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Numerical Integration and Differentiation Growth And Numerical Integration and Differentiation Growth and Development Ra¨ulSantaeul`alia-Llopis MOVE-UAB and Barcelona GSE Spring 2017 Ra¨ulSantaeul`alia-Llopis(MOVE,UAB,BGSE) GnD: Numerical Integration and Differentiation Spring 2017 1 / 27 1 Numerical Differentiation One-Sided and Two-Sided Differentiation Computational Issues on Very Small Numbers 2 Numerical Integration Newton-Cotes Methods Gaussian Quadrature Monte Carlo Integration Quasi-Monte Carlo Integration Ra¨ulSantaeul`alia-Llopis(MOVE,UAB,BGSE) GnD: Numerical Integration and Differentiation Spring 2017 2 / 27 Numerical Differentiation The definition of the derivative at x∗ is 0 f (x∗ + h) − f (x∗) f (x∗) = lim h!0 h Ra¨ulSantaeul`alia-Llopis(MOVE,UAB,BGSE) GnD: Numerical Integration and Differentiation Spring 2017 3 / 27 • Hence, a natural way to numerically obtain the derivative is to use: f (x + h) − f (x ) f 0(x ) ≈ ∗ ∗ (1) ∗ h with a small h. We call (1) the one-sided derivative. • Another way to numerically obtain the derivative is to use: f (x + h) − f (x − h) f 0(x ) ≈ ∗ ∗ (2) ∗ 2h with a small h. We call (2) the two-sided derivative. We can show that the two-sided numerical derivative has a smaller error than the one-sided numerical derivative. We can see this in 3 steps. Ra¨ulSantaeul`alia-Llopis(MOVE,UAB,BGSE) GnD: Numerical Integration and Differentiation Spring 2017 4 / 27 • Step 1, use a Taylor expansion of order 3 around x∗ to obtain 0 1 00 2 1 000 3 f (x) = f (x∗) + f (x∗)(x − x∗) + f (x∗)(x − x∗) + f (x∗)(x − x∗) + O3(x) (3) 2 6 • Step 2, evaluate the expansion (3) at x = x∗ + h 0 1 00 2 1 000 3 f (x∗ + h) = f (x∗) + f (x∗)h + f (x∗)(h) + f (x∗)(h) + O3(x∗ + h) (4) 2 6 and rearrange to obtain the one-sided derivative formula: f (x∗ + h) − f (x∗) 0 1 00 1 000 2 O3(x∗ + h) = f (x∗) + f (x∗)(h) + f (x∗)(h) + (5) h 2 6 h • Step 3, evaluate the expansion (3) at x = x∗ − h 0 1 00 2 1 000 3 f (x∗ − h) = f (x∗) + f (x∗)(−h) + f (x∗)(−h) + f (x∗)(−h) + O3(x∗ − h) (6) 2 6 and combine (4) and (6) to obtain the two-sided derivative formula: f (x∗ + h) − f (x∗ − h) 0 1 000 2 O3(x∗ + h) − O3(x∗ − h) = f (x∗) + f (x∗)(h) + (7) 2h 6 2h Comparing (5) and (7) shows the error associated with the two-sided formula is smaller. Ra¨ulSantaeul`alia-Llopis(MOVE,UAB,BGSE) GnD: Numerical Integration and Differentiation Spring 2017 5 / 27 Computational Issues on Very Small Numbers • Exact arithmetic and computer arithmetic do not always give the same answers. This is not an issue of programming skills but a matter of computer precision. • For example, compute y = (1:0e − 20 + 1:0) − 1:0 and y = 1:0e − 20 + (1:0 − 1:0) where 1:0e − 20 is the computer's shorthand for 10−20. Exact arithmetic says the two statements above are identical because addition and substraction are associative. A computer, however, would evaluate the statements differently. The first statement would, incorrectly, likely result in x = 0 whereas the second one would, correctly, result in x = 10−20. • Usually, if one is using double precision in Fortran, numbers can be not precise if we reach 10−16. Ra¨ulSantaeul`alia-Llopis(MOVE,UAB,BGSE) GnD: Numerical Integration and Differentiation Spring 2017 6 / 27 For this reason, practice suggets for the two-sided formula an alternative 0 f (x∗ + h) − f (x∗ − h) f (x∗) ≈ (8) (x∗ + h) − (x∗ − h) with a small h. Note that the denominator might not be precisely 2h. That is why this formula helps to avoid trouble associated with a small h. Ra¨ulSantaeul`alia-Llopis(MOVE,UAB,BGSE) GnD: Numerical Integration and Differentiation Spring 2017 7 / 27 Numerical Integration • Goal: Compute the definite integral of a real-valued function f w.r.t. a weight function w over an interval I of <n, Z f (x) w(x) dx I • The weight function can be, for example • w(x) ≡ 1 ! the integral is the area under f • w(x) ≡ p.d.f. of a r.v. xe with support I ! the integral is E[f (xe)]. Ra¨ulSantaeul`alia-Llopis(MOVE,UAB,BGSE) GnD: Numerical Integration and Differentiation Spring 2017 8 / 27 • We study methods that approximate a definite integral with a weighted sum of function values, n Z X f (x) w(x) dx ≈ wi f (xi ) I i=0 Ra¨ulSantaeul`alia-Llopis(MOVE,UAB,BGSE) GnD: Numerical Integration and Differentiation Spring 2017 9 / 27 • We will see three classes of numerical integration (numerical quadrature) methods that differ on how the quadrature weights wi and the quadrature nodes xi are chosen. • Newton-Cotes methods approximate the integrand f between nodes using low-order polynomials and sum the integrals. • Gaussian Quadrature methods choose the nodes and weights that satisfy some moment-matching conditions. • Monte Carlo (and quasi-Monte Carlo) methods use equally weighted random or equidistributed nodes. Ra¨ulSantaeul`alia-Llopis(MOVE,UAB,BGSE) GnD: Numerical Integration and Differentiation Spring 2017 10 / 27 Newton-Cotes Methods • Univariate quadrature methods are designed to approximate the integral of a real-valued function f defined on a bounded interval [a; b] of the real line. • Two Newton-Cotes methods are widely used, • Trapezoid Rule • Simpson's Rule • Both rules are easy to implement and are typically adequate for computing the area under a continuous function. Ra¨ulSantaeul`alia-Llopis(MOVE,UAB,BGSE) GnD: Numerical Integration and Differentiation Spring 2017 11 / 27 Trapezoid Rule • First, partition the interval [a; b] into subintervals, (say, though not necessarily, equal length) • b−a Define the nodes xi = a + (i − 1)h for i = 1; :::; n with h = n−1 . • Second, approximate f over each subinterval [xi ; xi+1] using piecewise linear spline passing through (xi ; f (xi )) and (xi+1; f (xi+1)) Ra¨ulSantaeul`alia-Llopis(MOVE,UAB,BGSE) GnD: Numerical Integration and Differentiation Spring 2017 12 / 27 • Third, the area under each line segment defines a trapezoid approximates the area under f over the subinterval, Z xi+1 h f (x) dx ≈ [f (xi ) + f (xi+1)] xi 2 Summing up the areas of the trapezoides across the subintervals yields the Trapezoide rule: n Z b X f (x) dx ≈ wi f (xi ) a i h with w1 = wn = 2 and wi = h otherwise. Ra¨ulSantaeul`alia-Llopis(MOVE,UAB,BGSE) GnD: Numerical Integration and Differentiation Spring 2017 13 / 27 • Remarks: • It is simple and robust. • First-order exact: if not for rounding error, it will exactly compute the integral of any first-order polynomial (a line) • If the integrand is smooth, the trapezoid rule yields an approximation error that shrinks quadratically with the width of the subintervals, O(h2). Ra¨ulSantaeul`alia-Llopis(MOVE,UAB,BGSE) GnD: Numerical Integration and Differentiation Spring 2017 14 / 27 Simpson's Rule • First, partition the interval [a; b] into an even number of subintervals, (say, equal length) • b−a Define the nodes xi = a + (i − 1)h for i = 1; :::; n with h = n−1 and n is odd. • Second, approximate f over the jth pair of subintervals [xj−1; xj ] and [xj ; xj+1] using a piecewise quadratic function that passes through (xj−1; f (xj−1)), (xj ; f (xj )) and (xj+1; f (xj+1)) Ra¨ulSantaeul`alia-Llopis(MOVE,UAB,BGSE) GnD: Numerical Integration and Differentiation Spring 2017 15 / 27 • Third, the area under this quadratic function approximates the area under f over the subinterval, Z xj+1 h f (x) dx ≈ [f (xj−1) + 4f (xj ) + f (xj+1))] xj−1 3 Summing up the areas of the quadratic approximants across subintervals yields the Simpson's Rule: n Z b X f (x) dx ≈ wi f (xi ) a i h h h with w1 = wn = 3 and otherwise, wi = 4 3 if i is even, and wi = 2 3 if i is odd. Ra¨ulSantaeul`alia-Llopis(MOVE,UAB,BGSE) GnD: Numerical Integration and Differentiation Spring 2017 16 / 27 • Remarks: • Easy to implement, as the Trapezoide rule. • Even thought it is based on locally quadratic approximations of the integrand, it is third-order exact: if not for rounding error, it will exactly compute the integral of any cubic polynomial. • If the integrand is smooth, the Simpson's rule yields an approximation error that shrinks at twice the geometric rate of the error associated with the trapezoid rule, O(h4). • Simpson's rule is preferred to the Trapezoid rule when f is smooth because it offers twice the degree of approximation. • If f exhibits discontinuities, the trapezoid rule will often be more accurate. • Newton-Cotes rules based on 4th and higher order piecewise polynomial approximations exist, but rarely used. Ra¨ulSantaeul`alia-Llopis(MOVE,UAB,BGSE) GnD: Numerical Integration and Differentiation Spring 2017 17 / 27 • Higher dimensional integration: generalizations of the univariate Newton-Cotes quadrature schemes through tensor product principles. • Suppose one wishes to integrate a real-valued function defined on a 2 rectangle f(x1; x2) ja1 ≤ x1 ≤ b1; a2 ≤ x2 ≤ b2g in < . • One way to proceed is to compute the Newton-Cotes nodes and weights: • f(x1i ; w1i ) ji = 1; :::; n1g for the real interval (a1; b1), and • f(x2j ; w2j ) jj = 1; :::; n2g for the real interval (a2; b2) • The tensor product Newton-Cotes rule for the rectangle would comprise of the n = n1n2 grid points of the form • f(x1i ; w2j ) j i = 1; :::; n1; j = 1; :::; n2g with associated weights, • fwij = w1i w2j j i = 1; :::; n1; j = 1; :::; n2g • This construction principle can be applied to higher dimensions using repeated tensor product operations.
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