DLALN2 - 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 DLALN2(

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

DOUBLE PRECISION CA, D1, D2, SCALE, SMIN, WI, WR, XNORM

DOUBLE PRECISION A( LDA, * ), B( LDB, * ), X( LDX, * )

DLALN2 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 complex. 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 DLALN2, 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. (See BIGNUM.)

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) DOUBLE PRECISION

The desired lower bound on the singular values of A. This should be a safe distance away from underflow or overflow, say, between (underflow/machine precision) and (machine precision * overflow ). (See BIGNUM and ULP.)

CA (input) DOUBLE PRECISION

The coefficient c, which A is multiplied by.

A (input) DOUBLE PRECISION 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) DOUBLE PRECISION

The 1,1 element in the diagonal matrix D.

D2 (input) DOUBLE PRECISION

The 2,2 element in the diagonal matrix D. Not used if NW=1.

B (input) DOUBLE PRECISION 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) DOUBLE PRECISION

The real part of the scalar "w".

WI (input) DOUBLE PRECISION

The imaginary part of the scalar "w". Not used if NW=1.

X (output) DOUBLE PRECISION array, dimension (LDX,NW)

The NA x NW matrix X (unknowns), as computed by DLALN2. 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) DOUBLE PRECISION

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) DOUBLE PRECISION

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.