Abdelrahman Said
659 lines
19 KiB
C
659 lines
19 KiB
C
/* -- translated by f2c (version 20240504).
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You must link the resulting object file with libf2c:
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on Microsoft Windows system, link with libf2c.lib;
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on Linux or Unix systems, link with .../path/to/libf2c.a -lm
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or, if you install libf2c.a in a standard place, with -lf2c -lm
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-- in that order, at the end of the command line, as in
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cc *.o -lf2c -lm
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Source for libf2c is in /netlib/f2c/libf2c.zip, e.g.,
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http://www.netlib.org/f2c/libf2c.zip
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*/
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#include "f2c.h"
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/* > \brief \b DLALN2 solves a 1-by-1 or 2-by-2 linear system of equations of the specified form.
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=========== DOCUMENTATION ===========
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Online html documentation available at
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http://www.netlib.org/lapack/explore-html/
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> \htmlonly
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> Download DLALN2 + dependencies
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> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaln2.
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f">
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> [TGZ]</a>
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> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaln2.
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f">
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> [ZIP]</a>
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> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaln2.
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f">
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> [TXT]</a>
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> \endhtmlonly
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Definition:
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===========
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SUBROUTINE DLALN2( LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B,
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LDB, WR, WI, X, LDX, SCALE, XNORM, INFO )
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LOGICAL LTRANS
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INTEGER INFO, LDA, LDB, LDX, NA, NW
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DOUBLE PRECISION CA, D1, D2, SCALE, SMIN, WI, WR, XNORM
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DOUBLE PRECISION A( LDA, * ), B( LDB, * ), X( LDX, * )
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> \par Purpose:
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=============
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>
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> \verbatim
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>
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> DLALN2 solves a system of the form (ca A - w D ) X = s B
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> or (ca A**T - w D) X = s B with possible scaling ("s") and
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> perturbation of A. (A**T means A-transpose.)
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>
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> A is an NA x NA real matrix, ca is a real scalar, D is an NA x NA
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> real diagonal matrix, w is a real or complex value, and X and B are
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> NA x 1 matrices -- real if w is real, complex if w is complex. NA
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> may be 1 or 2.
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>
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> If w is complex, X and B are represented as NA x 2 matrices,
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> the first column of each being the real part and the second
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> being the imaginary part.
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>
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> "s" is a scaling factor (.LE. 1), computed by DLALN2, which is
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> so chosen that X can be computed without overflow. X is further
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> scaled if necessary to assure that norm(ca A - w D)*norm(X) is less
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> than overflow.
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>
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> If both singular values of (ca A - w D) are less than SMIN,
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> SMIN*identity will be used instead of (ca A - w D). If only one
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> singular value is less than SMIN, one element of (ca A - w D) will be
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> perturbed enough to make the smallest singular value roughly SMIN.
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> If both singular values are at least SMIN, (ca A - w D) will not be
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> perturbed. In any case, the perturbation will be at most some small
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> multiple of max( SMIN, ulp*norm(ca A - w D) ). The singular values
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> are computed by infinity-norm approximations, and thus will only be
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> correct to a factor of 2 or so.
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>
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> Note: all input quantities are assumed to be smaller than overflow
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> by a reasonable factor. (See BIGNUM.)
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> \endverbatim
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Arguments:
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==========
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> \param[in] LTRANS
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> \verbatim
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> LTRANS is LOGICAL
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> =.TRUE.: A-transpose will be used.
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> =.FALSE.: A will be used (not transposed.)
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> \endverbatim
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>
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> \param[in] NA
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> \verbatim
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> NA is INTEGER
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> The size of the matrix A. It may (only) be 1 or 2.
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> \endverbatim
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>
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> \param[in] NW
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> \verbatim
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> NW is INTEGER
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> 1 if "w" is real, 2 if "w" is complex. It may only be 1
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> or 2.
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> \endverbatim
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>
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> \param[in] SMIN
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> \verbatim
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> SMIN is DOUBLE PRECISION
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> The desired lower bound on the singular values of A. This
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> should be a safe distance away from underflow or overflow,
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> say, between (underflow/machine precision) and (machine
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> precision * overflow ). (See BIGNUM and ULP.)
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> \endverbatim
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>
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> \param[in] CA
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> \verbatim
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> CA is DOUBLE PRECISION
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> The coefficient c, which A is multiplied by.
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> \endverbatim
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>
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> \param[in] A
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> \verbatim
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> A is DOUBLE PRECISION array, dimension (LDA,NA)
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> The NA x NA matrix A.
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> \endverbatim
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>
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> \param[in] LDA
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> \verbatim
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> LDA is INTEGER
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> The leading dimension of A. It must be at least NA.
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> \endverbatim
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>
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> \param[in] D1
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> \verbatim
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> D1 is DOUBLE PRECISION
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> The 1,1 element in the diagonal matrix D.
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> \endverbatim
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>
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> \param[in] D2
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> \verbatim
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> D2 is DOUBLE PRECISION
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> The 2,2 element in the diagonal matrix D. Not used if NW=1.
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> \endverbatim
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>
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> \param[in] B
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> \verbatim
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> B is DOUBLE PRECISION array, dimension (LDB,NW)
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> The NA x NW matrix B (right-hand side). If NW=2 ("w" is
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> complex), column 1 contains the real part of B and column 2
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> contains the imaginary part.
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> \endverbatim
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>
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> \param[in] LDB
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> \verbatim
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> LDB is INTEGER
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> The leading dimension of B. It must be at least NA.
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> \endverbatim
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>
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> \param[in] WR
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> \verbatim
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> WR is DOUBLE PRECISION
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> The real part of the scalar "w".
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> \endverbatim
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>
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> \param[in] WI
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> \verbatim
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> WI is DOUBLE PRECISION
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> The imaginary part of the scalar "w". Not used if NW=1.
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> \endverbatim
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>
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> \param[out] X
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> \verbatim
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> X is DOUBLE PRECISION array, dimension (LDX,NW)
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> The NA x NW matrix X (unknowns), as computed by DLALN2.
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> If NW=2 ("w" is complex), on exit, column 1 will contain
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> the real part of X and column 2 will contain the imaginary
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> part.
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> \endverbatim
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>
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> \param[in] LDX
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> \verbatim
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> LDX is INTEGER
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> The leading dimension of X. It must be at least NA.
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> \endverbatim
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>
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> \param[out] SCALE
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> \verbatim
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> SCALE is DOUBLE PRECISION
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> The scale factor that B must be multiplied by to insure
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> that overflow does not occur when computing X. Thus,
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> (ca A - w D) X will be SCALE*B, not B (ignoring
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> perturbations of A.) It will be at most 1.
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> \endverbatim
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>
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> \param[out] XNORM
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> \verbatim
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> XNORM is DOUBLE PRECISION
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> The infinity-norm of X, when X is regarded as an NA x NW
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> real matrix.
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> \endverbatim
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>
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> \param[out] INFO
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> \verbatim
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> INFO is INTEGER
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> An error flag. It will be set to zero if no error occurs,
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> a negative number if an argument is in error, or a positive
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> number if ca A - w D had to be perturbed.
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> The possible values are:
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> = 0: No error occurred, and (ca A - w D) did not have to be
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> perturbed.
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> = 1: (ca A - w D) had to be perturbed to make its smallest
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> (or only) singular value greater than SMIN.
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> NOTE: In the interests of speed, this routine does not
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> check the inputs for errors.
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> \endverbatim
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Authors:
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========
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> \author Univ. of Tennessee
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> \author Univ. of California Berkeley
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> \author Univ. of Colorado Denver
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> \author NAG Ltd.
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> \date September 2012
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> \ingroup doubleOTHERauxiliary
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=====================================================================
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Subroutine */ int igraphdlaln2_(logical *ltrans, integer *na, integer *nw,
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doublereal *smin, doublereal *ca, doublereal *a, integer *lda,
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doublereal *d1, doublereal *d2, doublereal *b, integer *ldb,
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doublereal *wr, doublereal *wi, doublereal *x, integer *ldx,
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doublereal *scale, doublereal *xnorm, integer *info)
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{
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/* Initialized data */
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IGRAPH_F77_SAVE logical zswap[4] = { FALSE_,FALSE_,TRUE_,TRUE_ };
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IGRAPH_F77_SAVE logical rswap[4] = { FALSE_,TRUE_,FALSE_,TRUE_ };
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IGRAPH_F77_SAVE integer ipivot[16] /* was [4][4] */ = { 1,2,3,4,2,1,4,3,3,4,1,2,
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4,3,2,1 };
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/* System generated locals */
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integer a_dim1, a_offset, b_dim1, b_offset, x_dim1, x_offset;
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doublereal d__1, d__2, d__3, d__4, d__5, d__6;
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IGRAPH_F77_SAVE doublereal equiv_0[4], equiv_1[4];
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/* Local variables */
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integer j;
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#define ci (equiv_0)
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#define cr (equiv_1)
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doublereal bi1, bi2, br1, br2, xi1, xi2, xr1, xr2, ci21, ci22, cr21, cr22,
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li21, csi, ui11, lr21, ui12, ui22;
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#define civ (equiv_0)
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doublereal csr, ur11, ur12, ur22;
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#define crv (equiv_1)
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doublereal bbnd, cmax, ui11r, ui12s, temp, ur11r, ur12s, u22abs;
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integer icmax;
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doublereal bnorm, cnorm, smini;
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extern doublereal igraphdlamch_(char *);
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extern /* Subroutine */ int igraphdladiv_(doublereal *, doublereal *,
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doublereal *, doublereal *, doublereal *, doublereal *);
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doublereal bignum, smlnum;
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/* -- LAPACK auxiliary routine (version 3.4.2) --
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-- LAPACK is a software package provided by Univ. of Tennessee, --
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-- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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September 2012
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=====================================================================
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Parameter adjustments */
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a_dim1 = *lda;
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a_offset = 1 + a_dim1;
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a -= a_offset;
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b_dim1 = *ldb;
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b_offset = 1 + b_dim1;
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b -= b_offset;
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x_dim1 = *ldx;
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x_offset = 1 + x_dim1;
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x -= x_offset;
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/* Function Body
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Compute BIGNUM */
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smlnum = 2. * igraphdlamch_("Safe minimum");
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bignum = 1. / smlnum;
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smini = max(*smin,smlnum);
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/* Don't check for input errors */
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*info = 0;
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/* Standard Initializations */
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*scale = 1.;
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if (*na == 1) {
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/* 1 x 1 (i.e., scalar) system C X = B */
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if (*nw == 1) {
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/* Real 1x1 system.
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C = ca A - w D */
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csr = *ca * a[a_dim1 + 1] - *wr * *d1;
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cnorm = abs(csr);
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/* If | C | < SMINI, use C = SMINI */
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if (cnorm < smini) {
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csr = smini;
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cnorm = smini;
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*info = 1;
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}
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/* Check scaling for X = B / C */
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bnorm = (d__1 = b[b_dim1 + 1], abs(d__1));
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if (cnorm < 1. && bnorm > 1.) {
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if (bnorm > bignum * cnorm) {
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*scale = 1. / bnorm;
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}
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}
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/* Compute X */
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x[x_dim1 + 1] = b[b_dim1 + 1] * *scale / csr;
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*xnorm = (d__1 = x[x_dim1 + 1], abs(d__1));
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} else {
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/* Complex 1x1 system (w is complex)
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C = ca A - w D */
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csr = *ca * a[a_dim1 + 1] - *wr * *d1;
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csi = -(*wi) * *d1;
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cnorm = abs(csr) + abs(csi);
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/* If | C | < SMINI, use C = SMINI */
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if (cnorm < smini) {
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csr = smini;
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csi = 0.;
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cnorm = smini;
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*info = 1;
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}
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/* Check scaling for X = B / C */
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bnorm = (d__1 = b[b_dim1 + 1], abs(d__1)) + (d__2 = b[(b_dim1 <<
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1) + 1], abs(d__2));
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if (cnorm < 1. && bnorm > 1.) {
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if (bnorm > bignum * cnorm) {
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*scale = 1. / bnorm;
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}
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}
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/* Compute X */
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d__1 = *scale * b[b_dim1 + 1];
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d__2 = *scale * b[(b_dim1 << 1) + 1];
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igraphdladiv_(&d__1, &d__2, &csr, &csi, &x[x_dim1 + 1], &x[(x_dim1 << 1)
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+ 1]);
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*xnorm = (d__1 = x[x_dim1 + 1], abs(d__1)) + (d__2 = x[(x_dim1 <<
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1) + 1], abs(d__2));
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}
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} else {
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/* 2x2 System
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Compute the real part of C = ca A - w D (or ca A**T - w D ) */
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cr[0] = *ca * a[a_dim1 + 1] - *wr * *d1;
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cr[3] = *ca * a[(a_dim1 << 1) + 2] - *wr * *d2;
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if (*ltrans) {
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cr[2] = *ca * a[a_dim1 + 2];
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cr[1] = *ca * a[(a_dim1 << 1) + 1];
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} else {
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cr[1] = *ca * a[a_dim1 + 2];
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cr[2] = *ca * a[(a_dim1 << 1) + 1];
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}
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if (*nw == 1) {
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/* Real 2x2 system (w is real)
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Find the largest element in C */
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cmax = 0.;
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icmax = 0;
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for (j = 1; j <= 4; ++j) {
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if ((d__1 = crv[j - 1], abs(d__1)) > cmax) {
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cmax = (d__1 = crv[j - 1], abs(d__1));
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icmax = j;
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}
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/* L10: */
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}
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/* If norm(C) < SMINI, use SMINI*identity. */
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if (cmax < smini) {
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/* Computing MAX */
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d__3 = (d__1 = b[b_dim1 + 1], abs(d__1)), d__4 = (d__2 = b[
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b_dim1 + 2], abs(d__2));
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bnorm = max(d__3,d__4);
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if (smini < 1. && bnorm > 1.) {
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if (bnorm > bignum * smini) {
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*scale = 1. / bnorm;
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}
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}
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temp = *scale / smini;
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x[x_dim1 + 1] = temp * b[b_dim1 + 1];
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x[x_dim1 + 2] = temp * b[b_dim1 + 2];
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*xnorm = temp * bnorm;
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*info = 1;
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return 0;
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}
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/* Gaussian elimination with complete pivoting. */
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ur11 = crv[icmax - 1];
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cr21 = crv[ipivot[(icmax << 2) - 3] - 1];
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ur12 = crv[ipivot[(icmax << 2) - 2] - 1];
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cr22 = crv[ipivot[(icmax << 2) - 1] - 1];
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ur11r = 1. / ur11;
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lr21 = ur11r * cr21;
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ur22 = cr22 - ur12 * lr21;
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/* If smaller pivot < SMINI, use SMINI */
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if (abs(ur22) < smini) {
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ur22 = smini;
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*info = 1;
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}
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if (rswap[icmax - 1]) {
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br1 = b[b_dim1 + 2];
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br2 = b[b_dim1 + 1];
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} else {
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br1 = b[b_dim1 + 1];
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br2 = b[b_dim1 + 2];
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}
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br2 -= lr21 * br1;
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/* Computing MAX */
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d__2 = (d__1 = br1 * (ur22 * ur11r), abs(d__1)), d__3 = abs(br2);
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bbnd = max(d__2,d__3);
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if (bbnd > 1. && abs(ur22) < 1.) {
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if (bbnd >= bignum * abs(ur22)) {
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*scale = 1. / bbnd;
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}
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}
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xr2 = br2 * *scale / ur22;
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xr1 = *scale * br1 * ur11r - xr2 * (ur11r * ur12);
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if (zswap[icmax - 1]) {
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x[x_dim1 + 1] = xr2;
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x[x_dim1 + 2] = xr1;
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} else {
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x[x_dim1 + 1] = xr1;
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x[x_dim1 + 2] = xr2;
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}
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/* Computing MAX */
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d__1 = abs(xr1), d__2 = abs(xr2);
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*xnorm = max(d__1,d__2);
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/* Further scaling if norm(A) norm(X) > overflow */
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if (*xnorm > 1. && cmax > 1.) {
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if (*xnorm > bignum / cmax) {
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temp = cmax / bignum;
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x[x_dim1 + 1] = temp * x[x_dim1 + 1];
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x[x_dim1 + 2] = temp * x[x_dim1 + 2];
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*xnorm = temp * *xnorm;
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*scale = temp * *scale;
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}
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}
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} else {
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/* Complex 2x2 system (w is complex)
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Find the largest element in C */
|
|
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|
ci[0] = -(*wi) * *d1;
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|
ci[1] = 0.;
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|
ci[2] = 0.;
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|
ci[3] = -(*wi) * *d2;
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|
cmax = 0.;
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|
icmax = 0;
|
|
|
|
for (j = 1; j <= 4; ++j) {
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|
if ((d__1 = crv[j - 1], abs(d__1)) + (d__2 = civ[j - 1], abs(
|
|
d__2)) > cmax) {
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|
cmax = (d__1 = crv[j - 1], abs(d__1)) + (d__2 = civ[j - 1]
|
|
, abs(d__2));
|
|
icmax = j;
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|
}
|
|
/* L20: */
|
|
}
|
|
|
|
/* If norm(C) < SMINI, use SMINI*identity. */
|
|
|
|
if (cmax < smini) {
|
|
/* Computing MAX */
|
|
d__5 = (d__1 = b[b_dim1 + 1], abs(d__1)) + (d__2 = b[(b_dim1
|
|
<< 1) + 1], abs(d__2)), d__6 = (d__3 = b[b_dim1 + 2],
|
|
abs(d__3)) + (d__4 = b[(b_dim1 << 1) + 2], abs(d__4));
|
|
bnorm = max(d__5,d__6);
|
|
if (smini < 1. && bnorm > 1.) {
|
|
if (bnorm > bignum * smini) {
|
|
*scale = 1. / bnorm;
|
|
}
|
|
}
|
|
temp = *scale / smini;
|
|
x[x_dim1 + 1] = temp * b[b_dim1 + 1];
|
|
x[x_dim1 + 2] = temp * b[b_dim1 + 2];
|
|
x[(x_dim1 << 1) + 1] = temp * b[(b_dim1 << 1) + 1];
|
|
x[(x_dim1 << 1) + 2] = temp * b[(b_dim1 << 1) + 2];
|
|
*xnorm = temp * bnorm;
|
|
*info = 1;
|
|
return 0;
|
|
}
|
|
|
|
/* Gaussian elimination with complete pivoting. */
|
|
|
|
ur11 = crv[icmax - 1];
|
|
ui11 = civ[icmax - 1];
|
|
cr21 = crv[ipivot[(icmax << 2) - 3] - 1];
|
|
ci21 = civ[ipivot[(icmax << 2) - 3] - 1];
|
|
ur12 = crv[ipivot[(icmax << 2) - 2] - 1];
|
|
ui12 = civ[ipivot[(icmax << 2) - 2] - 1];
|
|
cr22 = crv[ipivot[(icmax << 2) - 1] - 1];
|
|
ci22 = civ[ipivot[(icmax << 2) - 1] - 1];
|
|
if (icmax == 1 || icmax == 4) {
|
|
|
|
/* Code when off-diagonals of pivoted C are real */
|
|
|
|
if (abs(ur11) > abs(ui11)) {
|
|
temp = ui11 / ur11;
|
|
/* Computing 2nd power */
|
|
d__1 = temp;
|
|
ur11r = 1. / (ur11 * (d__1 * d__1 + 1.));
|
|
ui11r = -temp * ur11r;
|
|
} else {
|
|
temp = ur11 / ui11;
|
|
/* Computing 2nd power */
|
|
d__1 = temp;
|
|
ui11r = -1. / (ui11 * (d__1 * d__1 + 1.));
|
|
ur11r = -temp * ui11r;
|
|
}
|
|
lr21 = cr21 * ur11r;
|
|
li21 = cr21 * ui11r;
|
|
ur12s = ur12 * ur11r;
|
|
ui12s = ur12 * ui11r;
|
|
ur22 = cr22 - ur12 * lr21;
|
|
ui22 = ci22 - ur12 * li21;
|
|
} else {
|
|
|
|
/* Code when diagonals of pivoted C are real */
|
|
|
|
ur11r = 1. / ur11;
|
|
ui11r = 0.;
|
|
lr21 = cr21 * ur11r;
|
|
li21 = ci21 * ur11r;
|
|
ur12s = ur12 * ur11r;
|
|
ui12s = ui12 * ur11r;
|
|
ur22 = cr22 - ur12 * lr21 + ui12 * li21;
|
|
ui22 = -ur12 * li21 - ui12 * lr21;
|
|
}
|
|
u22abs = abs(ur22) + abs(ui22);
|
|
|
|
/* If smaller pivot < SMINI, use SMINI */
|
|
|
|
if (u22abs < smini) {
|
|
ur22 = smini;
|
|
ui22 = 0.;
|
|
*info = 1;
|
|
}
|
|
if (rswap[icmax - 1]) {
|
|
br2 = b[b_dim1 + 1];
|
|
br1 = b[b_dim1 + 2];
|
|
bi2 = b[(b_dim1 << 1) + 1];
|
|
bi1 = b[(b_dim1 << 1) + 2];
|
|
} else {
|
|
br1 = b[b_dim1 + 1];
|
|
br2 = b[b_dim1 + 2];
|
|
bi1 = b[(b_dim1 << 1) + 1];
|
|
bi2 = b[(b_dim1 << 1) + 2];
|
|
}
|
|
br2 = br2 - lr21 * br1 + li21 * bi1;
|
|
bi2 = bi2 - li21 * br1 - lr21 * bi1;
|
|
/* Computing MAX */
|
|
d__1 = (abs(br1) + abs(bi1)) * (u22abs * (abs(ur11r) + abs(ui11r))
|
|
), d__2 = abs(br2) + abs(bi2);
|
|
bbnd = max(d__1,d__2);
|
|
if (bbnd > 1. && u22abs < 1.) {
|
|
if (bbnd >= bignum * u22abs) {
|
|
*scale = 1. / bbnd;
|
|
br1 = *scale * br1;
|
|
bi1 = *scale * bi1;
|
|
br2 = *scale * br2;
|
|
bi2 = *scale * bi2;
|
|
}
|
|
}
|
|
|
|
igraphdladiv_(&br2, &bi2, &ur22, &ui22, &xr2, &xi2);
|
|
xr1 = ur11r * br1 - ui11r * bi1 - ur12s * xr2 + ui12s * xi2;
|
|
xi1 = ui11r * br1 + ur11r * bi1 - ui12s * xr2 - ur12s * xi2;
|
|
if (zswap[icmax - 1]) {
|
|
x[x_dim1 + 1] = xr2;
|
|
x[x_dim1 + 2] = xr1;
|
|
x[(x_dim1 << 1) + 1] = xi2;
|
|
x[(x_dim1 << 1) + 2] = xi1;
|
|
} else {
|
|
x[x_dim1 + 1] = xr1;
|
|
x[x_dim1 + 2] = xr2;
|
|
x[(x_dim1 << 1) + 1] = xi1;
|
|
x[(x_dim1 << 1) + 2] = xi2;
|
|
}
|
|
/* Computing MAX */
|
|
d__1 = abs(xr1) + abs(xi1), d__2 = abs(xr2) + abs(xi2);
|
|
*xnorm = max(d__1,d__2);
|
|
|
|
/* Further scaling if norm(A) norm(X) > overflow */
|
|
|
|
if (*xnorm > 1. && cmax > 1.) {
|
|
if (*xnorm > bignum / cmax) {
|
|
temp = cmax / bignum;
|
|
x[x_dim1 + 1] = temp * x[x_dim1 + 1];
|
|
x[x_dim1 + 2] = temp * x[x_dim1 + 2];
|
|
x[(x_dim1 << 1) + 1] = temp * x[(x_dim1 << 1) + 1];
|
|
x[(x_dim1 << 1) + 2] = temp * x[(x_dim1 << 1) + 2];
|
|
*xnorm = temp * *xnorm;
|
|
*scale = temp * *scale;
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
return 0;
|
|
|
|
/* End of DLALN2 */
|
|
|
|
} /* igraphdlaln2_ */
|
|
|
|
#undef crv
|
|
#undef civ
|
|
#undef cr
|
|
#undef ci
|
|
|
|
|