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, 316 insertions, 0 deletions

diff --git a/SRC/csysv_rk.f b/SRC/csysv_rk.f
new file mode 100644
index 00000000..5cfd358b
--- /dev/null
+++ b/SRC/csysv_rk.f
@@ -0,0 +1,316 @@
+*> \brief <b> CSYSV_RK computes the solution to system of linear equations A * X = B for SY matrices</b>
+*
+*  =========== DOCUMENTATION ===========
+*
+* Online html documentation available at
+*            http://www.netlib.org/lapack/explore-html/
+*
+*> \htmlonly
+*> Download CSYSV_RK + dependencies
+*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/csysv_rk.f">
+*> [TGZ]</a>
+*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/csysv_rk.f">
+*> [ZIP]</a>
+*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/csysv_rk.f">
+*> [TXT]</a>
+*> \endhtmlonly
+*
+*  Definition:
+*  ===========
+*
+*       SUBROUTINE CSYSV_RK( UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB,
+*                            WORK, LWORK, INFO )
+*
+*       .. Scalar Arguments ..
+*       CHARACTER          UPLO
+*       INTEGER            INFO, LDA, LDB, LWORK, N, NRHS
+*       ..
+*       .. Array Arguments ..
+*       INTEGER            IPIV( * )
+*       COMPLEX            A( LDA, * ), B( LDB, * ), E( * ), WORK( * )
+*       ..
+*
+*
+*> \par Purpose:
+*  =============
+*>
+*> \verbatim
+*> CSYSV_RK computes the solution to a complex system of linear
+*> equations A * X = B, where A is an N-by-N symmetric matrix
+*> and X and B are N-by-NRHS matrices.
+*>
+*> The bounded Bunch-Kaufman (rook) diagonal pivoting method is used
+*> to factor A as
+*>    A = P*U*D*(U**T)*(P**T),  if UPLO = 'U', or
+*>    A = P*L*D*(L**T)*(P**T),  if UPLO = 'L',
+*> where U (or L) is unit upper (or lower) triangular matrix,
+*> U**T (or L**T) is the transpose of U (or L), P is a permutation
+*> matrix, P**T is the transpose of P, and D is symmetric and block
+*> diagonal with 1-by-1 and 2-by-2 diagonal blocks.
+*>
+*> CSYTRF_RK is called to compute the factorization of a complex
+*> symmetric matrix.  The factored form of A is then used to solve
+*> the system of equations A * X = B by calling BLAS3 routine CSYTRS_3.
+*> \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 triangle of A is stored;
+*>          = 'L':  Lower triangle of A is stored.
+*> \endverbatim
+*>
+*> \param[in] N
+*> \verbatim
+*>          N is INTEGER
+*>          The number of linear equations, i.e., the order of the
+*>          matrix A.  N >= 0.
+*> \endverbatim
+*>
+*> \param[in] NRHS
+*> \verbatim
+*>          NRHS is INTEGER
+*>          The number of right hand sides, i.e., the number of columns
+*>          of the matrix B.  NRHS >= 0.
+*> \endverbatim
+*>
+*> \param[in,out] A
+*> \verbatim
+*>          A is COMPLEX 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, if INFO = 0, diagonal of the block diagonal
+*>          matrix D and factors U or L  as computed by CSYTRF_RK:
+*>            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.
+*>
+*>          For more info see the description of CSYTRF_RK routine.
+*> \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 COMPLEX array, dimension (N)
+*>          On exit, contains the output computed by the factorization
+*>          routine CSYTRF_RK, i.e. 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.
+*>
+*>          For more info see the description of CSYTRF_RK routine.
+*> \endverbatim
+*>
+*> \param[out] IPIV
+*> \verbatim
+*>          IPIV is INTEGER array, dimension (N)
+*>          Details of the interchanges and the block structure of D,
+*>          as determined by CSYTRF_RK.
+*>
+*>          For more info see the description of CSYTRF_RK routine.
+*> \endverbatim
+*>
+*> \param[in,out] B
+*> \verbatim
+*>          B is COMPLEX array, dimension (LDB,NRHS)
+*>          On entry, the N-by-NRHS right hand side matrix B.
+*>          On exit, if INFO = 0, the N-by-NRHS solution matrix X.
+*> \endverbatim
+*>
+*> \param[in] LDB
+*> \verbatim
+*>          LDB is INTEGER
+*>          The leading dimension of the array B.  LDB >= max(1,N).
+*> \endverbatim
+*>
+*> \param[out] WORK
+*> \verbatim
+*>          WORK is COMPLEX array, dimension ( MAX(1,LWORK) ).
+*>          Work array used in the factorization stage.
+*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
+*> \endverbatim
+*>
+*> \param[in] LWORK
+*> \verbatim
+*>          LWORK is INTEGER
+*>          The length of WORK.  LWORK >= 1. For best performance
+*>          of factorization stage LWORK >= max(1,N*NB), where NB is
+*>          the optimal blocksize for CSYTRF_RK.
+*>
+*>          If LWORK = -1, then a workspace query is assumed;
+*>          the routine only calculates the optimal size of the WORK
+*>          array for factorization stage, returns this value as
+*>          the first entry of the WORK array, and no error message
+*>          related to LWORK is issued by XERBLA.
+*> \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 complexSYsolve
+*
+*> \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 CSYSV_RK( UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB, WORK,
+     $                     LWORK, INFO )
+*
+*  -- LAPACK driver 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, LDA, LDB, LWORK, N, NRHS
+*     ..
+*     .. Array Arguments ..
+      INTEGER            IPIV( * )
+      COMPLEX            A( LDA, * ), B( LDB, * ), E( * ), WORK( * )
+*     ..
+*
+*  =====================================================================
+*
+*     .. Local Scalars ..
+      LOGICAL            LQUERY
+      INTEGER            LWKOPT
+*     ..
+*     .. External Functions ..
+      LOGICAL            LSAME
+      EXTERNAL           LSAME
+*     ..
+*     .. External Subroutines ..
+      EXTERNAL           XERBLA, CSYTRF_RK, CSYTRS_3
+*     ..
+*     .. Intrinsic Functions ..
+      INTRINSIC          MAX
+*     ..
+*     .. Executable Statements ..
+*
+*     Test the input parameters.
+*
+      INFO = 0
+      LQUERY = ( LWORK.EQ.-1 )
+      IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
+         INFO = -1
+      ELSE IF( N.LT.0 ) THEN
+         INFO = -2
+      ELSE IF( NRHS.LT.0 ) THEN
+         INFO = -3
+      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
+         INFO = -5
+      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
+         INFO = -9
+      ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
+         INFO = -11
+      END IF
+*
+      IF( INFO.EQ.0 ) THEN
+         IF( N.EQ.0 ) THEN
+            LWKOPT = 1
+         ELSE
+            CALL CSYTRF_RK( UPLO, N, A, LDA, E, IPIV, WORK, -1, INFO )
+            LWKOPT = WORK(1)
+         END IF
+         WORK( 1 ) = LWKOPT
+      END IF
+*
+      IF( INFO.NE.0 ) THEN
+         CALL XERBLA( 'CSYSV_RK ', -INFO )
+         RETURN
+      ELSE IF( LQUERY ) THEN
+         RETURN
+      END IF
+*
+*     Compute the factorization A = U*D*U**T or A = L*D*L**T.
+*
+      CALL CSYTRF_RK( UPLO, N, A, LDA, E, IPIV, WORK, LWORK, INFO )
+*
+      IF( INFO.EQ.0 ) THEN
+*
+*        Solve the system A*X = B with BLAS3 solver, overwriting B with X.
+*
+         CALL CSYTRS_3( UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB, INFO )
+*
+      END IF
+*
+      WORK( 1 ) = LWKOPT
+*
+      RETURN
+*
+*     End of CSYSV_RK
+*
+      END