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DLALN2(l)			       )			     DLALN2(l)

NAME
       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

SYNOPSIS
       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, * )

PURPOSE
       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 com‐
       plex.  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  cho‐
       sen  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	 over‐
       flow.

       If  both singular values of (ca A - w D) are less than SMIN, SMIN*iden‐
       tity 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 com‐
	       plex), column 1 contains the real part of B and column  2  con‐
	       tains 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  num‐
	       ber  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			     DLALN2(l)
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