Algebraic equations Basic numerical methods Robert Maˇr´ık May 9, 2006 Contents 1 Algebraic equations – general theory 2 2 Basic numerical methods for algebraic equations 10 Exercise - Integer roots ................................ 14 Exercise - Bisection of the interval .......................... 27 3 Lagrange interpolation formula 53 ⊳⊳ ⊳ ⊲ ⊲⊲ c Robert Maˇr´ık, 2006 × 1 Algebraic equations – general theory We are able to solve linear and quadratic equations: b • If a 6 and ax b , then x − . a −b ± b − ac • If a 6=0 and ax + bx=0 c , then= x . ; pa2 2 4 =0 + + =0 1 2 = Now we wish to be able to solve higher order equations. 2 ⊳⊳ ⊳ ⊲ ⊲⊲ Algebraic equations – general theory c Robert Maˇr´ık, 2006 × Definition (polynomial). Let n be a nonnegative integer and n n− n− Pn x a x a x a x ··· an− x an− x an (1) 1 2 2 0 1 2 2 1 be an n-degree polynomial( )= + with real+ coefficients+ a ,+a , ... a+n, where+a 6 (i.e., the highest power really appears in the polynomial). 0 1 0 = 0 The coefficient a is called a leading coefficient of the polynomial Pn x and the coefficient n an is an absolute term of the polynomial Pn x . The term a x is called a leading term of 0 ( ) the polynomial Pn x . ( ) 0 ( ) Example 1. The polynomial y x x − x 6 5 is a -degree polynomial with the leading=2 term+ x 3and+ 10the absolute term . 6 6 2 10 ⊳⊳ ⊳ ⊲ ⊲⊲ Algebraic equations – general theory c Robert Maˇr´ık, 2006 × Definition (polynomial). Let n be a nonnegative integer and n n− n− Pn x a x a x a x ··· an− x an− x an (1) 1 2 2 0 1 2 2 1 be an n-degree polynomial( )= + with real+ coefficients+ a ,+a , ... a+n, where+a 6 (i.e., the highest power really appears in the polynomial). 0 1 0 = 0 The coefficient a is called a leading coefficient of the polynomial Pn x and the coefficient n an is an absolute term of the polynomial Pn x . The term a x is called a leading term of 0 ( ) the polynomial Pn x . ( ) 0 ( ) Example 1. The polynomial y x x − x 6 5 is a -degree polynomial with the leading=2 term+ x 3and+ 10the absolute term . 6 6 2 10 ⊳⊳ ⊳ ⊲ ⊲⊲ Algebraic equations – general theory c Robert Maˇr´ık, 2006 × Definition (polynomial). Let n be a nonnegative integer and n n− n− Pn x a x a x a x ··· an− x an− x an (1) 1 2 2 0 1 2 2 1 be an n-degree polynomial( )= + with real+ coefficients+ a ,+a , ... a+n, where+a 6 (i.e., the highest power really appears in the polynomial). 0 1 0 = 0 The coefficient a is called a leading coefficient of the polynomial Pn x and the coefficient n an is an absolute term of the polynomial Pn x . The term a x is called a leading term of 0 ( ) the polynomial Pn x . ( ) 0 ( ) Example 1. The polynomial y x x − x 6 5 is a -degree polynomial with the leading=2 term+ x 3and+ 10the absolute term . 6 6 2 10 ⊳⊳ ⊳ ⊲ ⊲⊲ Algebraic equations – general theory c Robert Maˇr´ık, 2006 × Definition (algebraic equation). Under an n-degree algebraic equation we understand the equation of the form Pn x , i.e. n n− n− a x a( x)=0 a x ··· an− x an− x an (2) 1 2 2 0 + 1 + 2 + + 2 + 1 + =0 Definition (zero of an algebraic equation). Under a zero of equation (2) (or a zero of polynomial (1)) we understand the number c which satisfies Pn c , i.e., which substituted for x converts (2) into a valid relation. ( )=0 A zero of an algebraic equation is sometimes called also a root of this equation or simply a solution. Example 2. The numbers x and x − are zeros of the polynomial =1 P x =x 2 x −x− : (3) 3 2 Really, a quick evaluation shows P ( )= and+2P − 2. The number x is not a zero of this polynomial, since P . (1) = 0 ( 2)=0 =3 (3) = 40 ⊳⊳ ⊳ ⊲ ⊲⊲ Algebraic equations – general theory c Robert Maˇr´ık, 2006 × Definition (algebraic equation). Under an n-degree algebraic equation we understand the equation of the form Pn x , i.e. n n− n− a x a( x)=0 a x ··· an− x an− x an (2) 1 2 2 0 + 1 + 2 + + 2 + 1 + =0 Definition (zero of an algebraic equation). Under a zero of equation (2) (or a zero of polynomial (1)) we understand the number c which satisfies Pn c , i.e., which substituted for x converts (2) into a valid relation. ( )=0 A zero of an algebraic equation is sometimes called also a root of this equation or simply a solution. Example 2. The numbers x and x − are zeros of the polynomial =1 P x =x 2 x −x− : (3) 3 2 Really, a quick evaluation shows P ( )= and+2P − 2. The number x is not a zero of this polynomial, since P . (1) = 0 ( 2)=0 =3 (3) = 40 ⊳⊳ ⊳ ⊲ ⊲⊲ Algebraic equations – general theory c Robert Maˇr´ık, 2006 × Definition (algebraic equation). Under an n-degree algebraic equation we understand the equation of the form Pn x , i.e. n n− n− a x a( x)=0 a x ··· an− x an− x an (2) 1 2 2 0 + 1 + 2 + + 2 + 1 + =0 Definition (zero of an algebraic equation). Under a zero of equation (2) (or a zero of polynomial (1)) we understand the number c which satisfies Pn c , i.e., which substituted for x converts (2) into a valid relation. ( )=0 A zero of an algebraic equation is sometimes called also a root of this equation or simply a solution. Example 2. The numbers x and x − are zeros of the polynomial =1 P x =x 2 x −x− : (3) 3 2 Really, a quick evaluation shows P ( )= and+2P − 2. The number x is not a zero of this polynomial, since P . (1) = 0 ( 2)=0 =3 (3) = 40 ⊳⊳ ⊳ ⊲ ⊲⊲ Algebraic equations – general theory c Robert Maˇr´ık, 2006 × Theorem 1 (the basic theorem of algebra). Every algebraic equation has a solution in the set of complex numbers. Theorem 2 (Bezout). The number c is a zero of polynomial (1) (of equation (2)) if and E only if the linear polynomial x−c is a factor of this polynomial, i.e., if and only if there exists an n− degree polynomial Qn− x with property ( ) 1 ( 1) Pn x ( )x−c Qn− x : (4) ( )=( ) 1( ) Definition (linear factor corresponding to the zero of algebraic equation). If c is a zero of polynomial (1) (of equation (2)), then the linear polynomial x−c with an independent variable x is said to be a linear factor of the polynomial (1) corresponding to the zero x c. ( ) = Example 3. Polynomial (3) can be written in each of the following equivalent forms y x− x x ; y x x − ; y x− x x : 2 2 The reader=( can check1)( these+3 + relations 2) by=( multiplying+ 2)( the1) parentheses.=( 1)( + 1)( + 2) ⊳⊳ ⊳ ⊲ ⊲⊲ Algebraic equations – general theory c Robert Maˇr´ık, 2006 × Theorem 1 (the basic theorem of algebra). Every algebraic equation has a solution in the set of complex numbers. Theorem 2 (Bezout). The number c is a zero of polynomial (1) (of equation (2)) if and E only if the linear polynomial x−c is a factor of this polynomial, i.e., if and only if there exists an n− degree polynomial Qn− x with property ( ) 1 ( 1) Pn x ( )x−c Qn− x : (4) ( )=( ) 1( ) Definition (linear factor corresponding to the zero of algebraic equation). If c is a zero of polynomial (1) (of equation (2)), then the linear polynomial x−c with an independent variable x is said to be a linear factor of the polynomial (1) corresponding to the zero x c. ( ) = Example 3. Polynomial (3) can be written in each of the following equivalent forms y x− x x ; y x x − ; y x− x x : 2 2 The reader=( can check1)( these+3 + relations 2) by=( multiplying+ 2)( the1) parentheses.=( 1)( + 1)( + 2) ⊳⊳ ⊳ ⊲ ⊲⊲ Algebraic equations – general theory c Robert Maˇr´ık, 2006 × Theorem 1 (the basic theorem of algebra). Every algebraic equation has a solution in the set of complex numbers. Theorem 2 (Bezout). The number c is a zero of polynomial (1) (of equation (2)) if and E only if the linear polynomial x−c is a factor of this polynomial, i.e., if and only if there exists an n− degree polynomial Qn− x with property ( ) 1 ( 1) Pn x ( )x−c Qn− x : (4) ( )=( ) 1( ) Definition (linear factor corresponding to the zero of algebraic equation). If c is a zero of polynomial (1) (of equation (2)), then the linear polynomial x−c with an independent variable x is said to be a linear factor of the polynomial (1) corresponding to the zero x c. ( ) = Example 3. Polynomial (3) can be written in each of the following equivalent forms y x− x x ; y x x − ; y x− x x : 2 2 The reader=( can check1)( these+3 + relations 2) by=( multiplying+ 2)( the1) parentheses.=( 1)( + 1)( + 2) ⊳⊳ ⊳ ⊲ ⊲⊲ Algebraic equations – general theory c Robert Maˇr´ık, 2006 × Theorem 1 (the basic theorem of algebra). Every algebraic equation has a solution in the set of complex numbers. Theorem 2 (Bezout). The number c is a zero of polynomial (1) (of equation (2)) if and E only if the linear polynomial x−c is a factor of this polynomial, i.e., if and only if there exists an n− degree polynomial Qn− x with property ( ) 1 ( 1) Pn x ( )x−c Qn− x : (4) ( )=( ) 1( ) Definition (linear factor corresponding to the zero of algebraic equation).