(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 = [1,..., 2m] 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 u = Pn (24), i = 1, 2,..., m using your function evalp.
(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 = [1,..., 2m] 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 u = Pn (24), i = 1, 2,..., m using your function evalp.
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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Question
Please do question 1a of the python coding.
![=
Let ₁, 1, 2, ,n+ 1 different nodes and let yi R, i = 1,2,...,n + 1. The
interpolating polynomial is written in Newton's form as:
Pn(x) = a₁ + a₂(x − x₁) + a₂(x − x₁)(x − x₂)+. ·+an+1(x-x₁)(x − xn+1);
where the coefficients a₁, 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
1.
for i=1: k-1 do
ak= (akai)/(æk — xi)
end for
end for
If the coefficients a₁, 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
8 = an+1
for in-1:1 do
s= a + (2x₁)s
end for
Pn (2) = 8
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 = [1,...,m] 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 u₂ = 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, 2π]
with 6 points x₁, i = 1,...,6. Using your function newtinterp compute the interpolating
polynomial p5, that interpolates function f at the nodes 2₁, at 101 equi-distributed points
2₁ € [0, 2π].
(c)
Verify your results and compare them with those obtained by using the PYTHON 'S
functions polyfit, polyval and CubicSpline.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba18de34-fc06-47a6-b1ea-c54726b84874%2F400a0835-2f74-4320-9ee7-3d6ad36515e8%2Fuit49rj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:=
Let ₁, 1, 2, ,n+ 1 different nodes and let yi R, i = 1,2,...,n + 1. The
interpolating polynomial is written in Newton's form as:
Pn(x) = a₁ + a₂(x − x₁) + a₂(x − x₁)(x − x₂)+. ·+an+1(x-x₁)(x − xn+1);
where the coefficients a₁, 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
1.
for i=1: k-1 do
ak= (akai)/(æk — xi)
end for
end for
If the coefficients a₁, 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
8 = an+1
for in-1:1 do
s= a + (2x₁)s
end for
Pn (2) = 8
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 = [1,...,m] 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 u₂ = 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, 2π]
with 6 points x₁, i = 1,...,6. Using your function newtinterp compute the interpolating
polynomial p5, that interpolates function f at the nodes 2₁, at 101 equi-distributed points
2₁ € [0, 2π].
(c)
Verify your results and compare them with those obtained by using the PYTHON 'S
functions polyfit, polyval and CubicSpline.
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