Added (S,D,C,Z) (SY,HE) routines, drivers for new rook code

Close #82

Added routines for new factorization code for symmetric indefinite
( or Hermitian indefinite ) matrices with bounded Bunch-Kaufman
( rook ) pivoting algorithm.

New more efficient storage format for factors U ( or L ),
block-diagonal matrix D, and pivoting information stored in IPIV:

factor L is stored explicitly in lower triangle of A;
diagonal of D is stored on the diagonal of A;
subdiagonal elements of D are stored in array E;
IPIV format is the same as in *_ROOK routines, but differs
from SY Bunch-Kaufman routines (e.g. *SYTRF).
The factorization output of these new rook _RK routines is not
compatible
with the existing _ROOK routines and vice versa. This new factorization
format is designed in such a way, that there is a possibility in the
future
to write new Bunch-Kaufman routines that conform to this new
factorization format.
Then the future Bunch-Kaufman routines could share solver
*TRS_3,inversion *TRI_3
and condition estimator *CON_3.

To convert between the factorization formats in both ways the following
routines
are developed:

CONVERSION ROUTINES BETWEEN FACTORIZATION FORMATS

DOUBLE PRECISION (symmetric indefinite matrices):

new file:   SRC/dsyconvf.f
new file:   SRC/dsyconvf_rook.f
REAL (symmetric indefinite matrices):

new file:   SRC/csyconvf.f
new file:   SRC/csyconvf_rook.f
COMPLEX*16 (symmetric indefinite and Hermitian indefinite matrices):

new file:   SRC/zsyconvf.f
new file:   SRC/zsyconvf_rook.f
COMPLEX (symmetric indefinite and Hermitian indefinite matrices):

new file:   SRC/ssyconvf.f
new file:   SRC/ssyconvf_rook.f
*SYCONVF routine converts between old Bunch-Kaufman storage format (
denote (L1,D1,IPIV1) )
that is used by *SYTRF and new rook storage format ( denote (L2,D2,
IPIV2))
that is used by *SYTRF_RK

*SYCONVF_ROOK routine between old rook storage format ( denote
(L1,D1,IPIV2) )
that is used by *SYTRF_ROOK and new rook storage format ( denote
(L2,D2, IPIV2))
that is used by *SYTRF_RK

ROUTINES AND DRIVERS

DOUBLE PRECISION (symmetric indefinite matrices):

new file:   SRC/dsytf2_rk.f BLAS2 unblocked factorization
new file:   SRC/dlasyf_rk.f BLAS3 auxiliary blocked partial
factorization
new file:   SRC/dsytrf_rk.f BLAS3 blocked factorization
new file:   SRC/dsytrs_3.f BLAS3 solver
new file:   SRC/dsycon_3.f BLAS3 condition number estimator
new file:   SRC/dsytri_3.f BLAS3 inversion, sets the size of work array
and calls *sytri_3x
new file:   SRC/dsytri_3x.f BLAS3 auxiliary inversion, actually
computes blocked inversion
new file:   SRC/dsysv_rk.f BLAS3 solver driver
REAL (symmetric indefinite matrices):

new file:   SRC/ssytf2_rk.f BLAS2 unblocked factorization
new file:   SRC/slasyf_rk.f BLAS3 auxiliary blocked partial
factorization
new file:   SRC/ssytrf_rk.f BLAS3 blocked factorization
new file:   SRC/ssytrs_3.f BLAS3 solver
new file:   SRC/ssycon_3.f BLAS3 condition number estimator
new file:   SRC/ssytri_3.f BLAS3 inversion, sets the size of work array
and calls *sytri_3x
new file:   SRC/ssytri_3x.f BLAS3 auxiliary inversion, actually
computes blocked inversion
new file:   SRC/ssysv_rk.f BLAS3 solver driver
COMPLEX*16 (symmetric indefinite matrices):

new file:   SRC/zsytf2_rk.f BLAS2 unblocked factorization
new file:   SRC/zlasyf_rk.f BLAS3 auxiliary blocked partial
factorization
new file:   SRC/zsytrf_rk.f BLAS3 blocked factorization
new file:   SRC/zsytrs_3.f BLAS3 solver
new file:   SRC/zsycon_3.f BLAS3 condition number estimator
new file:   SRC/zsytri_3.f BLAS3 inversion, sets the size of work array
and calls *sytri_3x
new file:   SRC/zsytri_3x.f BLAS3 auxiliary inversion, actually
computes blocked inversion
new file:   SRC/zsysv_rk.f BLAS3 solver driver
COMPLEX*16 (Hermitian indefinite matrices):

new file:   SRC/zhetf2_rk.f BLAS2 unblocked factorization
new file:   SRC/zlahef_rk.f BLAS3 auxiliary blocked partial
factorization
new file:   SRC/zhetrf_rk.f BLAS3 blocked factorization
new file:   SRC/zhetrs_3.f BLAS3 solver
new file:   SRC/zhecon_3.f BLAS3 condition number estimator
new file:   SRC/zhetri_3.f BLAS3 inversion, sets the size of work array
and calls *sytri_3x
new file:   SRC/zhetri_3x.f BLAS3 auxiliary inversion, actually
computes blocked inversion
new file:   SRC/zhesv_rk.f BLAS3 solver driver
COMPLEX (symmetric indefinite matrices):

new file:   SRC/csytf2_rk.f BLAS2 unblocked factorization
new file:   SRC/clasyf_rk.f BLAS3 auxiliary blocked partial
factorization
new file:   SRC/csytrf_rk.f BLAS3 blocked factorization
new file:   SRC/csytrs_3.f BLAS3 solver
new file:   SRC/csycon_3.f BLAS3 condition number estimator
new file:   SRC/csytri_3.f BLAS3 inversion, sets the size of work array
and calls *sytri_3x
new file:   SRC/csytri_3x.f BLAS3 auxiliary inversion, actually
computes blocked inversion
new file:   SRC/csysv_rk.f BLAS3 solver driver
COMPLEX (Hermitian indefinite matrices):

new file:   SRC/chetf2_rk.f BLAS2 unblocked factorization
new file:   SRC/clahef_rk.f BLAS3 auxiliary blocked partial
factorization
new file:   SRC/chetrf_rk.f BLAS3 blocked factorization
new file:   SRC/chetrs_3.f BLAS3 solver
new file:   SRC/checon_3.f BLAS3 condition number estimator
new file:   SRC/chetri_3.f BLAS3 inversion, sets the size of work array
and calls *sytri_3x
new file:   SRC/chetri_3x.f BLAS3 auxiliary inversion, actually
computes blocked inversion
new file:   SRC/chesv_rk.f BLAS3 solver driver
MISC

modified:   SRC/CMakeLists.txt
modified:   SRC/Makefile
TEST CODE

modified:   TESTING/LIN/CMakeLists.txt
modified:   TESTING/LIN/Makefile

modified:   TESTING/LIN/aladhd.f
modified:   TESTING/LIN/alaerh.f
modified:   TESTING/LIN/alahd.f
DOUBLE PRECISION (symmetric indefinite matrices):

modified:   TESTING/LIN/dchkaa.f
modified:   TESTING/LIN/derrsy.f
modified:   TESTING/LIN/derrsyx.f
modified:   TESTING/LIN/derrvx.f
modified:   TESTING/LIN/derrvxx.f

modified:   TESTING/dtest.in

new file:   TESTING/LIN/dchksy_rk.f
new file:   TESTING/LIN/ddrvsy_rk.f
new file:   TESTING/LIN/dsyt01_3.f
REAL (symmetric indefinite matrices):

modified:   TESTING/LIN/schkaa.f
modified:   TESTING/LIN/serrsy.f
modified:   TESTING/LIN/serrsyx.f
modified:   TESTING/LIN/serrvx.f
modified:   TESTING/LIN/serrvxx.f

modified:   TESTING/stest.in

new file:   TESTING/LIN/schksy_rk.f
new file:   TESTING/LIN/sdrvsy_rk.f
new file:   TESTING/LIN/ssyt01_3.f
COMPLEX*16 (symmetric indefinite and Hermitian indefinite matrices):

modified:   TESTING/LIN/zchkaa.f
modified:   TESTING/LIN/zerrsy.f
modified:   TESTING/LIN/zerrsyx.f
modified:   TESTING/LIN/zerrhe.f
modified:   TESTING/LIN/zerrhex.f
modified:   TESTING/LIN/zerrvx.f
modified:   TESTING/LIN/zerrvxx.f

modified:   TESTING/ztest.in

new file:   TESTING/LIN/zchksy_rk.f
new file:   TESTING/LIN/zdrvsy_rk.f
new file:   TESTING/LIN/zsyt01_3.f
new file:   TESTING/LIN/zchkhe_rk.f
new file:   TESTING/LIN/zdrvhe_rk.f
new file:   TESTING/LIN/zhet01_3.f
COMPLEX (symmetric indefinite and Hermitian indefinite matrices):

modified:   TESTING/LIN/cchkaa.f
modified:   TESTING/LIN/cerrsy.f
modified:   TESTING/LIN/cerrsyx.f
modified:   TESTING/LIN/cerrhe.f
modified:   TESTING/LIN/cerrhex.f
modified:   TESTING/LIN/cerrvx.f
modified:   TESTING/LIN/cerrvxx.f

modified:   TESTING/ctest.in

new file:   TESTING/LIN/cchksy_rk.f
new file:   TESTING/LIN/cdrvsy_rk.f
new file:   TESTING/LIN/csyt01_3.f
new file:   TESTING/LIN/cchkhe_rk.f
new file:   TESTING/LIN/cdrvhe_rk.f
new file:   TESTING/LIN/chet01_3.f

1 files changed, 965 insertions, 0 deletions

diff --git a/SRC/dlasyf_rk.f b/SRC/dlasyf_rk.f
new file mode 100644
index 00000000..cbc13deb
--- /dev/null
+++ b/SRC/dlasyf_rk.f
@@ -0,0 +1,965 @@
+*> \brief \b DLASYF_RK computes a partial factorization of a real symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method.
+*
+*  =========== DOCUMENTATION ===========
+*
+* Online html documentation available at
+*            http://www.netlib.org/lapack/explore-html/
+*
+*> \htmlonly
+*> Download DLASYF_RK + dependencies
+*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasyf_rk.f">
+*> [TGZ]</a>
+*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasyf_rk.f">
+*> [ZIP]</a>
+*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasyf_rk.f">
+*> [TXT]</a>
+*> \endhtmlonly
+*
+*  Definition:
+*  ===========
+*
+*       SUBROUTINE DLASYF_RK( UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW,
+*                             INFO )
+*
+*       .. Scalar Arguments ..
+*       CHARACTER          UPLO
+*       INTEGER            INFO, KB, LDA, LDW, N, NB
+*       ..
+*       .. Array Arguments ..
+*       INTEGER            IPIV( * )
+*       DOUBLE PRECISION   A( LDA, * ), E( * ), W( LDW, * )
+*       ..
+*
+*
+*> \par Purpose:
+*  =============
+*>
+*> \verbatim
+*> DLASYF_RK computes a partial factorization of a real symmetric
+*> matrix A using the bounded Bunch-Kaufman (rook) diagonal
+*> pivoting method. The partial factorization has the form:
+*>
+*> A  =  ( I  U12 ) ( A11  0  ) (  I       0    )  if UPLO = 'U', or:
+*>       ( 0  U22 ) (  0   D  ) ( U12**T U22**T )
+*>
+*> A  =  ( L11  0 ) (  D   0  ) ( L11**T L21**T )  if UPLO = 'L',
+*>       ( L21  I ) (  0  A22 ) (  0       I    )
+*>
+*> where the order of D is at most NB. The actual order is returned in
+*> the argument KB, and is either NB or NB-1, or N if N <= NB.
+*>
+*> DLASYF_RK is an auxiliary routine called by DSYTRF_RK. It uses
+*> blocked code (calling Level 3 BLAS) to update the submatrix
+*> A11 (if UPLO = 'U') or A22 (if UPLO = 'L').
+*> \endverbatim
+*
+*  Arguments:
+*  ==========
+*
+*> \param[in] UPLO
+*> \verbatim
+*>          UPLO is CHARACTER*1
+*>          Specifies whether the upper or lower triangular part of the
+*>          symmetric matrix A is stored:
+*>          = 'U':  Upper triangular
+*>          = 'L':  Lower triangular
+*> \endverbatim
+*>
+*> \param[in] N
+*> \verbatim
+*>          N is INTEGER
+*>          The order of the matrix A.  N >= 0.
+*> \endverbatim
+*>
+*> \param[in] NB
+*> \verbatim
+*>          NB is INTEGER
+*>          The maximum number of columns of the matrix A that should be
+*>          factored.  NB should be at least 2 to allow for 2-by-2 pivot
+*>          blocks.
+*> \endverbatim
+*>
+*> \param[out] KB
+*> \verbatim
+*>          KB is INTEGER
+*>          The number of columns of A that were actually factored.
+*>          KB is either NB-1 or NB, or N if N <= NB.
+*> \endverbatim
+*>
+*> \param[in,out] A
+*> \verbatim
+*>          A is DOUBLE PRECISION array, dimension (LDA,N)
+*>          On entry, the symmetric matrix A.
+*>            If UPLO = 'U': the leading N-by-N upper triangular part
+*>            of A contains the upper triangular part of the matrix A,
+*>            and the strictly lower triangular part of A is not
+*>            referenced.
+*>
+*>            If UPLO = 'L': the leading N-by-N lower triangular part
+*>            of A contains the lower triangular part of the matrix A,
+*>            and the strictly upper triangular part of A is not
+*>            referenced.
+*>
+*>          On exit, contains:
+*>            a) ONLY diagonal elements of the symmetric block diagonal
+*>               matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
+*>               (superdiagonal (or subdiagonal) elements of D
+*>                are stored on exit in array E), and
+*>            b) If UPLO = 'U': factor U in the superdiagonal part of A.
+*>               If UPLO = 'L': factor L in the subdiagonal part of A.
+*> \endverbatim
+*>
+*> \param[in] LDA
+*> \verbatim
+*>          LDA is INTEGER
+*>          The leading dimension of the array A.  LDA >= max(1,N).
+*> \endverbatim
+*>
+*> \param[out] E
+*> \verbatim
+*>          E is DOUBLE PRECISION array, dimension (N)
+*>          On exit, contains the superdiagonal (or subdiagonal)
+*>          elements of the symmetric block diagonal matrix D
+*>          with 1-by-1 or 2-by-2 diagonal blocks, where
+*>          If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
+*>          If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
+*>
+*>          NOTE: For 1-by-1 diagonal block D(k), where
+*>          1 <= k <= N, the element E(k) is set to 0 in both
+*>          UPLO = 'U' or UPLO = 'L' cases.
+*> \endverbatim
+*>
+*> \param[out] IPIV
+*> \verbatim
+*>          IPIV is INTEGER array, dimension (N)
+*>          IPIV describes the permutation matrix P in the factorization
+*>          of matrix A as follows. The absolute value of IPIV(k)
+*>          represents the index of row and column that were
+*>          interchanged with the k-th row and column. The value of UPLO
+*>          describes the order in which the interchanges were applied.
+*>          Also, the sign of IPIV represents the block structure of
+*>          the symmetric block diagonal matrix D with 1-by-1 or 2-by-2
+*>          diagonal blocks which correspond to 1 or 2 interchanges
+*>          at each factorization step.
+*>
+*>          If UPLO = 'U',
+*>          ( in factorization order, k decreases from N to 1 ):
+*>            a) A single positive entry IPIV(k) > 0 means:
+*>               D(k,k) is a 1-by-1 diagonal block.
+*>               If IPIV(k) != k, rows and columns k and IPIV(k) were
+*>               interchanged in the submatrix A(1:N,N-KB+1:N);
+*>               If IPIV(k) = k, no interchange occurred.
+*>
+*>
+*>            b) A pair of consecutive negative entries
+*>               IPIV(k) < 0 and IPIV(k-1) < 0 means:
+*>               D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
+*>               (NOTE: negative entries in IPIV appear ONLY in pairs).
+*>               1) If -IPIV(k) != k, rows and columns
+*>                  k and -IPIV(k) were interchanged
+*>                  in the matrix A(1:N,N-KB+1:N).
+*>                  If -IPIV(k) = k, no interchange occurred.
+*>               2) If -IPIV(k-1) != k-1, rows and columns
+*>                  k-1 and -IPIV(k-1) were interchanged
+*>                  in the submatrix A(1:N,N-KB+1:N).
+*>                  If -IPIV(k-1) = k-1, no interchange occurred.
+*>
+*>            c) In both cases a) and b) is always ABS( IPIV(k) ) <= k.
+*>
+*>            d) NOTE: Any entry IPIV(k) is always NONZERO on output.
+*>
+*>          If UPLO = 'L',
+*>          ( in factorization order, k increases from 1 to N ):
+*>            a) A single positive entry IPIV(k) > 0 means:
+*>               D(k,k) is a 1-by-1 diagonal block.
+*>               If IPIV(k) != k, rows and columns k and IPIV(k) were
+*>               interchanged in the submatrix A(1:N,1:KB).
+*>               If IPIV(k) = k, no interchange occurred.
+*>
+*>            b) A pair of consecutive negative entries
+*>               IPIV(k) < 0 and IPIV(k+1) < 0 means:
+*>               D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
+*>               (NOTE: negative entries in IPIV appear ONLY in pairs).
+*>               1) If -IPIV(k) != k, rows and columns
+*>                  k and -IPIV(k) were interchanged
+*>                  in the submatrix A(1:N,1:KB).
+*>                  If -IPIV(k) = k, no interchange occurred.
+*>               2) If -IPIV(k+1) != k+1, rows and columns
+*>                  k-1 and -IPIV(k-1) were interchanged
+*>                  in the submatrix A(1:N,1:KB).
+*>                  If -IPIV(k+1) = k+1, no interchange occurred.
+*>
+*>            c) In both cases a) and b) is always ABS( IPIV(k) ) >= k.
+*>
+*>            d) NOTE: Any entry IPIV(k) is always NONZERO on output.
+*> \endverbatim
+*>
+*> \param[out] W
+*> \verbatim
+*>          W is DOUBLE PRECISION array, dimension (LDW,NB)
+*> \endverbatim
+*>
+*> \param[in] LDW
+*> \verbatim
+*>          LDW is INTEGER
+*>          The leading dimension of the array W.  LDW >= max(1,N).
+*> \endverbatim
+*>
+*> \param[out] INFO
+*> \verbatim
+*>          INFO is INTEGER
+*>          = 0: successful exit
+*>
+*>          < 0: If INFO = -k, the k-th argument had an illegal value
+*>
+*>          > 0: If INFO = k, the matrix A is singular, because:
+*>                 If UPLO = 'U': column k in the upper
+*>                 triangular part of A contains all zeros.
+*>                 If UPLO = 'L': column k in the lower
+*>                 triangular part of A contains all zeros.
+*>
+*>               Therefore D(k,k) is exactly zero, and superdiagonal
+*>               elements of column k of U (or subdiagonal elements of
+*>               column k of L ) are all zeros. The factorization has
+*>               been completed, but the block diagonal matrix D is
+*>               exactly singular, and division by zero will occur if
+*>               it is used to solve a system of equations.
+*>
+*>               NOTE: INFO only stores the first occurrence of
+*>               a singularity, any subsequent occurrence of singularity
+*>               is not stored in INFO even though the factorization
+*>               always completes.
+*> \endverbatim
+*
+*  Authors:
+*  ========
+*
+*> \author Univ. of Tennessee
+*> \author Univ. of California Berkeley
+*> \author Univ. of Colorado Denver
+*> \author NAG Ltd.
+*
+*> \date November 2016
+*
+*> \ingroup doubleSYcomputational
+*
+*> \par Contributors:
+*  ==================
+*>
+*> \verbatim
+*>
+*>  November 2016,  Igor Kozachenko,
+*>                  Computer Science Division,
+*>                  University of California, Berkeley
+*>
+*>  September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
+*>                  School of Mathematics,
+*>                  University of Manchester
+*>
+*> \endverbatim
+*
+*  =====================================================================
+      SUBROUTINE DLASYF_RK( UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW,
+     $                      INFO )
+*
+*  -- LAPACK computational routine (version 3.7.0) --
+*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
+*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
+*     November 2016
+*
+*     .. Scalar Arguments ..
+      CHARACTER          UPLO
+      INTEGER            INFO, KB, LDA, LDW, N, NB
+*     ..
+*     .. Array Arguments ..
+      INTEGER            IPIV( * )
+      DOUBLE PRECISION   A( LDA, * ), E( * ), W( LDW, * )
+*     ..
+*
+*  =====================================================================
+*
+*     .. Parameters ..
+      DOUBLE PRECISION   ZERO, ONE
+      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
+      DOUBLE PRECISION   EIGHT, SEVTEN
+      PARAMETER          ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
+*     ..
+*     .. Local Scalars ..
+      LOGICAL            DONE
+      INTEGER            IMAX, ITEMP, J, JB, JJ, JMAX, K, KK, KW, KKW,
+     $                   KP, KSTEP, P, II
+      DOUBLE PRECISION   ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22,
+     $                   DTEMP, R1, ROWMAX, T, SFMIN
+*     ..
+*     .. External Functions ..
+      LOGICAL            LSAME
+      INTEGER            IDAMAX
+      DOUBLE PRECISION   DLAMCH
+      EXTERNAL           LSAME, IDAMAX, DLAMCH
+*     ..
+*     .. External Subroutines ..
+      EXTERNAL           DCOPY, DGEMM, DGEMV, DSCAL, DSWAP
+*     ..
+*     .. Intrinsic Functions ..
+      INTRINSIC          ABS, MAX, MIN, SQRT
+*     ..
+*     .. Executable Statements ..
+*
+      INFO = 0
+*
+*     Initialize ALPHA for use in choosing pivot block size.
+*
+      ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
+*
+*     Compute machine safe minimum
+*
+      SFMIN = DLAMCH( 'S' )
+*
+      IF( LSAME( UPLO, 'U' ) ) THEN
+*
+*        Factorize the trailing columns of A using the upper triangle
+*        of A and working backwards, and compute the matrix W = U12*D
+*        for use in updating A11
+*
+*        Initilize the first entry of array E, where superdiagonal
+*        elements of D are stored
+*
+         E( 1 ) = ZERO
+*
+*        K is the main loop index, decreasing from N in steps of 1 or 2
+*
+         K = N
+   10    CONTINUE
+*
+*        KW is the column of W which corresponds to column K of A
+*
+         KW = NB + K - N
+*
+*        Exit from loop
+*
+         IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 )
+     $      GO TO 30
+*
+         KSTEP = 1
+         P = K
+*
+*        Copy column K of A to column KW of W and update it
+*
+         CALL DCOPY( K, A( 1, K ), 1, W( 1, KW ), 1 )
+         IF( K.LT.N )
+     $      CALL DGEMV( 'No transpose', K, N-K, -ONE, A( 1, K+1 ),
+     $                  LDA, W( K, KW+1 ), LDW, ONE, W( 1, KW ), 1 )
+*
+*        Determine rows and columns to be interchanged and whether
+*        a 1-by-1 or 2-by-2 pivot block will be used
+*
+         ABSAKK = ABS( W( K, KW ) )
+*
+*        IMAX is the row-index of the largest off-diagonal element in
+*        column K, and COLMAX is its absolute value.
+*        Determine both COLMAX and IMAX.
+*
+         IF( K.GT.1 ) THEN
+            IMAX = IDAMAX( K-1, W( 1, KW ), 1 )
+            COLMAX = ABS( W( IMAX, KW ) )
+         ELSE
+            COLMAX = ZERO
+         END IF
+*
+         IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
+*
+*           Column K is zero or underflow: set INFO and continue
+*
+            IF( INFO.EQ.0 )
+     $         INFO = K
+            KP = K
+            CALL DCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
+*
+*           Set E( K ) to zero
+*
+            IF( K.GT.1 )
+     $         E( K ) = ZERO
+*
+         ELSE
+*
+*           ============================================================
+*
+*           Test for interchange
+*
+*           Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
+*           (used to handle NaN and Inf)
+*
+            IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
+*
+*              no interchange, use 1-by-1 pivot block
+*
+               KP = K
+*
+            ELSE
+*
+               DONE = .FALSE.
+*
+*              Loop until pivot found
+*
+   12          CONTINUE
+*
+*                 Begin pivot search loop body
+*
+*
+*                 Copy column IMAX to column KW-1 of W and update it
+*
+                  CALL DCOPY( IMAX, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 )
+                  CALL DCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA,
+     $                        W( IMAX+1, KW-1 ), 1 )
+*
+                  IF( K.LT.N )
+     $               CALL DGEMV( 'No transpose', K, N-K, -ONE,
+     $                           A( 1, K+1 ), LDA, W( IMAX, KW+1 ), LDW,
+     $                           ONE, W( 1, KW-1 ), 1 )
+*
+*                 JMAX is the column-index of the largest off-diagonal
+*                 element in row IMAX, and ROWMAX is its absolute value.
+*                 Determine both ROWMAX and JMAX.
+*
+                  IF( IMAX.NE.K ) THEN
+                     JMAX = IMAX + IDAMAX( K-IMAX, W( IMAX+1, KW-1 ),
+     $                                     1 )
+                     ROWMAX = ABS( W( JMAX, KW-1 ) )
+                  ELSE
+                     ROWMAX = ZERO
+                  END IF
+*
+                  IF( IMAX.GT.1 ) THEN
+                     ITEMP = IDAMAX( IMAX-1, W( 1, KW-1 ), 1 )
+                     DTEMP = ABS( W( ITEMP, KW-1 ) )
+                     IF( DTEMP.GT.ROWMAX ) THEN
+                        ROWMAX = DTEMP
+                        JMAX = ITEMP
+                     END IF
+                  END IF
+*
+*                 Equivalent to testing for
+*                 ABS( W( IMAX, KW-1 ) ).GE.ALPHA*ROWMAX
+*                 (used to handle NaN and Inf)
+*
+                  IF( .NOT.(ABS( W( IMAX, KW-1 ) ).LT.ALPHA*ROWMAX ) )
+     $            THEN
+*
+*                    interchange rows and columns K and IMAX,
+*                    use 1-by-1 pivot block
+*
+                     KP = IMAX
+*
+*                    copy column KW-1 of W to column KW of W
+*
+                     CALL DCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
+*
+                     DONE = .TRUE.
+*
+*                 Equivalent to testing for ROWMAX.EQ.COLMAX,
+*                 (used to handle NaN and Inf)
+*
+                  ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
+     $            THEN
+*
+*                    interchange rows and columns K-1 and IMAX,
+*                    use 2-by-2 pivot block
+*
+                     KP = IMAX
+                     KSTEP = 2
+                     DONE = .TRUE.
+                  ELSE
+*
+*                    Pivot not found: set params and repeat
+*
+                     P = IMAX
+                     COLMAX = ROWMAX
+                     IMAX = JMAX
+*
+*                    Copy updated JMAXth (next IMAXth) column to Kth of W
+*
+                     CALL DCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
+*
+                  END IF
+*
+*                 End pivot search loop body
+*
+               IF( .NOT. DONE ) GOTO 12
+*
+            END IF
+*
+*           ============================================================
+*
+            KK = K - KSTEP + 1
+*
+*           KKW is the column of W which corresponds to column KK of A
+*
+            KKW = NB + KK - N
+*
+            IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
+*
+*              Copy non-updated column K to column P
+*
+               CALL DCOPY( K-P, A( P+1, K ), 1, A( P, P+1 ), LDA )
+               CALL DCOPY( P, A( 1, K ), 1, A( 1, P ), 1 )
+*
+*              Interchange rows K and P in last N-K+1 columns of A
+*              and last N-K+2 columns of W
+*
+               CALL DSWAP( N-K+1, A( K, K ), LDA, A( P, K ), LDA )
+               CALL DSWAP( N-KK+1, W( K, KKW ), LDW, W( P, KKW ), LDW )
+            END IF
+*
+*           Updated column KP is already stored in column KKW of W
+*
+            IF( KP.NE.KK ) THEN
+*
+*              Copy non-updated column KK to column KP
+*
+               A( KP, K ) = A( KK, K )
+               CALL DCOPY( K-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ),
+     $                     LDA )
+               CALL DCOPY( KP, A( 1, KK ), 1, A( 1, KP ), 1 )
+*
+*              Interchange rows KK and KP in last N-KK+1 columns
+*              of A and W
+*
+               CALL DSWAP( N-KK+1, A( KK, KK ), LDA, A( KP, KK ), LDA )
+               CALL DSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ),
+     $                     LDW )
+            END IF
+*
+            IF( KSTEP.EQ.1 ) THEN
+*
+*              1-by-1 pivot block D(k): column KW of W now holds
+*
+*              W(k) = U(k)*D(k)
+*
+*              where U(k) is the k-th column of U
+*
+*              Store U(k) in column k of A
+*
+               CALL DCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
+               IF( K.GT.1 ) THEN
+                  IF( ABS( A( K, K ) ).GE.SFMIN ) THEN
+                     R1 = ONE / A( K, K )
+                     CALL DSCAL( K-1, R1, A( 1, K ), 1 )
+                  ELSE IF( A( K, K ).NE.ZERO ) THEN
+                     DO 14 II = 1, K - 1
+                        A( II, K ) = A( II, K ) / A( K, K )
+   14                CONTINUE
+                  END IF
+*
+*                 Store the superdiagonal element of D in array E
+*
+                  E( K ) = ZERO
+*
+               END IF
+*
+            ELSE
+*
+*              2-by-2 pivot block D(k): columns KW and KW-1 of W now
+*              hold
+*
+*              ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
+*
+*              where U(k) and U(k-1) are the k-th and (k-1)-th columns
+*              of U
+*
+               IF( K.GT.2 ) THEN
+*
+*                 Store U(k) and U(k-1) in columns k and k-1 of A
+*
+                  D12 = W( K-1, KW )
+                  D11 = W( K, KW ) / D12
+                  D22 = W( K-1, KW-1 ) / D12
+                  T = ONE / ( D11*D22-ONE )
+                  DO 20 J = 1, K - 2
+                     A( J, K-1 ) = T*( (D11*W( J, KW-1 )-W( J, KW ) ) /
+     $                             D12 )
+                     A( J, K ) = T*( ( D22*W( J, KW )-W( J, KW-1 ) ) /
+     $                           D12 )
+   20             CONTINUE
+               END IF
+*
+*              Copy diagonal elements of D(K) to A,
+*              copy superdiagonal element of D(K) to E(K) and
+*              ZERO out superdiagonal entry of A
+*
+               A( K-1, K-1 ) = W( K-1, KW-1 )
+               A( K-1, K ) = ZERO
+               A( K, K ) = W( K, KW )
+               E( K ) = W( K-1, KW )
+               E( K-1 ) = ZERO
+*
+            END IF
+*
+*           End column K is nonsingular
+*
+         END IF
+*
+*        Store details of the interchanges in IPIV
+*
+         IF( KSTEP.EQ.1 ) THEN
+            IPIV( K ) = KP
+         ELSE
+            IPIV( K ) = -P
+            IPIV( K-1 ) = -KP
+         END IF
+*
+*        Decrease K and return to the start of the main loop
+*
+         K = K - KSTEP
+         GO TO 10
+*
+   30    CONTINUE
+*
+*        Update the upper triangle of A11 (= A(1:k,1:k)) as
+*
+*        A11 := A11 - U12*D*U12**T = A11 - U12*W**T
+*
+*        computing blocks of NB columns at a time
+*
+         DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB
+            JB = MIN( NB, K-J+1 )
+*
+*           Update the upper triangle of the diagonal block
+*
+            DO 40 JJ = J, J + JB - 1
+               CALL DGEMV( 'No transpose', JJ-J+1, N-K, -ONE,
+     $                     A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, ONE,
+     $                     A( J, JJ ), 1 )
+   40       CONTINUE
+*
+*           Update the rectangular superdiagonal block
+*
+            IF( J.GE.2 )
+     $         CALL DGEMM( 'No transpose', 'Transpose', J-1, JB,
+     $                  N-K, -ONE, A( 1, K+1 ), LDA, W( J, KW+1 ),
+     $                  LDW, ONE, A( 1, J ), LDA )
+   50    CONTINUE
+*
+*        Set KB to the number of columns factorized
+*
+         KB = N - K
+*
+      ELSE
+*
+*        Factorize the leading columns of A using the lower triangle
+*        of A and working forwards, and compute the matrix W = L21*D
+*        for use in updating A22
+*
+*        Initilize the unused last entry of the subdiagonal array E.
+*
+         E( N ) = ZERO
+*
+*        K is the main loop index, increasing from 1 in steps of 1 or 2
+*
+         K = 1
+   70   CONTINUE
+*
+*        Exit from loop
+*
+         IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N )
+     $      GO TO 90
+*
+         KSTEP = 1
+         P = K
+*
+*        Copy column K of A to column K of W and update it
+*
+         CALL DCOPY( N-K+1, A( K, K ), 1, W( K, K ), 1 )
+         IF( K.GT.1 )
+     $      CALL DGEMV( 'No transpose', N-K+1, K-1, -ONE, A( K, 1 ),
+     $                  LDA, W( K, 1 ), LDW, ONE, W( K, K ), 1 )
+*
+*        Determine rows and columns to be interchanged and whether
+*        a 1-by-1 or 2-by-2 pivot block will be used
+*
+         ABSAKK = ABS( W( K, K ) )
+*
+*        IMAX is the row-index of the largest off-diagonal element in
+*        column K, and COLMAX is its absolute value.
+*        Determine both COLMAX and IMAX.
+*
+         IF( K.LT.N ) THEN
+            IMAX = K + IDAMAX( N-K, W( K+1, K ), 1 )
+            COLMAX = ABS( W( IMAX, K ) )
+         ELSE
+            COLMAX = ZERO
+         END IF
+*
+         IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
+*
+*           Column K is zero or underflow: set INFO and continue
+*
+            IF( INFO.EQ.0 )
+     $         INFO = K
+            KP = K
+            CALL DCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
+*
+*           Set E( K ) to zero
+*
+            IF( K.LT.N )
+     $         E( K ) = ZERO
+*
+         ELSE
+*
+*           ============================================================
+*
+*           Test for interchange
+*
+*           Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
+*           (used to handle NaN and Inf)
+*
+            IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
+*
+*              no interchange, use 1-by-1 pivot block
+*
+               KP = K
+*
+            ELSE
+*
+               DONE = .FALSE.
+*
+*              Loop until pivot found
+*
+   72          CONTINUE
+*
+*                 Begin pivot search loop body
+*
+*
+*                 Copy column IMAX to column K+1 of W and update it
+*
+                  CALL DCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1)
+                  CALL DCOPY( N-IMAX+1, A( IMAX, IMAX ), 1,
+     $                        W( IMAX, K+1 ), 1 )
+                  IF( K.GT.1 )
+     $               CALL DGEMV( 'No transpose', N-K+1, K-1, -ONE,
+     $                           A( K, 1 ), LDA, W( IMAX, 1 ), LDW,
+     $                           ONE, W( K, K+1 ), 1 )
+*
+*                 JMAX is the column-index of the largest off-diagonal
+*                 element in row IMAX, and ROWMAX is its absolute value.
+*                 Determine both ROWMAX and JMAX.
+*
+                  IF( IMAX.NE.K ) THEN
+                     JMAX = K - 1 + IDAMAX( IMAX-K, W( K, K+1 ), 1 )
+                     ROWMAX = ABS( W( JMAX, K+1 ) )
+                  ELSE
+                     ROWMAX = ZERO
+                  END IF
+*
+                  IF( IMAX.LT.N ) THEN
+                     ITEMP = IMAX + IDAMAX( N-IMAX, W( IMAX+1, K+1 ), 1)
+                     DTEMP = ABS( W( ITEMP, K+1 ) )
+                     IF( DTEMP.GT.ROWMAX ) THEN
+                        ROWMAX = DTEMP
+                        JMAX = ITEMP
+                     END IF
+                  END IF
+*
+*                 Equivalent to testing for
+*                 ABS( W( IMAX, K+1 ) ).GE.ALPHA*ROWMAX
+*                 (used to handle NaN and Inf)
+*
+                  IF( .NOT.( ABS( W( IMAX, K+1 ) ).LT.ALPHA*ROWMAX ) )
+     $            THEN
+*
+*                    interchange rows and columns K and IMAX,
+*                    use 1-by-1 pivot block
+*
+                     KP = IMAX
+*
+*                    copy column K+1 of W to column K of W
+*
+                     CALL DCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
+*
+                     DONE = .TRUE.
+*
+*                 Equivalent to testing for ROWMAX.EQ.COLMAX,
+*                 (used to handle NaN and Inf)
+*
+                  ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
+     $            THEN
+*
+*                    interchange rows and columns K+1 and IMAX,
+*                    use 2-by-2 pivot block
+*
+                     KP = IMAX
+                     KSTEP = 2
+                     DONE = .TRUE.
+                  ELSE
+*
+*                    Pivot not found: set params and repeat
+*
+                     P = IMAX
+                     COLMAX = ROWMAX
+                     IMAX = JMAX
+*
+*                    Copy updated JMAXth (next IMAXth) column to Kth of W
+*
+                     CALL DCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
+*
+                  END IF
+*
+*                 End pivot search loop body
+*
+               IF( .NOT. DONE ) GOTO 72
+*
+            END IF
+*
+*           ============================================================
+*
+            KK = K + KSTEP - 1
+*
+            IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
+*
+*              Copy non-updated column K to column P
+*
+               CALL DCOPY( P-K, A( K, K ), 1, A( P, K ), LDA )
+               CALL DCOPY( N-P+1, A( P, K ), 1, A( P, P ), 1 )
+*
+*              Interchange rows K and P in first K columns of A
+*              and first K+1 columns of W
+*
+               CALL DSWAP( K, A( K, 1 ), LDA, A( P, 1 ), LDA )
+               CALL DSWAP( KK, W( K, 1 ), LDW, W( P, 1 ), LDW )
+            END IF
+*
+*           Updated column KP is already stored in column KK of W
+*
+            IF( KP.NE.KK ) THEN
+*
+*              Copy non-updated column KK to column KP
+*
+               A( KP, K ) = A( KK, K )
+               CALL DCOPY( KP-K-1, A( K+1, KK ), 1, A( KP, K+1 ), LDA )
+               CALL DCOPY( N-KP+1, A( KP, KK ), 1, A( KP, KP ), 1 )
+*
+*              Interchange rows KK and KP in first KK columns of A and W
+*
+               CALL DSWAP( KK, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
+               CALL DSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW )
+            END IF
+*
+            IF( KSTEP.EQ.1 ) THEN
+*
+*              1-by-1 pivot block D(k): column k of W now holds
+*
+*              W(k) = L(k)*D(k)
+*
+*              where L(k) is the k-th column of L
+*
+*              Store L(k) in column k of A
+*
+               CALL DCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
+               IF( K.LT.N ) THEN
+                  IF( ABS( A( K, K ) ).GE.SFMIN ) THEN
+                     R1 = ONE / A( K, K )
+                     CALL DSCAL( N-K, R1, A( K+1, K ), 1 )
+                  ELSE IF( A( K, K ).NE.ZERO ) THEN
+                     DO 74 II = K + 1, N
+                        A( II, K ) = A( II, K ) / A( K, K )
+   74                CONTINUE
+                  END IF
+*
+*                 Store the subdiagonal element of D in array E
+*
+                  E( K ) = ZERO
+*
+               END IF
+*
+            ELSE
+*
+*              2-by-2 pivot block D(k): columns k and k+1 of W now hold
+*
+*              ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
+*
+*              where L(k) and L(k+1) are the k-th and (k+1)-th columns
+*              of L
+*
+               IF( K.LT.N-1 ) THEN
+*
+*                 Store L(k) and L(k+1) in columns k and k+1 of A
+*
+                  D21 = W( K+1, K )
+                  D11 = W( K+1, K+1 ) / D21
+                  D22 = W( K, K ) / D21
+                  T = ONE / ( D11*D22-ONE )
+                  DO 80 J = K + 2, N
+                     A( J, K ) = T*( ( D11*W( J, K )-W( J, K+1 ) ) /
+     $                           D21 )
+                     A( J, K+1 ) = T*( ( D22*W( J, K+1 )-W( J, K ) ) /
+     $                             D21 )
+   80             CONTINUE
+               END IF
+*
+*              Copy diagonal elements of D(K) to A,
+*              copy subdiagonal element of D(K) to E(K) and
+*              ZERO out subdiagonal entry of A
+*
+               A( K, K ) = W( K, K )
+               A( K+1, K ) = ZERO
+               A( K+1, K+1 ) = W( K+1, K+1 )
+               E( K ) = W( K+1, K )
+               E( K+1 ) = ZERO
+*
+            END IF
+*
+*           End column K is nonsingular
+*
+         END IF
+*
+*        Store details of the interchanges in IPIV
+*
+         IF( KSTEP.EQ.1 ) THEN
+            IPIV( K ) = KP
+         ELSE
+            IPIV( K ) = -P
+            IPIV( K+1 ) = -KP
+         END IF
+*
+*        Increase K and return to the start of the main loop
+*
+         K = K + KSTEP
+         GO TO 70
+*
+   90    CONTINUE
+*
+*        Update the lower triangle of A22 (= A(k:n,k:n)) as
+*
+*        A22 := A22 - L21*D*L21**T = A22 - L21*W**T
+*
+*        computing blocks of NB columns at a time
+*
+         DO 110 J = K, N, NB
+            JB = MIN( NB, N-J+1 )
+*
+*           Update the lower triangle of the diagonal block
+*
+            DO 100 JJ = J, J + JB - 1
+               CALL DGEMV( 'No transpose', J+JB-JJ, K-1, -ONE,
+     $                     A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, ONE,
+     $                     A( JJ, JJ ), 1 )
+  100       CONTINUE
+*
+*           Update the rectangular subdiagonal block
+*
+            IF( J+JB.LE.N )
+     $         CALL DGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB,
+     $                     K-1, -ONE, A( J+JB, 1 ), LDA, W( J, 1 ),
+     $                     LDW, ONE, A( J+JB, J ), LDA )
+  110    CONTINUE
+*
+*        Set KB to the number of columns factorized
+*
+         KB = K - 1
+*
+      END IF
+*
+      RETURN
+*
+*     End of DLASYF_RK
+*
+      END