IMSL_SP_BDFAC

The IMSL_SP_BDFAC procedure computes the LU factorization of a matrix stored in band storage mode.

Note
This routine requires an IDL Advanced Math and Stats license. For more information, contact your ITT Visual Information Solutions sales or technical support representative.

Syntax

IMSL_SP_BDFAC, nlca, nuca, n_rows, a, pivot, factor [, BLK_FACTOR=value] [, CONDITION=variable] [, /DOUBLE]

Arguments

a

Array of size (nlca + nuca + 1) x n containing the n x n banded coefficient matrix in band storage mode A(i,j). See Band Storage Format for a description of band storage mode.

factor

A named variable that will contain an array of size (2*nlca + nuca + 1) x n_rows containing the LU factorization of A with column pivoting. The keywords FACTOR and CONDITION cannot be used together.

n_rows

Number of rows in a.

nlca

Number of lower codiagonals in a.

nuca

Number of upper codiagonals in a.

pivot

A named variable that will contain a one-dimensional array containing the pivot sequence. The keywords PIVOT and CONDITION cannot be used together.

Keywords

BLK_FACTOR

The blocking factor. This keyword must be set no larger than 32. Default: BLK_FACTOR = 1.

CONDITION

Named variable into which an estimate of the L₁ condition number is stored. The keyword CONDITION cannot be used with arguments pivot or factor.

DOUBLE

If present and nonzero, double precision is used.

Discussion

The IMSL_SP_BDFAC function computes the LU factorization of A with based on the blocked LU factorization algorithm given in Du Croz, et al, (1990). Level-3 BLAS invocations were replaced by in-line loops. The blocking factor BLK_FACTOR has the default value of 1, but can be reset to any positive value not exceeding 32.

An estimate of the L₁ condition number of A is computed using Higham's modifications to Hager's method, as given in Higham (1988). If the estimated condition number is greater than 1/ε (where ε is the machine precision), a warning message is issued. This indicates that very small changes in A may produce large changes in the solution x.

Example

Consider the 1000 x 1000 banded matrix below:

This example computes the solution to Ax₁ = b₁ and Ax₂ = b₂, where b₁ and b₂ are random vectors. The factorization is computed just once, using IMSL_SP_BDFAC, and the solutions are computed using IMSL_SP_BDSOL.

n_rows = 1000L 
nlca = 1L 
nuca = 1L 
a = DBLARR(n_rows*(nlca+nuca+1)) 
a(1:n_rows-1) = 4 
a(n_rows:2*n_rows-1) = -1 
a(2*n_rows:*) = 4 
; Fill A with the values of the bands. 
seed = 123L 
b1 = RANDOMU(seed, n_rows) 
b2 = RANDOMU(seed, n_rows) 
; Fill random vectors 
IMSL_SP_BDFAC, nlca, nuca, n_rows, a, pivot, factor 
; Compute the factorization using IMSL_SP_BDFAC. 
x1 = IMSL_SP_BDSOL(b1, nlca, nuca, Factor = factor, Pivot = pivot) 
; Compute solution of Ax1 = b1 above, and output residual below. 
PM, TOTAL(ABS(IMSL_SP_MVMUL(n_rows, n_rows, nlca, nuca, $ 
   a, x1)-b1)) 
 
   1.2367884e-13 
 
x2 = IMSL_SP_BDSOL(b2, nlca, nuca, Factor = factor, Pivot = pivot) 
; Compute the solution of Ax2 = b2 above, and output residual. 
PM, TOTAL(ABS(IMSL_SP_MVMUL(n_rows, n_rows, nlca, nuca, $ 
   a, x2)-b2)) 
 
   9.1537888e-14

Version History


6.4	Introduced