Systems of Linear Algebraic Equations - Numerical Methods in Engineering with MATLAB - page 41

EXAMPLE 2.4
An n × n Vandermode matrix A is definedby

v n − j
i

A i j = , i = 1 , 2 ,..., n , j = 1 , 2 ,..., n

where v is avector. In MATLAB a Vandermode matrix can be generatedbythe com-
mand vander(v) . Use the function gauss to compute the solution of Ax

= b , where

A is the 6 × 6 Vandermode matrix generated from the vector

1

0 T

v =

. 0 1 . 21 . 4 1 . 61 . 82 .

and

0 1 0 1 0 1 T

b =

Also evaluate the accuracy of the solution (Vandermode matrices tend to be ill-
conditioned).

Solution We used the program shown below. After constructing A and b , the output
format waschanged to long so that the solutionwouldbe printed to 14 decimal
places. Here are the results:

% Example 2.4 (Gauss elimination)
A = vander(1:0.2:2);
b=[010101]';
format long
[x,det] = gauss(A,b)
x=
1.0e+004 *
0.04166666666701
-0.31250000000246
0.92500000000697
-1.35000000000972
0.97093333334002
-0.27510000000181

det =
-1.132462079991823e-006

As the determinant isquite small relative to the elements of A (you may wantto
print A to verify this), weexpect detectable roundoff error. Inspection of x leads usto
suspect that the exact solutionis

1250

2751 T x = / 3 − 3125 9250 − 13500 29128 / 3 −

in which case the numerical solutionwouldbe accurate to 9decimal places.