generate_test_coordinate (complex)

IMSL_SYMMETRIC_STORAGE,
For coordinate representation, return only values for the diagonal and lower triangle. This option is not allowed if IMSL_D_MATRIX is specified.

Description

The same nomenclature as Østerby and Zlatev (1982) is used. Test matrices of class E(n, c), to which we will generally refer to as E-matrices, are symmetric, positive definite matrices of order n with (6.0, 0.0) in the diagonal, (‑1.0, 1.0) in the superdiagonal and (‑1.0, ‑1.0) subdiagonal. In addition there are two bands at a distance c from the diagonal with (‑1.0, 1.0) in the upper codiagonal and (‑1.0, ‑1.0) in the lower codiagonal. More precisely:

ai,i = 6	0 ≤ i < n
ai,i+1 = ‑1 ‑ i	0 ≤ i < n ‑1
ai+1,i = ‑1 ‑ i	0 ≤ i < n ‑1
ai,i+c = ‑1 + i	0 ≤ i < n ‑c
ai+c,i = ‑1 + i	0 ≤ i < n ‑c

for any n ≥ 3 and 2 ≤ c ≤ n ‑ 1.

Test matrices of class D(n, c) are square matrices of order n with a full diagonal, three bands at a distance c above the diagonal and reappearing cyclically under the diagonal, and a 10 × 10 triangle of elements in the upper-right corner. More precisely:

ai,i = 1	0 ≤ i < n
ai,i+c = i + 2	0 ≤ i < n - c
ai,i-n+c = i + 2	n ‑ c ≤ i < n
ai,i+c+1 = ‑(i + 1)	0 ≤ i < n ‑ c ‑ 1
ai,i+c+1 = ‑(i + 1)	n ‑ c ‑ 1≤ i < n
ai,i+c+2 = 16	0 ≤ i < n ‑ c ‑ 2
ai,i-n+c+2 = 16	n ‑ c ‑ 2≤ i < n
ai,n-11+i+j = 100j	1 ≤ i< 11 ‑ j, 0 ≤ j < 10

for any n ≥ 14 and 1 ≤ c ≤ n ‑ 13.

The sparsity pattern of D(20, 5) is as follows:

By default imsl_c_generate_test_coordinate returns an E-matrix in coordinate representation. By specifying the IMSL_SYMMETRIC_STORAGE option, only the diagonal and lower triangle are returned. The scalar nz will contain the number of non-zeros in this representation.

The option IMSL_D_MATRIX will return a matrix of class D(n, c). Since D-matrices are not symmetric, the IMSL_SYMMETRIC_STORAGE option is not allowed.

Examples

Example 1

This example generates the matrix

and prints the result.

#include <imsl.h>

#include <stdio.h>

int main()

{

int i, n = 5, c = 3, nz;

Imsl_c_sparse_elem *a;

a = imsl_c_generate_test_coordinate (n, c, &nz,

0);

printf ("row col val\n");

for (i=0; i<nz; i++)

printf (" %d %d (%5.1f, %5.1f)\n",

a[i].row, a[i].col, a[i].val.re, a[i].val.im);

}

Output

row col val

0 0 ( 6.0, 0.0)

1 1 ( 6.0, 0.0)

2 2 ( 6.0, 0.0)

3 3 ( 6.0, 0.0)

4 4 ( 6.0, 0.0)

1 0 ( -1.0, -1.0)

2 1 ( -1.0, -1.0)

3 2 ( -1.0, -1.0)

4 3 ( -1.0, -1.0)

0 1 ( -1.0, 1.0)

1 2 ( -1.0, 1.0)

2 3 ( -1.0, 1.0)

3 4 ( -1.0, 1.0)

3 0 ( -1.0, -1.0)

4 1 ( -1.0, -1.0)

0 3 ( -1.0, 1.0)

1 4 ( -1.0, 1.0)

Example 2

In this example, the matrix E(5, 3) is returned in symmetric storage and printed.