1. Let i, i = 1,2,..., n + 1 different nodes and let yi ER, 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 ai, 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 = = (akai)/(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,...,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 u = Pn (zi), i = 1, 2, ..., m using your function evalp. -
1. Let i, i = 1,2,..., n + 1 different nodes and let yi ER, 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 ai, 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 = = (akai)/(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,...,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 u = Pn (zi), 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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![1.
Let i,
i = 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) + a3(x − x₁)(x − x₂)+...+an+1(x − x₁)(x − Xn+1),
where the coefficients ai, 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 =
end for
end for
(akai)/(xk - Xi)
If the coefficients a,, 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, the step and c 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, ..., %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 ui= Pn (Zi), i = 1, 2,..., m using your function evalp.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F01b4eff1-1c92-4d86-9563-264f29556131%2Fc51f9179-bcd1-4db6-92da-e05527bdc266%2F6n3yv8m9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1.
Let i,
i = 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) + a3(x − x₁)(x − x₂)+...+an+1(x − x₁)(x − Xn+1),
where the coefficients ai, 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 =
end for
end for
(akai)/(xk - Xi)
If the coefficients a,, 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, the step and c 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, ..., %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 ui= Pn (Zi), i = 1, 2,..., m using your function evalp.
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