Compute, using divided differences, the value of the piecewise
cubic Her-
mite interpolating polynomial at x = 11=10 given nodes at xi = i,
for i = 1; : : : ; 10,
and values and derivatives at the nodes from the function f(x) =
1=x.
Remember iterative formula for divided differences:
Homework Answers
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Compute, using divided differences, the value of the piecewise cubic Her- mite interpolating polynomial at x = 11=10 giv...
4. For the following table, answer the questions. (1) Find the cubic Newton’s interpolating polynomial using...
4. For the following table, answer the questions.
(1) Find the cubic Newton’s interpolating polynomial using the
first four data points and estimate the function value at x=2.5
with the interpolating polynomial.
(2) Find the quartic Newton’s interpolating polynomial using the
five data points and estimate the function value at x=2.5 with the
interpolating polynomial.
(3) Find the bases functions of Lagrange interpolation, Li(x)
(i=1,2,…,5), and estimate the function value at x=2.5 with the
Lagrange interpolating polynomial.
3 5 1...
Consider polynomial interpolation of the function f(x)=1/(1+25x^2) on the interval [-1,1] by (1) ...
Consider polynomial interpolation of the function f(x)=1/(1+25x^2) on the interval [-1,1] by (1) an interpolating polynomial determined by m equidistant interpolation points, (2) an interpolating polynomial determined by interpolation at the m zeros of the Chebyshev polynomial T_m(x), and (3) by interpolating by cubic splines instead of by a polynomial. Estimate the approximation error by evaluation max_i |f(z_i)-p(z_i)| for many points z_i on [-1,1]. For instance, you could use 10m points z_i. The cubic spline interpolant can be determined in...
**********************matlab code please******************* 1. Interpolation error of polynomial fit Using 11 equi-dist...
**********************matlab code please*******************
1. Interpolation error of polynomial fit Using 11 equi-distributed points (10 equal segments) in the interval [-1 1], Using Newton's form find and plot the interpolating polynomial p(x) for the function f(x) -1/(125x2). Comment on the large discrepancies between p(x) and the function f(x) that the data came from Write down an expression for the error in the interpolating polynomial above? Which part of the expression is responsible for the large errors observed?
1. Interpolation error of...
6. Consider f(x)-sinx and evenly spaced nodes 0-0 < xīく… < Zn-2T. Let P(z) be the piecewise cubic interpolant given values and first derivatives of f at the nodes. (a) In the case n = 100, use...
6. Consider f(x)-sinx and evenly spaced nodes 0-0 < xīく… < Zn-2T. Let P(z) be the piecewise cubic interpolant given values and first derivatives of f at the nodes. (a) In the case n = 100, use calculus and the error formula 4! where 1 E [xi,Ti+1], to bound the absolute error lf(1)-P(1) (b) For arbitrary x E [0.2 , use error bounds to determine n ensuring that If(x)- P(x) s 10-10
6. Consider f(x)-sinx and evenly spaced nodes 0-0
matlab matlab For this problem you will test a few interpolation approaches for the application of generating interpolated data. We'll do this by interpolating data that is sampled from a k...
matlab
matlab
For this problem you will test a few interpolation approaches for the application of generating interpolated data. We'll do this by interpolating data that is sampled from a known mathematical function. The same function can then be used to compute true values at the interpolated points to use in error Consider the following mathematical function (Runge's function): 1+25r2 Write a function mfile that uses this formula to generate a set of data use those points along approaches outlined...
MATLAB Create a function that provides a definite integration using Simpson's Rule Problem Summar This example demo...
MATLAB Create a function that provides a definite integration
using Simpson's Rule
Problem Summar This example demonstrates using instructor-provided and randomized inputs to assess a function problem. Custom numerical tolerances are used to assess the output. Simpson's Rule approximates the definite integral of a function f(x) on the interval a,a according to the following formula + f (ati) This approximation is in general more accurate than the trapezoidal rule, which itself is more accurate than the leftright-hand rules. The increased...
Midterm Examination CSCI 3321 Summer, 2015 &n
Midterm Examination CSCI 3321 Summer, 2015 ___________ Your Name … .. Please answer each question by entering the most nearly correct answer (a, b, c, d) in the blank on the left. ____1. When approximating ex by a truncated Taylor series (expanded about x=0), how many terms would be required to keep the absolute value of the error below 10-3 over the interval [-1,1] ? a. 1 b. 2 c. 3 d. 4 ____2. The polynomial that...
4. For the following table, answer the questions.
(1) Find the cubic Newton’s interpolating polynomial using the
first four data points and estimate the function value at x=2.5
with the interpolating polynomial.
(2) Find the quartic Newton’s interpolating polynomial using the
five data points and estimate the function value at x=2.5 with the
interpolating polynomial.
(3) Find the bases functions of Lagrange interpolation, Li(x)
(i=1,2,…,5), and estimate the function value at x=2.5 with the
Lagrange interpolating polynomial.
3 5 1...
**********************matlab code please*******************
1. Interpolation error of polynomial fit Using 11 equi-distributed points (10 equal segments) in the interval [-1 1], Using Newton's form find and plot the interpolating polynomial p(x) for the function f(x) -1/(125x2). Comment on the large discrepancies between p(x) and the function f(x) that the data came from Write down an expression for the error in the interpolating polynomial above? Which part of the expression is responsible for the large errors observed?
1. Interpolation error of...
6. Consider f(x)-sinx and evenly spaced nodes 0-0 < xīく… < Zn-2T. Let P(z) be the piecewise cubic interpolant given values and first derivatives of f at the nodes. (a) In the case n = 100, use calculus and the error formula 4! where 1 E [xi,Ti+1], to bound the absolute error lf(1)-P(1) (b) For arbitrary x E [0.2 , use error bounds to determine n ensuring that If(x)- P(x) s 10-10
6. Consider f(x)-sinx and evenly spaced nodes 0-0
matlab
matlab
For this problem you will test a few interpolation approaches for the application of generating interpolated data. We'll do this by interpolating data that is sampled from a known mathematical function. The same function can then be used to compute true values at the interpolated points to use in error Consider the following mathematical function (Runge's function): 1+25r2 Write a function mfile that uses this formula to generate a set of data use those points along approaches outlined...
MATLAB Create a function that provides a definite integration
using Simpson's Rule
Problem Summar This example demonstrates using instructor-provided and randomized inputs to assess a function problem. Custom numerical tolerances are used to assess the output. Simpson's Rule approximates the definite integral of a function f(x) on the interval a,a according to the following formula + f (ati) This approximation is in general more accurate than the trapezoidal rule, which itself is more accurate than the leftright-hand rules. The increased...
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