slaln2(3)

NAME

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 pertur

bation of A

SYNOPSIS

SUBROUTINE SLALN2( LTRANS, NA, NW, SMIN, CA, A,  LDA,  D1,
D2, B, LDB, WR, WI, X, LDX, SCALE, XNORM, INFO )
    LOGICAL        LTRANS
    INTEGER        INFO, LDA, LDB, LDX, NA, NW
    REAL           CA, D1, D2, SCALE, SMIN, WI, WR, XNORM
    REAL           A( LDA, * ), B( LDB, * ), X( LDX, * )

PURPOSE

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 pertur

bation of A. (A' means A-transpose.) A is an NA x NA real ma

trix, 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 complex. NA may be 1 or 2.

If w is complex, X and B are represented as NA x 2 matri

ces, 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 on

ly 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 approxima

tions, 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. (See BIGNUM.)

ARGUMENTS

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 on

ly be 1 or 2.

SMIN (input) REAL

The desired lower bound on the singular values of

A. This should be a safe distance away from underflow or over

flow, 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 contains 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 imaginary part.

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 posi

tive 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(3)