Question

Problem 1:

The matrix e^A is defined by the Mclaurin series

e^A=1+A/1!+A^2/2!+A^3/3!+........+A^n/n!

This computation can be quite difficult, as this problem illustrates.

a)Enter the function

function E=matexp(A)

E=zeros(size(A));

F=eye(size(A));

k=1;

while norm (E+F-E,1) > 0

E=E+F;

F=A*F/k;

k=k+a;

end

that estimates e^A using Mclaurin series.

b)Apply matexp to the matrix a=[99 -100; 137 -138] to obtaion matrix A_Mclaurin

c)A_Mclaurin is far from the correct result.ompute the true value of e^A using the MATLAB function expm as follows:

>>showdemo expmdemo

The function uses the Pade approximation

d)Explain why the result in part (b) is so far from the correct result. Use MATLAB help by typing

Problem 1 The matrix exponential e is defined by the McLaurin series A A2 A n! This computation can be quite difficult, as this problem illustrates. a. Enter the function function E matexp(A) % MATEXP Taylor series for exp(A) E zeros size(A)) F- eye(size(A)) while normCE+FE.1 0 E-EF end that estimates e using the McLaurin series. b. Apply matexp to the matrix 99-100 37-138 to obtain the matrix A McLaurin c. A McLaurin is far from the correct result. Compute the true value of e using the MATLAB function expm as follows: Atrue expm(A) The function uses the Padé approximation d. Explain why the result in part (b) is so far from the correct result. Use MATLAB help by typing > showdemo expmdemo

Answer 1

%%Matlab code for matrix exponential by McLaurin series clear all close all %Here matrix A A-99-100; 137-138); tApplying matexp on A [E, F,k]-matexp(A); fprintf 'A matrix is In disp (A) fprintf 'Using matexp AMaclaurin is \n') disp(E) fprintf'Using expm AMaclaurin is n) A_trueexpm(A) disp(A true) help expm %function for matrix exponential function [E, F,k] matexp(A) zeros ( size (A) ) ; eye ( size (A)); E = F = while norm (E+F-E,1) 0 k+ 1 ; k = ,,8888888888 88 End of Code 움88888888888888 A matrix is 99 -100 137 -138 Using matexp AMaclaurin is 12.5843312471568 -5.15944115258847 11.2745588069514 -1.26246584 103961 Using expm A Maclaurin is 1. 36214820109438 -0.994268759922901 1.36214820109437 -0.994268759922897

EXPM Matrix exponential EXPM(A) is the matrix exponential of A and is computed using a scaling and squaring algorithm with a Pade approximation. Although it is not computed this way, if A has a full set of eigenvectors V with corresponding eigenvalues D then V,DI-EIG(A) and EXPM(A)-v diag(exp diag(DDZ EXP(A) computes the exponential of A element-by-element. See also LOGM, SQRTM, FUNM. Reference page in Doc Center doc expnm Other functions named expm gpuArray/expm sym/expm Published with MATLAB R2018a

%%Matlab code for matrix exponential by McLaurin series
clear all
close all

%Here matrix A
A=[99 -100; 137 -138];
%Applying matexp on A

[E,F,k]=matexp(A);
fprintf('A matrix is \n')
disp(A)

fprintf('Using matexp A_Maclaurin is \n')
disp(E)

fprintf('Using expm A_Maclaurin is \n')
A_true= expm(A);
disp(A_true)

help expm

%function for matrix exponential
function [E,F,k]=matexp(A)

    E = zeros(size(A));
    F = eye(size(A));
    k = 1;

    while norm(E+F-E,1) > 0

        E = E+F;
        F = A*F/k;
        k = k+1;

    end

end

%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%