SLALN2 (3lapack)

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

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 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 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.  (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 only 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 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 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 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.