SLALN2(l) ) SLALN2(l)
SLALN2 - solve a system of the form (ca A - w D ) X = s B or (ca A' - w D) X = s B with
possible scaling ("s") and perturbation of A
SUBROUTINE SLALN2( LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B, LDB, WR, WI, X, LDX,
SCALE, XNORM, INFO )
INTEGER INFO, LDA, LDB, LDX, NA, NW
REAL CA, D1, D2, SCALE, SMIN, WI, WR, XNORM
REAL A( LDA, * ), B( LDB, * ), X( LDX, * )
SLALN2 solves a system of the form (ca A - w D ) X = s B or (ca A' - w D) X = s B with
possible scaling ("s") and perturbation of A. (A' means A-transpose.) A is an NA x NA
real matrix, ca is a real scalar, D is an NA x NA real diagonal matrix, w is a real or
complex value, and X and B are NA x 1 matrices -- real if w is real, complex if w is com-
plex. NA may be 1 or 2.
If w is complex, X and B are represented as NA x 2 matrices, the first column of each
being the real part and the second being the imaginary part.
"s" is a scaling factor (.LE. 1), computed by SLALN2, which is so chosen that X can be
computed without overflow. X is further scaled if necessary to assure that norm(ca A - w
D)*norm(X) is less than overflow.
If both singular values of (ca A - w D) are less than SMIN, SMIN*identity will be used
instead of (ca A - w D). If only one singular value is less than SMIN, one element of (ca
A - w D) will be perturbed enough to make the smallest singular value roughly SMIN. If
both singular values are at least SMIN, (ca A - w D) will not be perturbed. In any case,
the perturbation will be at most some small multiple of max( SMIN, ulp*norm(ca A - w D) ).
The singular values are computed by infinity-norm approximations, and thus will only be
correct to a factor of 2 or so.
Note: all input quantities are assumed to be smaller than overflow by a reasonable factor.
LTRANS (input) LOGICAL
=.TRUE.: A-transpose will be used.
=.FALSE.: A will be used (not transposed.)
NA (input) INTEGER
The size of the matrix A. It may (only) be 1 or 2.
NW (input) INTEGER
1 if "w" is real, 2 if "w" is complex. It may only be 1 or 2.
SMIN (input) REAL
The desired lower bound on the singular values of A. This should be a safe dis-
tance away from underflow or overflow, say, between (underflow/machine precision)
and (machine precision * overflow ). (See BIGNUM and ULP.)
CA (input) REAL
The coefficient c, which A is multiplied by.
A (input) REAL array, dimension (LDA,NA)
The NA x NA matrix A.
LDA (input) INTEGER
The leading dimension of A. It must be at least NA.
D1 (input) REAL
The 1,1 element in the diagonal matrix D.
D2 (input) REAL
The 2,2 element in the diagonal matrix D. Not used if NW=1.
B (input) REAL array, dimension (LDB,NW)
The NA x NW matrix B (right-hand side). If NW=2 ("w" is complex), column 1 con-
tains the real part of B and column 2 contains the imaginary part.
LDB (input) INTEGER
The leading dimension of B. It must be at least NA.
WR (input) REAL
The real part of the scalar "w".
WI (input) REAL
The imaginary part of the scalar "w". Not used if NW=1.
X (output) REAL array, dimension (LDX,NW)
The NA x NW matrix X (unknowns), as computed by SLALN2. If NW=2 ("w" is complex),
on exit, column 1 will contain the real part of X and column 2 will contain the
LDX (input) INTEGER
The leading dimension of X. It must be at least NA.
SCALE (output) REAL
The scale factor that B must be multiplied by to insure that overflow does not
occur when computing X. Thus, (ca A - w D) X will be SCALE*B, not B (ignoring
perturbations of A.) It will be at most 1.
XNORM (output) REAL
The infinity-norm of X, when X is regarded as an NA x NW real matrix.
INFO (output) INTEGER
An error flag. It will be set to zero if no error occurs, a negative number if an
argument is in error, or a positive number if ca A - w D had to be perturbed.
The possible values are:
= 0: No error occurred, and (ca A - w D) did not have to be perturbed. = 1: (ca A
- w D) had to be perturbed to make its smallest (or only) singular value greater
than SMIN. NOTE: In the interests of speed, this routine does not check the
inputs for errors.
LAPACK version 3.0 15 June 2000 SLALN2(l)