.TH SLALN2 1 "November 2006" " LAPACK auxiliary routine (version 3.1) " " LAPACK auxiliary routine (version 3.1) " .SH NAME SLALN2 - a system of the form (ca A - w D ) X = s B or (ca A\(aq - w D) X = s B with possible scaling ("s") and perturbation of A .SH SYNOPSIS .TP 19 SUBROUTINE SLALN2( LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B, LDB, WR, WI, X, LDX, SCALE, XNORM, INFO ) .TP 19 .ti +4 LOGICAL LTRANS .TP 19 .ti +4 INTEGER INFO, LDA, LDB, LDX, NA, NW .TP 19 .ti +4 REAL CA, D1, D2, SCALE, SMIN, WI, WR, XNORM .TP 19 .ti +4 REAL A( LDA, * ), B( LDB, * ), X( LDX, * ) .SH PURPOSE SLALN2 solves a system of the form (ca A - w D ) X = s B or (ca A\(aq - w D) X = s B with possible scaling ("s") and perturbation of A. (A\(aq 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. .br 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. .br "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. .br 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. .br Note: all input quantities are assumed to be smaller than overflow by a reasonable factor. (See BIGNUM.) .br .SH ARGUMENTS .TP 8 LTRANS (input) LOGICAL =.TRUE.: A-transpose will be used. .br =.FALSE.: A will be used (not transposed.) .TP 8 NA (input) INTEGER The size of the matrix A. It may (only) be 1 or 2. .TP 8 NW (input) INTEGER 1 if "w" is real, 2 if "w" is complex. It may only be 1 or 2. .TP 8 SMIN (input) REAL 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.) .TP 8 CA (input) REAL The coefficient c, which A is multiplied by. .TP 8 A (input) REAL array, dimension (LDA,NA) The NA x NA matrix A. .TP 8 LDA (input) INTEGER The leading dimension of A. It must be at least NA. .TP 8 D1 (input) REAL The 1,1 element in the diagonal matrix D. .TP 8 D2 (input) REAL The 2,2 element in the diagonal matrix D. Not used if NW=1. .TP 8 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. .TP 8 LDB (input) INTEGER The leading dimension of B. It must be at least NA. .TP 8 WR (input) REAL The real part of the scalar "w". .TP 8 WI (input) REAL The imaginary part of the scalar "w". Not used if NW=1. .TP 8 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. .TP 8 LDX (input) INTEGER The leading dimension of X. It must be at least NA. .TP 8 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. .TP 8 XNORM (output) REAL The infinity-norm of X, when X is regarded as an NA x NW real matrix. .TP 8 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: .br = 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.