥ 1,
Logical Pitfalls Constraint propagation: Interpretation in inequality constraints Which bounds to use and the sense can be confusing.
Interval Arithmetic (IA) Fundamentals 2 2 I Consider an inequality constraint x1 − x2 ≤ 1 within the What is IA? box ([−3, 3], [−0.1, 1]). Variations • If we solve for x1, we obtain Underlying √ √ Rationale x1 ≤ [1, 2] orx 1 ≤ [− 2, −1]. Successes √ √ Famous Proofs Engineering Verifications • Here, our conclusion is that x1 ∈ [− 2, −1] ∪ [1, 2],
History and the computation and logic are straightforward. Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
25/31
25 / 31 • solving√ for x1, we obtain x1 ∈ (−∞, −1] ∪ [1, ∞). • [1, 2] must be replaced√ by [1, ∞); this depends on ≥ and monotonicity of ·. • The interpretation of the interval arithmetic result is different for ≥ than for ≤.
Logical Pitfalls Constraint propagation: Interpretation in inequality constraints Which bounds to use and the sense can be confusing.
Interval Arithmetic (IA) Fundamentals 2 2 I Consider an inequality constraint x1 − x2 ≤ 1 within the What is IA? box ([−3, 3], [−0.1, 1]). Variations • If we solve for x1, we obtain Underlying √ √ Rationale x1 ≤ [1, 2] orx 1 ≤ [− 2, −1]. Successes √ √ Famous Proofs Engineering Verifications • Here, our conclusion is that x1 ∈ [− 2, −1] ∪ [1, 2],
History and the computation and logic are straightforward. Early Moore Karlsruhe 2 2 Russian I If the equality instead had been reversed, x1 − x2 ≥ 1, Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
25/31
25 / 31 √ • [1, 2] must be replaced√ by [1, ∞); this depends on ≥ and monotonicity of ·. • The interpretation of the interval arithmetic result is different for ≥ than for ≤.
Logical Pitfalls Constraint propagation: Interpretation in inequality constraints Which bounds to use and the sense can be confusing.
Interval Arithmetic (IA) Fundamentals 2 2 I Consider an inequality constraint x1 − x2 ≤ 1 within the What is IA? box ([−3, 3], [−0.1, 1]). Variations • If we solve for x1, we obtain Underlying √ √ Rationale x1 ≤ [1, 2] orx 1 ≤ [− 2, −1]. Successes √ √ Famous Proofs Engineering Verifications • Here, our conclusion is that x1 ∈ [− 2, −1] ∪ [1, 2],
History and the computation and logic are straightforward. Early Moore Karlsruhe 2 2 Russian I If the equality instead had been reversed, x1 − x2 ≥ 1, Logical Pitfalls • solving for x1, we obtain x1 ∈ (−∞, −1] ∪ [1, ∞). Constraints Simplex representations Existence Verification
The IEEE Standard
25/31
25 / 31 • The interpretation of the interval arithmetic result is different for ≥ than for ≤.
Logical Pitfalls Constraint propagation: Interpretation in inequality constraints Which bounds to use and the sense can be confusing.
Interval Arithmetic (IA) Fundamentals 2 2 I Consider an inequality constraint x1 − x2 ≤ 1 within the What is IA? box ([−3, 3], [−0.1, 1]). Variations • If we solve for x1, we obtain Underlying √ √ Rationale x1 ≤ [1, 2] orx 1 ≤ [− 2, −1]. Successes √ √ Famous Proofs Engineering Verifications • Here, our conclusion is that x1 ∈ [− 2, −1] ∪ [1, 2],
History and the computation and logic are straightforward. Early Moore Karlsruhe 2 2 Russian I If the equality instead had been reversed, x1 − x2 ≥ 1, Logical Pitfalls • solving√ for x1, we obtain x1 ∈ (−∞, −1] ∪ [1, ∞). Constraints • [1, 2] must be replaced by [1, ∞); this depends on ≥ Simplex representations √ Existence Verification and monotonicity of ·. The IEEE Standard
25/31
25 / 31 Logical Pitfalls Constraint propagation: Interpretation in inequality constraints Which bounds to use and the sense can be confusing.
Interval Arithmetic (IA) Fundamentals 2 2 I Consider an inequality constraint x1 − x2 ≤ 1 within the What is IA? box ([−3, 3], [−0.1, 1]). Variations • If we solve for x1, we obtain Underlying √ √ Rationale x1 ≤ [1, 2] orx 1 ≤ [− 2, −1]. Successes √ √ Famous Proofs Engineering Verifications • Here, our conclusion is that x1 ∈ [− 2, −1] ∪ [1, 2],
History and the computation and logic are straightforward. Early Moore Karlsruhe 2 2 Russian I If the equality instead had been reversed, x1 − x2 ≥ 1, Logical Pitfalls • solving√ for x1, we obtain x1 ∈ (−∞, −1] ∪ [1, ∞). Constraints • [1, 2] must be replaced by [1, ∞); this depends on ≥ Simplex representations √ Existence Verification and monotonicity of ·. The IEEE • Standard The interpretation of the interval arithmetic result is different for ≥ than for ≤.
25/31
25 / 31 I Suppose we have a simplex S = hP0, P1,..., Pni represented in terms of its vertices n Pi = (x1,i ,... xn,i ) ∈ R , • Pi is only known to lie within a small box Pi , and • we wish to find a set of inequalities, that is, coefficients of + × + A ∈ Rn 1 n and b ∈ Rn 1 such that the set with Ax ≥ b encloses the actual simplex S as sharply as possible.
I For each row Ai,:x ≥ bi , suppose we have an enclosure Ai,: for the normal vector Ai,:, and we adjust bi , so
• Ai,:Pj ≥ bi for 1 ≤ i ≤ n + 1 and 0 ≤ j ≤ n. Then, I the feasible set of Ax ≥ b encloses S for any A ∈ A.
Logical Pitfalls A contrasting context with inequalities: Vertex and half-plane representation of a simplex
Interval Arithmetic (IA) Fundamentals
What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
26/31
26 / 31 • Pi is only known to lie within a small box Pi , and • we wish to find a set of inequalities, that is, coefficients of + × + A ∈ Rn 1 n and b ∈ Rn 1 such that the set with Ax ≥ b encloses the actual simplex S as sharply as possible.
I For each row Ai,:x ≥ bi , suppose we have an enclosure Ai,: for the normal vector Ai,:, and we adjust bi , so
• Ai,:Pj ≥ bi for 1 ≤ i ≤ n + 1 and 0 ≤ j ≤ n. Then, I the feasible set of Ax ≥ b encloses S for any A ∈ A.
Logical Pitfalls A contrasting context with inequalities: Vertex and half-plane representation of a simplex
Interval Arithmetic (IA) Fundamentals I Suppose we have a simplex S = hP , P ,..., P i What is IA? 0 1 n
Variations represented in terms of its vertices n Underlying Pi = (x1,i ,... xn,i ) ∈ R , Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
26/31
26 / 31 • we wish to find a set of inequalities, that is, coefficients of + × + A ∈ Rn 1 n and b ∈ Rn 1 such that the set with Ax ≥ b encloses the actual simplex S as sharply as possible.
I For each row Ai,:x ≥ bi , suppose we have an enclosure Ai,: for the normal vector Ai,:, and we adjust bi , so
• Ai,:Pj ≥ bi for 1 ≤ i ≤ n + 1 and 0 ≤ j ≤ n. Then, I the feasible set of Ax ≥ b encloses S for any A ∈ A.
Logical Pitfalls A contrasting context with inequalities: Vertex and half-plane representation of a simplex
Interval Arithmetic (IA) Fundamentals I Suppose we have a simplex S = hP , P ,..., P i What is IA? 0 1 n
Variations represented in terms of its vertices n Underlying Pi = (x1,i ,... xn,i ) ∈ R , Rationale • Pi is only known to lie within a small box Pi , and Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
26/31
26 / 31 I For each row Ai,:x ≥ bi , suppose we have an enclosure Ai,: for the normal vector Ai,:, and we adjust bi , so
• Ai,:Pj ≥ bi for 1 ≤ i ≤ n + 1 and 0 ≤ j ≤ n. Then, I the feasible set of Ax ≥ b encloses S for any A ∈ A.
Logical Pitfalls A contrasting context with inequalities: Vertex and half-plane representation of a simplex
Interval Arithmetic (IA) Fundamentals I Suppose we have a simplex S = hP , P ,..., P i What is IA? 0 1 n
Variations represented in terms of its vertices n Underlying Pi = (x1,i ,... xn,i ) ∈ R , Rationale • Pi is only known to lie within a small box Pi , and Successes Famous Proofs Engineering Verifications • we wish to find a set of inequalities, that is, coefficients of n+1×n n+1 History A ∈ R and b ∈ R such that the set with Ax ≥ b Early Moore encloses the actual simplex S as sharply as possible. Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
26/31
26 / 31 • Ai,:Pj ≥ bi for 1 ≤ i ≤ n + 1 and 0 ≤ j ≤ n. Then, I the feasible set of Ax ≥ b encloses S for any A ∈ A.
Logical Pitfalls A contrasting context with inequalities: Vertex and half-plane representation of a simplex
Interval Arithmetic (IA) Fundamentals I Suppose we have a simplex S = hP , P ,..., P i What is IA? 0 1 n
Variations represented in terms of its vertices n Underlying Pi = (x1,i ,... xn,i ) ∈ R , Rationale • Pi is only known to lie within a small box Pi , and Successes Famous Proofs Engineering Verifications • we wish to find a set of inequalities, that is, coefficients of n+1×n n+1 History A ∈ R and b ∈ R such that the set with Ax ≥ b Early Moore encloses the actual simplex S as sharply as possible. Karlsruhe Russian I For each row Ai,:x ≥ bi , suppose we have an enclosure Logical Pitfalls Ai,: for the normal vector Ai,:, and we adjust bi , so Constraints Simplex representations Existence Verification
The IEEE Standard
26/31
26 / 31 I the feasible set of Ax ≥ b encloses S for any A ∈ A.
Logical Pitfalls A contrasting context with inequalities: Vertex and half-plane representation of a simplex
Interval Arithmetic (IA) Fundamentals I Suppose we have a simplex S = hP , P ,..., P i What is IA? 0 1 n
Variations represented in terms of its vertices n Underlying Pi = (x1,i ,... xn,i ) ∈ R , Rationale • Pi is only known to lie within a small box Pi , and Successes Famous Proofs Engineering Verifications • we wish to find a set of inequalities, that is, coefficients of n+1×n n+1 History A ∈ R and b ∈ R such that the set with Ax ≥ b Early Moore encloses the actual simplex S as sharply as possible. Karlsruhe Russian I For each row Ai,:x ≥ bi , suppose we have an enclosure Logical Pitfalls Ai,: for the normal vector Ai,:, and we adjust bi , so Constraints Simplex representations Existence Verification • Ai,:Pj ≥ bi for 1 ≤ i ≤ n + 1 and 0 ≤ j ≤ n. Then, The IEEE Standard
26/31
26 / 31 Logical Pitfalls A contrasting context with inequalities: Vertex and half-plane representation of a simplex
Interval Arithmetic (IA) Fundamentals I Suppose we have a simplex S = hP , P ,..., P i What is IA? 0 1 n
Variations represented in terms of its vertices n Underlying Pi = (x1,i ,... xn,i ) ∈ R , Rationale • Pi is only known to lie within a small box Pi , and Successes Famous Proofs Engineering Verifications • we wish to find a set of inequalities, that is, coefficients of n+1×n n+1 History A ∈ R and b ∈ R such that the set with Ax ≥ b Early Moore encloses the actual simplex S as sharply as possible. Karlsruhe Russian I For each row Ai,:x ≥ bi , suppose we have an enclosure Logical Pitfalls Ai,: for the normal vector Ai,:, and we adjust bi , so Constraints Simplex representations Existence Verification • Ai,:Pj ≥ bi for 1 ≤ i ≤ n + 1 and 0 ≤ j ≤ n. Then, The IEEE I the feasible set of Ax ≥ b encloses S for any A ∈ A. Standard
26/31
26 / 31 Left: An n-simplex S enclosed in the polyhedron Tn {Ax ≥ b} = i=0 Hi . Right: A verified floating-point enclosure Sfl of S. Pj . • (Thank you, Sam Karhbet.)
Simplex Representations Illustration (box sizes were exaggerated for clarity)
Interval Arithmetic (IA) Fundamentals
What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
27/31
27 / 31 Right: A verified floating-point enclosure Sfl of S. Pj . • (Thank you, Sam Karhbet.)
Simplex Representations Illustration (box sizes were exaggerated for clarity)
Interval Arithmetic (IA) Fundamentals
What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Left: An n-simplex S enclosed in the polyhedron Existence Verification Tn {Ax ≥ b} = i=0 Hi . The IEEE Standard
27/31
27 / 31 • (Thank you, Sam Karhbet.)
Simplex Representations Illustration (box sizes were exaggerated for clarity)
Interval Arithmetic (IA) Fundamentals
What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Left: An n-simplex S enclosed in the polyhedron Existence Verification Tn {Ax ≥ b} = i=0 Hi . The IEEE Standard Right: A verified floating-point enclosure Sfl of S. Pj .
27/31
27 / 31 Simplex Representations Illustration (box sizes were exaggerated for clarity)
Interval Arithmetic (IA) Fundamentals
What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Left: An n-simplex S enclosed in the polyhedron Existence Verification Tn {Ax ≥ b} = i=0 Hi . The IEEE Standard Right: A verified floating-point enclosure Sfl of S. Pj . • (Thank you, Sam Karhbet.) 27/31
27 / 31 Theorem (Brouwer fixed point theorem) If g is a continuous mapping from a compact convex set x into itself, there is a fixed-point x ∈ x of g, i.e. g(x) = x. n n I If we evaluate g : x ⊂ R → R over an interval vector x and the interval value g(x) ⊆ x, this proves existence of a fixed point of g in x. I Example (thank you, John Pryce) √ Consider g(x) = x − 1 + 0.9, with a fixed point at x ≈ 1.0127 and x ≈ 1.7873. • On x ∈ [1.5, 2], an interval evaluation gives g(x) ⊆ [1.6071, 1.9001] ⊂ [1.5, 2], and we correctly conclude g has a fixed point in [1.6071, 1.9001]. However, ··· √ • if x = [0, 1], x − 1 = p[−1, 0] evaluates to [0, 0], so g(x) = [0.9, 0.9] ⊂ x, for an incorrect conclusion.
Logical Pitfalls Use in existence-uniqueness theory: Care must be taken with partial evaluation and the continuity hypothesis.
Interval Arithmetic (IA) Fundamentals
What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
28/31
28 / 31 n n I If we evaluate g : x ⊂ R → R over an interval vector x and the interval value g(x) ⊆ x, this proves existence of a fixed point of g in x. I Example (thank you, John Pryce) √ Consider g(x) = x − 1 + 0.9, with a fixed point at x ≈ 1.0127 and x ≈ 1.7873. • On x ∈ [1.5, 2], an interval evaluation gives g(x) ⊆ [1.6071, 1.9001] ⊂ [1.5, 2], and we correctly conclude g has a fixed point in [1.6071, 1.9001]. However, ··· √ • if x = [0, 1], x − 1 = p[−1, 0] evaluates to [0, 0], so g(x) = [0.9, 0.9] ⊂ x, for an incorrect conclusion.
Logical Pitfalls Use in existence-uniqueness theory: Care must be taken with partial evaluation and the continuity hypothesis.
Interval Arithmetic (IA) Theorem (Brouwer fixed point theorem) Fundamentals If g is a continuous mapping from a compact convex set x What is IA? into itself, there is a fixed-point x ∈ x of g, i.e. g(x) = x. Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
28/31
28 / 31 I Example (thank you, John Pryce) √ Consider g(x) = x − 1 + 0.9, with a fixed point at x ≈ 1.0127 and x ≈ 1.7873. • On x ∈ [1.5, 2], an interval evaluation gives g(x) ⊆ [1.6071, 1.9001] ⊂ [1.5, 2], and we correctly conclude g has a fixed point in [1.6071, 1.9001]. However, ··· √ • if x = [0, 1], x − 1 = p[−1, 0] evaluates to [0, 0], so g(x) = [0.9, 0.9] ⊂ x, for an incorrect conclusion.
Logical Pitfalls Use in existence-uniqueness theory: Care must be taken with partial evaluation and the continuity hypothesis.
Interval Arithmetic (IA) Theorem (Brouwer fixed point theorem) Fundamentals If g is a continuous mapping from a compact convex set x What is IA? into itself, there is a fixed-point x ∈ x of g, i.e. g(x) = x. n n Variations I If we evaluate g : x ⊂ R → R over an interval vector x Underlying Rationale and the interval value g(x) ⊆ x, this proves existence of
Successes a fixed point of g in x. Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
28/31
28 / 31 • On x ∈ [1.5, 2], an interval evaluation gives g(x) ⊆ [1.6071, 1.9001] ⊂ [1.5, 2], and we correctly conclude g has a fixed point in [1.6071, 1.9001]. However, ··· √ • if x = [0, 1], x − 1 = p[−1, 0] evaluates to [0, 0], so g(x) = [0.9, 0.9] ⊂ x, for an incorrect conclusion.
Logical Pitfalls Use in existence-uniqueness theory: Care must be taken with partial evaluation and the continuity hypothesis.
Interval Arithmetic (IA) Theorem (Brouwer fixed point theorem) Fundamentals If g is a continuous mapping from a compact convex set x What is IA? into itself, there is a fixed-point x ∈ x of g, i.e. g(x) = x. n n Variations I If we evaluate g : x ⊂ R → R over an interval vector x Underlying Rationale and the interval value g(x) ⊆ x, this proves existence of
Successes a fixed point of g in x. Famous Proofs I Example (thank you, John Pryce) Engineering Verifications √ History Consider g(x) = x − 1 + 0.9, with a fixed point at Early Moore x ≈ 1.0127 and x ≈ 1.7873. Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
28/31
28 / 31 √ • if x = [0, 1], x − 1 = p[−1, 0] evaluates to [0, 0], so g(x) = [0.9, 0.9] ⊂ x, for an incorrect conclusion.
Logical Pitfalls Use in existence-uniqueness theory: Care must be taken with partial evaluation and the continuity hypothesis.
Interval Arithmetic (IA) Theorem (Brouwer fixed point theorem) Fundamentals If g is a continuous mapping from a compact convex set x What is IA? into itself, there is a fixed-point x ∈ x of g, i.e. g(x) = x. n n Variations I If we evaluate g : x ⊂ R → R over an interval vector x Underlying Rationale and the interval value g(x) ⊆ x, this proves existence of
Successes a fixed point of g in x. Famous Proofs I Example (thank you, John Pryce) Engineering Verifications √ History Consider g(x) = x − 1 + 0.9, with a fixed point at Early Moore x ≈ 1.0127 and x ≈ 1.7873. Karlsruhe Russian • On x ∈ [1.5, 2], an interval evaluation gives Logical Pitfalls g(x) ⊆ [1.6071, 1.9001] ⊂ [1.5, 2], and we correctly Constraints Simplex representations conclude g has a fixed point in [1.6071, 1.9001]. Existence Verification
The IEEE However, ··· Standard
28/31
28 / 31 Logical Pitfalls Use in existence-uniqueness theory: Care must be taken with partial evaluation and the continuity hypothesis.
Interval Arithmetic (IA) Theorem (Brouwer fixed point theorem) Fundamentals If g is a continuous mapping from a compact convex set x What is IA? into itself, there is a fixed-point x ∈ x of g, i.e. g(x) = x. n n Variations I If we evaluate g : x ⊂ R → R over an interval vector x Underlying Rationale and the interval value g(x) ⊆ x, this proves existence of
Successes a fixed point of g in x. Famous Proofs I Example (thank you, John Pryce) Engineering Verifications √ History Consider g(x) = x − 1 + 0.9, with a fixed point at Early Moore x ≈ 1.0127 and x ≈ 1.7873. Karlsruhe Russian • On x ∈ [1.5, 2], an interval evaluation gives Logical Pitfalls g(x) ⊆ [1.6071, 1.9001] ⊂ [1.5, 2], and we correctly Constraints Simplex representations conclude g has a fixed point in [1.6071, 1.9001]. Existence Verification
The IEEE However, ··· Standard √ • if x = [0, 1], x − 1 = p[−1, 0] evaluates to [0, 0], so g(x) = [0.9, 0.9] ⊂ x, for an incorrect conclusion. 28/31
28 / 31 I There are situations where a condition must hold for every element of a computed interval, and other situations where a any element of a computed interval (or interval vector) may be chosen.
I Simple partial evaluation ignores continuity conditions that are necessary for rigorous existence / uniqueness proofs.
Logical Pitfalls Summary
Interval Arithmetic (IA) Fundamentals
What is IA? I Bounds obtained from interval arithmetic have different Variations interpretations in constraint propagation, depending on Underlying Rationale the sense of the inequality.
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
29/31
29 / 31 I Simple partial evaluation ignores continuity conditions that are necessary for rigorous existence / uniqueness proofs.
Logical Pitfalls Summary
Interval Arithmetic (IA) Fundamentals
What is IA? I Bounds obtained from interval arithmetic have different Variations interpretations in constraint propagation, depending on Underlying Rationale the sense of the inequality.
Successes I There are situations where a condition must hold for Famous Proofs Engineering Verifications every element of a computed interval, and other History Early situations where a any element of a computed interval Moore Karlsruhe (or interval vector) may be chosen. Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
29/31
29 / 31 Logical Pitfalls Summary
Interval Arithmetic (IA) Fundamentals
What is IA? I Bounds obtained from interval arithmetic have different Variations interpretations in constraint propagation, depending on Underlying Rationale the sense of the inequality.
Successes I There are situations where a condition must hold for Famous Proofs Engineering Verifications every element of a computed interval, and other History Early situations where a any element of a computed interval Moore Karlsruhe (or interval vector) may be chosen. Russian I Simple partial evaluation ignores continuity conditions Logical Pitfalls Constraints that are necessary for rigorous existence / uniqueness Simplex representations Existence Verification proofs. The IEEE Standard
29/31
29 / 31 I Defines an optional binding to the IEEE 754-2008 standard for floating point arithmetic.
I Specifies how extended interval arithmetic is handled, from various special cases. Example (The underlying set is R, not R.) 1 [2, 3] , ∞ ← . 2 [0, 4]
I Contains a decoration system for tracking continuity of an expression, if extended interval arithmetic has been used, etc. This can be viewed as a generalization of IEEE 754 exception handling.
I Thank you, John Pryce, IEEE 1788 technical editor and a leader in development of the decoration system.
IEEE 1788-2015 Standard for Interval Arithmetic
Interval Arithmetic (IA) I Defines basic interval arithmetic, specifying accuracy, Fundamentals required elementary functions, etc. What is IA?
Variations
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
30/31
30 / 31 I Specifies how extended interval arithmetic is handled, from various special cases. Example (The underlying set is R, not R.) 1 [2, 3] , ∞ ← . 2 [0, 4]
I Contains a decoration system for tracking continuity of an expression, if extended interval arithmetic has been used, etc. This can be viewed as a generalization of IEEE 754 exception handling.
I Thank you, John Pryce, IEEE 1788 technical editor and a leader in development of the decoration system.
IEEE 1788-2015 Standard for Interval Arithmetic
Interval Arithmetic (IA) I Defines basic interval arithmetic, specifying accuracy, Fundamentals required elementary functions, etc. What is IA? I Defines an optional binding to the IEEE 754-2008 Variations standard for floating point arithmetic. Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
30/31
30 / 31 Example (The underlying set is R, not R.) 1 [2, 3] , ∞ ← . 2 [0, 4]
I Contains a decoration system for tracking continuity of an expression, if extended interval arithmetic has been used, etc. This can be viewed as a generalization of IEEE 754 exception handling.
I Thank you, John Pryce, IEEE 1788 technical editor and a leader in development of the decoration system.
IEEE 1788-2015 Standard for Interval Arithmetic
Interval Arithmetic (IA) I Defines basic interval arithmetic, specifying accuracy, Fundamentals required elementary functions, etc. What is IA? I Defines an optional binding to the IEEE 754-2008 Variations standard for floating point arithmetic. Underlying Rationale I Specifies how extended interval arithmetic is handled, Successes from various special cases. Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
30/31
30 / 31 I Contains a decoration system for tracking continuity of an expression, if extended interval arithmetic has been used, etc. This can be viewed as a generalization of IEEE 754 exception handling.
I Thank you, John Pryce, IEEE 1788 technical editor and a leader in development of the decoration system.
IEEE 1788-2015 Standard for Interval Arithmetic
Interval Arithmetic (IA) I Defines basic interval arithmetic, specifying accuracy, Fundamentals required elementary functions, etc. What is IA? I Defines an optional binding to the IEEE 754-2008 Variations standard for floating point arithmetic. Underlying Rationale I Specifies how extended interval arithmetic is handled, Successes from various special cases. Famous Proofs Engineering Verifications Example (The underlying set is R, not R.) History Early 1 [2, 3] Moore , ∞ ← . Karlsruhe 2 [0, 4] Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
30/31
30 / 31 I Thank you, John Pryce, IEEE 1788 technical editor and a leader in development of the decoration system.
IEEE 1788-2015 Standard for Interval Arithmetic
Interval Arithmetic (IA) I Defines basic interval arithmetic, specifying accuracy, Fundamentals required elementary functions, etc. What is IA? I Defines an optional binding to the IEEE 754-2008 Variations standard for floating point arithmetic. Underlying Rationale I Specifies how extended interval arithmetic is handled, Successes from various special cases. Famous Proofs Engineering Verifications Example (The underlying set is R, not R.) History Early 1 [2, 3] Moore , ∞ ← . Karlsruhe 2 [0, 4] Russian
Logical Pitfalls I Contains a decoration system for tracking continuity of Constraints an expression, if extended interval arithmetic has been Simplex representations Existence Verification used, etc. This can be viewed as a generalization of The IEEE Standard IEEE 754 exception handling.
30/31
30 / 31 IEEE 1788-2015 Standard for Interval Arithmetic
Interval Arithmetic (IA) I Defines basic interval arithmetic, specifying accuracy, Fundamentals required elementary functions, etc. What is IA? I Defines an optional binding to the IEEE 754-2008 Variations standard for floating point arithmetic. Underlying Rationale I Specifies how extended interval arithmetic is handled, Successes from various special cases. Famous Proofs Engineering Verifications Example (The underlying set is R, not R.) History Early 1 [2, 3] Moore , ∞ ← . Karlsruhe 2 [0, 4] Russian
Logical Pitfalls I Contains a decoration system for tracking continuity of Constraints an expression, if extended interval arithmetic has been Simplex representations Existence Verification used, etc. This can be viewed as a generalization of The IEEE Standard IEEE 754 exception handling. I Thank you, John Pryce, IEEE 1788 technical editor and a leader in development of the decoration system. 30/31
30 / 31 JInterval (Java) by Dmitry Nadezhin and Sergei Zhilin. See https://java.net/projects/jinterval C++ by Marco Nehmeier (J. Wolff v. Gudenberg). See https://github.com/nehmeier/libieeep1788
Conformance in Progress ValidatedNumerics.jl (Julia) by David P. Sanders and Luis Benet (UNAM) See https: //github.com/dpsanders/ValidatedNumerics.jl
IEEE 1788-2015 Standard Implementations
Interval Arithmetic (IA) Conforming Fundamentals
What is IA? Gnu Octave (Matlab-like) by Oliver Heimlich.
Variations See http://octave.sourceforge.net/interval/
Underlying Rationale
Successes Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
31/31
31 / 31 C++ by Marco Nehmeier (J. Wolff v. Gudenberg). See https://github.com/nehmeier/libieeep1788
Conformance in Progress ValidatedNumerics.jl (Julia) by David P. Sanders and Luis Benet (UNAM) See https: //github.com/dpsanders/ValidatedNumerics.jl
IEEE 1788-2015 Standard Implementations
Interval Arithmetic (IA) Conforming Fundamentals
What is IA? Gnu Octave (Matlab-like) by Oliver Heimlich.
Variations See http://octave.sourceforge.net/interval/
Underlying Rationale JInterval (Java) by Dmitry Nadezhin and Sergei Zhilin. Successes See https://java.net/projects/jinterval Famous Proofs Engineering Verifications
History Early Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
31/31
31 / 31 Conformance in Progress ValidatedNumerics.jl (Julia) by David P. Sanders and Luis Benet (UNAM) See https: //github.com/dpsanders/ValidatedNumerics.jl
IEEE 1788-2015 Standard Implementations
Interval Arithmetic (IA) Conforming Fundamentals
What is IA? Gnu Octave (Matlab-like) by Oliver Heimlich.
Variations See http://octave.sourceforge.net/interval/
Underlying Rationale JInterval (Java) by Dmitry Nadezhin and Sergei Zhilin. Successes See https://java.net/projects/jinterval Famous Proofs Engineering Verifications C++ by Marco Nehmeier (J. Wolff v. Gudenberg). History Early See https://github.com/nehmeier/libieeep1788 Moore Karlsruhe Russian
Logical Pitfalls Constraints Simplex representations Existence Verification
The IEEE Standard
31/31
31 / 31 IEEE 1788-2015 Standard Implementations
Interval Arithmetic (IA) Conforming Fundamentals
What is IA? Gnu Octave (Matlab-like) by Oliver Heimlich.
Variations See http://octave.sourceforge.net/interval/
Underlying Rationale JInterval (Java) by Dmitry Nadezhin and Sergei Zhilin. Successes See https://java.net/projects/jinterval Famous Proofs Engineering Verifications C++ by Marco Nehmeier (J. Wolff v. Gudenberg). History Early See https://github.com/nehmeier/libieeep1788 Moore Karlsruhe Russian Conformance in Progress Logical Pitfalls Constraints Simplex representations ValidatedNumerics.jl (Julia) by David P. Sanders and Existence Verification Luis Benet (UNAM) The IEEE Standard See https: //github.com/dpsanders/ValidatedNumerics.jl
31/31
31 / 31