Taylor polynomial The graphs of the approximation PM(x)and the function fx) a both pass through the center xo and they have the same slopes (N=1)and curvature (N=2)at it. The approximation is good near the center xo and that the distance between the curves grows as x moves away from xo. The accuracy of a Taylor polynomial is increased when we choose N large. Cor.3.1.If P(x)is the Taylor polynomial of degree N given in Theorem 3.1,then P(o)=f(xo) for k=0,1,....N. 华南师范大学数学科学学院谢删玲
Taylor polynomial ◼ The graphs of the approximation PN (x) and the function f(x) both pass through the center x0 and they have the same slopes (N=1) and curvature (N=2) at it. ◼ The approximation is good near the center x0 and that the distance between the curves grows as x moves away from x0 . ◼ The accuracy of a Taylor polynomial is increased when we choose N large. Cor. 3.1. If PN (x) is the Taylor polynomial of degree N given in Theorem 3.1, then 华南师范大学数学科学学院 谢骊玲 ( ) ( ) 0 0 ( ) ( ) k k P x f x N = for k=0,1,…,N
23/9/8 6 Example for Taylor Approximation(1) y=P(x)=1+x 十干干干干++++十+千4+++++++◆+++++++++◆++++十+++++++++++ .2 2 =P2x) =1+x+x2/2 华南师范大学数学科学学院谢细玲
华南师范大学数学科学学院 谢骊玲 2023/9/8 y=P1 (x)=1+x y=e x y=e x y=P2 (x) =1+x+x2 /2 Example for Taylor Approximation (1)
23/9/8 Example for Taylor Approximation(2) =P4x) MRN+ Error EN(x) y=P:(x) (N+1)川 y=P2(x) ++++++++++++H Example 3.2 3.3 华南师范大学数学科学学院谢蜘玲
华南师范大学数学科学学院 谢骊玲 2023/9/8 y=e x y=P4 (x) y=P3 (x) y=P2 (x) Example for Taylor Approximation (2) 1 | | | ( ) | ( 1)! N N MR E x N + = + Error Example 3.2 & 3.3
The graph of the error y=E(x)=ex-P(x) Example 3.3 y-E(x) 华南师范大学数学科学学院谢骊玲
华南师范大学数学科学学院 谢骊玲 Example 3.3 y=E9 (x) The graph of the error y=E9 (x)=e x -P9 (x)
Methods for Evaluating a Polynomial Nested Multiplication: (i)Algorithm to Evaluate P(x) (ii)Algorithm to Evaluate P(x) (iii)Algorithm to Evaluate I(x) B(N):=A(N) D(N-1):=N*A(N) IN+1)=AN)/N+1) FOR K-N-1 DOWNTO 0 DO FOR K-N-1 DOWNTO 1 DO FOR K-N DOWNTO 1 DO B(K):=A(K)+B(K+1)*X D(K-1):=K*A(K)+D(K)*X I(K)=A(K-1)/K+HK+1)*X PRNT“The value P(xis”,B(O) PRINT"The value P'(x)is",D(0) I(0)=C+H(1)*X PRNT“The value I(x)is”,IO) 华南师范大学数学科学学院谢删玲
Methods for Evaluating a Polynomial ◼ Nested Multiplication: 华南师范大学数学科学学院 谢骊玲 (i) Algorithm to Evaluate P(x) (ii) Algorithm to Evaluate P’(x) (iii) Algorithm to Evaluate I(x) B(N):=A(N) FOR K=N-1 DOWNTO 0 DO B(K):=A(K)+B(K+1)*X PRINT “The value P(x) is”, B(0) D(N-1):=N*A(N) FOR K=N-1 DOWNTO 1 DO D(K-1):=K*A(K)+D(K)*X PRINT “The value P’(x) is”, D(0) I(N+1):=A(N) /(N+1) FOR K=N DOWNTO 1 DO I(K):=A(K-1)/K+I(K+1)*X I(0):=C+I(1)*X PRINT “The value I(x) is”, I(0)