1. Let Xi, i = 1, 2,. ..., n + 1 different nodes and let yį € R, i = 1,2,. interpolating polynomial is written in Newton's form as: Pn(x) = = a₁ + a₂(x-x₁) + az(x − x₁)(x - x₂)+ ·+an+1(x-x₁)... (x − xn+1), where the coefficients a¿, i = 1, ..., n + 1 can be computed using the following algorithm: Algorithm 1 Newton's polynomial aiyi, i = 1,2,...,n+1 for k= 2: n + 1 do for i=1: k- 1 do ak = = (ak — α₂)/(xk — Xi) end for end for If the coefficients ai, i = 1,..., n+1 are known, then the value of the interpolating polynomial at the point z can be computed using Horner's formula: Algorithm 2 Horner's formula S = an+1 for in-1:1 do s = a₁ + (2x₁) s end for Pn (2) = ,n+ 1. The = S Remark: It is noted that in the loop conditions i= a b c of the previous pseudo-codes a is the starting value, b is the step and c is the last value. (a) Write PYTHON 's functions coefs and evalp implementing the previously described algorithms for the coefficients of the interpolating polynomial and it's evaluation at values z using Horner's formula. Write a function newtinterp with input arguments (x, y, z), where (xi, Yi), i = 1,..., n+1, the interpolation data points and z = [21, ..., Zm] the vector containing the m points on which we want to evaluate the interpolating polynomial. This will compute the coefficients of the interpolating polynomial a; using your function coefs and will return the values uį = Pn (zi), i=1,2,...,m using your function evalp. (b) Consider the function f(x) = sinx, x = [0, 2π], and a uniform partition of [0, 2] with 6 points xi, i = 1,...,6. Using your function newtinterp compute the interpolating polynomial p5, that interpolates function ƒ at the nodes ï¿, at 101 equi-distributed points Zį € [0, 2π].
1. Let Xi, i = 1, 2,. ..., n + 1 different nodes and let yį € R, i = 1,2,. interpolating polynomial is written in Newton's form as: Pn(x) = = a₁ + a₂(x-x₁) + az(x − x₁)(x - x₂)+ ·+an+1(x-x₁)... (x − xn+1), where the coefficients a¿, i = 1, ..., n + 1 can be computed using the following algorithm: Algorithm 1 Newton's polynomial aiyi, i = 1,2,...,n+1 for k= 2: n + 1 do for i=1: k- 1 do ak = = (ak — α₂)/(xk — Xi) end for end for If the coefficients ai, i = 1,..., n+1 are known, then the value of the interpolating polynomial at the point z can be computed using Horner's formula: Algorithm 2 Horner's formula S = an+1 for in-1:1 do s = a₁ + (2x₁) s end for Pn (2) = ,n+ 1. The = S Remark: It is noted that in the loop conditions i= a b c of the previous pseudo-codes a is the starting value, b is the step and c is the last value. (a) Write PYTHON 's functions coefs and evalp implementing the previously described algorithms for the coefficients of the interpolating polynomial and it's evaluation at values z using Horner's formula. Write a function newtinterp with input arguments (x, y, z), where (xi, Yi), i = 1,..., n+1, the interpolation data points and z = [21, ..., Zm] the vector containing the m points on which we want to evaluate the interpolating polynomial. This will compute the coefficients of the interpolating polynomial a; using your function coefs and will return the values uį = Pn (zi), i=1,2,...,m using your function evalp. (b) Consider the function f(x) = sinx, x = [0, 2π], and a uniform partition of [0, 2] with 6 points xi, i = 1,...,6. Using your function newtinterp compute the interpolating polynomial p5, that interpolates function ƒ at the nodes ï¿, at 101 equi-distributed points Zį € [0, 2π].
Operations Research : Applications and Algorithms
4th Edition
ISBN:9780534380588
Author:Wayne L. Winston
Publisher:Wayne L. Winston
Chapter2: Basic Linear Algebra
Section: Chapter Questions
Problem 15RP
Related questions
Question
![1.
Let Xi, i = 1, 2,. ..., n + 1 different nodes and let yį € R, i = 1,2, . ,n+ 1. The
interpolating polynomial is written in Newton's form as:
Pn(x) = a₁ + a₂(x − x₁) + a3(x − x₁)(x − x2) +
·+an+1(x-x₁)... (x - xn+1),
where the coefficients a¿, i = 1, ..., n + 1 can be computed using the following algorithm:
Algorithm 1 Newton's polynomial
aiyi, i = 1,2,...,n+1
for k= 2: n + 1 do
for i=1: k- 1 do
ak = = (ak — α₂)/(xk — xi)
end for
end for
If the coefficients ai, i = 1,..., n +1 are known, then the value of the interpolating polynomial at
the point z can be computed using Horner's formula:
Algorithm 2 Horner's formula
S = an+1
for in-1:1 do
s = a₁ + (2x₁) s
end for
Pn (2) =
= S
Remark: It is noted that in the loop conditions i= a: b:c of the previous pseudo-codes a is the
starting value, b is the step and c is the last value.
(a)
Write PYTHON 's functions coefs and evalp implementing the previously described
algorithms for the coefficients of the interpolating polynomial and it's evaluation at values z
using Horner's formula. Write a function newtinterp with input arguments (x, y, z), where
(xi, Yi), i = 1,..., n+1, the interpolation data points and z = [21, ..., Zm] the vector containing
the m points on which we want to evaluate the interpolating polynomial. This will compute
the coefficients of the interpolating polynomial a; using your function coefs and will return
the values uį = Pn (zi), i=1,2,...,m using your function evalp.
(b)
Consider the function f(x) = sinx, x = [0, 2π], and a uniform partition of [0, 2]
with 6 points xi, i = 1,...,6. Using your function newtinterp compute the interpolating
polynomial p5, that interpolates function ƒ at the nodes xį, at 101 equi-distributed points
Zį € [0, 2π].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F01b4eff1-1c92-4d86-9563-264f29556131%2F2d1cc45d-1be7-4e5d-b742-a6dd2eeb42e6%2F24io68w_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1.
Let Xi, i = 1, 2,. ..., n + 1 different nodes and let yį € R, i = 1,2, . ,n+ 1. The
interpolating polynomial is written in Newton's form as:
Pn(x) = a₁ + a₂(x − x₁) + a3(x − x₁)(x − x2) +
·+an+1(x-x₁)... (x - xn+1),
where the coefficients a¿, i = 1, ..., n + 1 can be computed using the following algorithm:
Algorithm 1 Newton's polynomial
aiyi, i = 1,2,...,n+1
for k= 2: n + 1 do
for i=1: k- 1 do
ak = = (ak — α₂)/(xk — xi)
end for
end for
If the coefficients ai, i = 1,..., n +1 are known, then the value of the interpolating polynomial at
the point z can be computed using Horner's formula:
Algorithm 2 Horner's formula
S = an+1
for in-1:1 do
s = a₁ + (2x₁) s
end for
Pn (2) =
= S
Remark: It is noted that in the loop conditions i= a: b:c of the previous pseudo-codes a is the
starting value, b is the step and c is the last value.
(a)
Write PYTHON 's functions coefs and evalp implementing the previously described
algorithms for the coefficients of the interpolating polynomial and it's evaluation at values z
using Horner's formula. Write a function newtinterp with input arguments (x, y, z), where
(xi, Yi), i = 1,..., n+1, the interpolation data points and z = [21, ..., Zm] the vector containing
the m points on which we want to evaluate the interpolating polynomial. This will compute
the coefficients of the interpolating polynomial a; using your function coefs and will return
the values uį = Pn (zi), i=1,2,...,m using your function evalp.
(b)
Consider the function f(x) = sinx, x = [0, 2π], and a uniform partition of [0, 2]
with 6 points xi, i = 1,...,6. Using your function newtinterp compute the interpolating
polynomial p5, that interpolates function ƒ at the nodes xį, at 101 equi-distributed points
Zį € [0, 2π].
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