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version 1.4, 2010/08/06 15:32:27 version 1.20, 2023/08/07 08:38:54
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   *> \brief \b DLALN2 solves a 1-by-1 or 2-by-2 linear system of equations of the specified form.
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at
   *            http://www.netlib.org/lapack/explore-html/
   *
   *> \htmlonly
   *> Download DLALN2 + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaln2.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaln2.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaln2.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DLALN2( LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B,
   *                          LDB, WR, WI, X, LDX, SCALE, XNORM, INFO )
   *
   *       .. Scalar Arguments ..
   *       LOGICAL            LTRANS
   *       INTEGER            INFO, LDA, LDB, LDX, NA, NW
   *       DOUBLE PRECISION   CA, D1, D2, SCALE, SMIN, WI, WR, XNORM
   *       ..
   *       .. Array Arguments ..
   *       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), X( LDX, * )
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DLALN2 solves a system of the form  (ca A - w D ) X = s B
   *> or (ca A**T - w D) X = s B   with possible scaling ("s") and
   *> perturbation of A.  (A**T 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 (<= 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.)
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] LTRANS
   *> \verbatim
   *>          LTRANS is LOGICAL
   *>          =.TRUE.:  A-transpose will be used.
   *>          =.FALSE.: A will be used (not transposed.)
   *> \endverbatim
   *>
   *> \param[in] NA
   *> \verbatim
   *>          NA is INTEGER
   *>          The size of the matrix A.  It may (only) be 1 or 2.
   *> \endverbatim
   *>
   *> \param[in] NW
   *> \verbatim
   *>          NW is INTEGER
   *>          1 if "w" is real, 2 if "w" is complex.  It may only be 1
   *>          or 2.
   *> \endverbatim
   *>
   *> \param[in] SMIN
   *> \verbatim
   *>          SMIN is 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.)
   *> \endverbatim
   *>
   *> \param[in] CA
   *> \verbatim
   *>          CA is DOUBLE PRECISION
   *>          The coefficient c, which A is multiplied by.
   *> \endverbatim
   *>
   *> \param[in] A
   *> \verbatim
   *>          A is DOUBLE PRECISION array, dimension (LDA,NA)
   *>          The NA x NA matrix A.
   *> \endverbatim
   *>
   *> \param[in] LDA
   *> \verbatim
   *>          LDA is INTEGER
   *>          The leading dimension of A.  It must be at least NA.
   *> \endverbatim
   *>
   *> \param[in] D1
   *> \verbatim
   *>          D1 is DOUBLE PRECISION
   *>          The 1,1 element in the diagonal matrix D.
   *> \endverbatim
   *>
   *> \param[in] D2
   *> \verbatim
   *>          D2 is DOUBLE PRECISION
   *>          The 2,2 element in the diagonal matrix D.  Not used if NA=1.
   *> \endverbatim
   *>
   *> \param[in] B
   *> \verbatim
   *>          B is 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.
   *> \endverbatim
   *>
   *> \param[in] LDB
   *> \verbatim
   *>          LDB is INTEGER
   *>          The leading dimension of B.  It must be at least NA.
   *> \endverbatim
   *>
   *> \param[in] WR
   *> \verbatim
   *>          WR is DOUBLE PRECISION
   *>          The real part of the scalar "w".
   *> \endverbatim
   *>
   *> \param[in] WI
   *> \verbatim
   *>          WI is DOUBLE PRECISION
   *>          The imaginary part of the scalar "w".  Not used if NW=1.
   *> \endverbatim
   *>
   *> \param[out] X
   *> \verbatim
   *>          X is 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.
   *> \endverbatim
   *>
   *> \param[in] LDX
   *> \verbatim
   *>          LDX is INTEGER
   *>          The leading dimension of X.  It must be at least NA.
   *> \endverbatim
   *>
   *> \param[out] SCALE
   *> \verbatim
   *>          SCALE is 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.
   *> \endverbatim
   *>
   *> \param[out] XNORM
   *> \verbatim
   *>          XNORM is DOUBLE PRECISION
   *>          The infinity-norm of X, when X is regarded as an NA x NW
   *>          real matrix.
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is 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.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee
   *> \author Univ. of California Berkeley
   *> \author Univ. of Colorado Denver
   *> \author NAG Ltd.
   *
   *> \ingroup doubleOTHERauxiliary
   *
   *  =====================================================================
       SUBROUTINE DLALN2( LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B,        SUBROUTINE DLALN2( LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B,
      $                   LDB, WR, WI, X, LDX, SCALE, XNORM, INFO )       $                   LDB, WR, WI, X, LDX, SCALE, XNORM, INFO )
 *  *
 *  -- LAPACK auxiliary routine (version 3.2) --  *  -- LAPACK auxiliary routine --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *     November 2006  
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       LOGICAL            LTRANS        LOGICAL            LTRANS
Line 15 Line 229
       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), X( LDX, * )        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 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.  
 *  
 * =====================================================================  * =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Line 257 Line 357
 *  *
 *        2x2 System  *        2x2 System
 *  *
 *        Compute the real part of  C = ca A - w D  (or  ca A' - w D )  *        Compute the real part of  C = ca A - w D  (or  ca A**T - w D )
 *  *
          CR( 1, 1 ) = CA*A( 1, 1 ) - WR*D1           CR( 1, 1 ) = CA*A( 1, 1 ) - WR*D1
          CR( 2, 2 ) = CA*A( 2, 2 ) - WR*D2           CR( 2, 2 ) = CA*A( 2, 2 ) - WR*D2
Removed from v.1.4  
changed lines
  Added in v.1.20

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