CXML
        	    
LAPACK version 3.0
dlaln2(3)

PURPOSE
  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

SYNTAX
  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, * )

DESCRIPTION
  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.)

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) 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.