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Update LAPACK to 3.9.0

This commit is contained in:
Martin Kroeker 2019-12-30 16:05:18 +01:00 committed by GitHub
parent d4859bb2cc
commit 49f80f0c16
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25 changed files with 902 additions and 129 deletions
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16
lapack-netlib/SRC/zporfsx.f
View File

@ -44,7 +44,7 @@
*> \verbatim
*>
*> ZPORFSX improves the computed solution to a system of linear
*> equations when the coefficient matrix is symmetric positive
*> equations when the coefficient matrix is Hermitian positive
*> definite, and provides error bounds and backward error estimates
*> for the solution. In addition to normwise error bound, the code
*> provides maximum componentwise error bound if possible. See
@ -103,7 +103,7 @@
*> \param[in] A
*> \verbatim
*> A is COMPLEX*16 array, dimension (LDA,N)
*> The symmetric matrix A. If UPLO = 'U', the leading N-by-N
*> The Hermitian 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
@ -134,7 +134,7 @@
*> \param[in,out] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (N)
*> The row scale factors for A. If EQUED = 'Y', A is multiplied on
*> The scale factors for A. If EQUED = 'Y', A is multiplied on
*> the left and right by diag(S). S is an input argument if FACT =
*> 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED
*> = 'Y', each element of S must be positive. If S is output, each
@ -262,7 +262,7 @@
*> information as described below. There currently are up to three
*> pieces of information returned for each right-hand side. If
*> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
*> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
*> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most
*> the first (:,N_ERR_BNDS) entries are returned.
*>
*> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
@ -298,14 +298,14 @@
*> \param[in] NPARAMS
*> \verbatim
*> NPARAMS is INTEGER
*> Specifies the number of parameters set in PARAMS. If .LE. 0, the
*> Specifies the number of parameters set in PARAMS. If <= 0, the
*> PARAMS array is never referenced and default values are used.
*> \endverbatim
*>
*> \param[in,out] PARAMS
*> \verbatim
*> PARAMS is DOUBLE PRECISION array, dimension NPARAMS
*> Specifies algorithm parameters. If an entry is .LT. 0.0, then
*> Specifies algorithm parameters. If an entry is < 0.0, then
*> that entry will be filled with default value used for that
*> parameter. Only positions up to NPARAMS are accessed; defaults
*> are used for higher-numbered parameters.
@ -313,9 +313,9 @@
*> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
*> refinement or not.
*> Default: 1.0D+0
*> = 0.0 : No refinement is performed, and no error bounds are
*> = 0.0: No refinement is performed, and no error bounds are
*> computed.
*> = 1.0 : Use the double-precision refinement algorithm,
*> = 1.0: Use the double-precision refinement algorithm,
*> possibly with doubled-single computations if the
*> compilation environment does not support DOUBLE
*> PRECISION.

14
lapack-netlib/SRC/zposvxx.f
View File

@ -45,7 +45,7 @@
*>
*> ZPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
*> to compute the solution to a complex*16 system of linear equations
*> A * X = B, where A is an N-by-N symmetric positive definite matrix
*> A * X = B, where A is an N-by-N Hermitian positive definite matrix
*> and X and B are N-by-NRHS matrices.
*>
*> If requested, both normwise and maximum componentwise error bounds
@ -157,7 +157,7 @@
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX*16 array, dimension (LDA,N)
*> On entry, the symmetric matrix A, except if FACT = 'F' and EQUED =
*> On entry, the Hermitian matrix A, except if FACT = 'F' and EQUED =
*> 'Y', then A must contain the equilibrated matrix
*> diag(S)*A*diag(S). If UPLO = 'U', the leading N-by-N upper
*> triangular part of A contains the upper triangular part of the
@ -365,7 +365,7 @@
*> information as described below. There currently are up to three
*> pieces of information returned for each right-hand side. If
*> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
*> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
*> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most
*> the first (:,N_ERR_BNDS) entries are returned.
*>
*> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
@ -401,14 +401,14 @@
*> \param[in] NPARAMS
*> \verbatim
*> NPARAMS is INTEGER
*> Specifies the number of parameters set in PARAMS. If .LE. 0, the
*> Specifies the number of parameters set in PARAMS. If <= 0, the
*> PARAMS array is never referenced and default values are used.
*> \endverbatim
*>
*> \param[in,out] PARAMS
*> \verbatim
*> PARAMS is DOUBLE PRECISION array, dimension NPARAMS
*> Specifies algorithm parameters. If an entry is .LT. 0.0, then
*> Specifies algorithm parameters. If an entry is < 0.0, then
*> that entry will be filled with default value used for that
*> parameter. Only positions up to NPARAMS are accessed; defaults
*> are used for higher-numbered parameters.
@ -416,9 +416,9 @@
*> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
*> refinement or not.
*> Default: 1.0D+0
*> = 0.0 : No refinement is performed, and no error bounds are
*> = 0.0: No refinement is performed, and no error bounds are
*> computed.
*> = 1.0 : Use the extra-precise refinement algorithm.
*> = 1.0: Use the extra-precise refinement algorithm.
*> (other values are reserved for future use)
*>
*> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual

4
lapack-netlib/SRC/zpotrf2.f
View File

@ -24,7 +24,7 @@
*>
*> \verbatim
*>
*> ZPOTRF2 computes the Cholesky factorization of a real symmetric
*> ZPOTRF2 computes the Cholesky factorization of a Hermitian
*> positive definite matrix A using the recursive algorithm.
*>
*> The factorization has the form
@ -63,7 +63,7 @@
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX*16 array, dimension (LDA,N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
*> On entry, the Hermitian 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

4
lapack-netlib/SRC/zstemr.f
View File

@ -250,13 +250,13 @@
*> \param[in,out] TRYRAC
*> \verbatim
*> TRYRAC is LOGICAL
*> If TRYRAC.EQ..TRUE., indicates that the code should check whether
*> If TRYRAC = .TRUE., indicates that the code should check whether
*> the tridiagonal matrix defines its eigenvalues to high relative
*> accuracy. If so, the code uses relative-accuracy preserving
*> algorithms that might be (a bit) slower depending on the matrix.
*> If the matrix does not define its eigenvalues to high relative
*> accuracy, the code can uses possibly faster algorithms.
*> If TRYRAC.EQ..FALSE., the code is not required to guarantee
*> If TRYRAC = .FALSE., the code is not required to guarantee
*> relatively accurate eigenvalues and can use the fastest possible
*> techniques.
*> On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix

9
lapack-netlib/SRC/zsycon_3.f
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@ -19,7 +19,7 @@
* ===========
*
* SUBROUTINE ZSYCON_3( UPLO, N, A, LDA, E, IPIV, ANORM, RCOND,
* WORK, IWORK, INFO )
* WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
@ -27,7 +27,7 @@
* DOUBLE PRECISION ANORM, RCOND
* ..
* .. Array Arguments ..
* INTEGER IPIV( * ), IWORK( * )
* INTEGER IPIV( * )
* COMPLEX*16 A( LDA, * ), E ( * ), WORK( * )
* ..
*
@ -129,11 +129,6 @@
*> WORK is COMPLEX*16 array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER

8
lapack-netlib/SRC/zsyconvf.f
View File

@ -294,7 +294,7 @@
*
* Convert PERMUTATIONS and IPIV
*
* Apply permutaions to submatrices of upper part of A
* Apply permutations to submatrices of upper part of A
* in factorization order where i decreases from N to 1
*
I = N
@ -347,7 +347,7 @@
*
* Revert PERMUTATIONS and IPIV
*
* Apply permutaions to submatrices of upper part of A
* Apply permutations to submatrices of upper part of A
* in reverse factorization order where i increases from 1 to N
*
I = 1
@ -438,7 +438,7 @@
*
* Convert PERMUTATIONS and IPIV
*
* Apply permutaions to submatrices of lower part of A
* Apply permutations to submatrices of lower part of A
* in factorization order where k increases from 1 to N
*
I = 1
@ -491,7 +491,7 @@
*
* Revert PERMUTATIONS and IPIV
*
* Apply permutaions to submatrices of lower part of A
* Apply permutations to submatrices of lower part of A
* in reverse factorization order where i decreases from N to 1
*
I = N

8
lapack-netlib/SRC/zsyconvf_rook.f
View File

@ -285,7 +285,7 @@
*
* Convert PERMUTATIONS
*
* Apply permutaions to submatrices of upper part of A
* Apply permutations to submatrices of upper part of A
* in factorization order where i decreases from N to 1
*
I = N
@ -336,7 +336,7 @@
*
* Revert PERMUTATIONS
*
* Apply permutaions to submatrices of upper part of A
* Apply permutations to submatrices of upper part of A
* in reverse factorization order where i increases from 1 to N
*
I = 1
@ -426,7 +426,7 @@
*
* Convert PERMUTATIONS
*
* Apply permutaions to submatrices of lower part of A
* Apply permutations to submatrices of lower part of A
* in factorization order where i increases from 1 to N
*
I = 1
@ -477,7 +477,7 @@
*
* Revert PERMUTATIONS
*
* Apply permutaions to submatrices of lower part of A
* Apply permutations to submatrices of lower part of A
* in reverse factorization order where i decreases from N to 1
*
I = N

10
lapack-netlib/SRC/zsyrfsx.f
View File

@ -271,7 +271,7 @@
*> information as described below. There currently are up to three
*> pieces of information returned for each right-hand side. If
*> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
*> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
*> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most
*> the first (:,N_ERR_BNDS) entries are returned.
*>
*> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
@ -307,14 +307,14 @@
*> \param[in] NPARAMS
*> \verbatim
*> NPARAMS is INTEGER
*> Specifies the number of parameters set in PARAMS. If .LE. 0, the
*> Specifies the number of parameters set in PARAMS. If <= 0, the
*> PARAMS array is never referenced and default values are used.
*> \endverbatim
*>
*> \param[in,out] PARAMS
*> \verbatim
*> PARAMS is DOUBLE PRECISION array, dimension NPARAMS
*> Specifies algorithm parameters. If an entry is .LT. 0.0, then
*> Specifies algorithm parameters. If an entry is < 0.0, then
*> that entry will be filled with default value used for that
*> parameter. Only positions up to NPARAMS are accessed; defaults
*> are used for higher-numbered parameters.
@ -322,9 +322,9 @@
*> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
*> refinement or not.
*> Default: 1.0D+0
*> = 0.0 : No refinement is performed, and no error bounds are
*> = 0.0: No refinement is performed, and no error bounds are
*> computed.
*> = 1.0 : Use the double-precision refinement algorithm,
*> = 1.0: Use the double-precision refinement algorithm,
*> possibly with doubled-single computations if the
*> compilation environment does not support DOUBLE
*> PRECISION.

6
lapack-netlib/SRC/zsysv_aa.f
View File

@ -42,7 +42,7 @@
*> matrices.
*>
*> Aasen's algorithm is used to factor A as
*> A = U * T * U**T, if UPLO = 'U', or
*> A = U**T * T * U, if UPLO = 'U', or
*> A = L * T * L**T, if UPLO = 'L',
*> where U (or L) is a product of permutation and unit upper (lower)
*> triangular matrices, and T is symmetric tridiagonal. The factored
@ -86,7 +86,7 @@
*>
*> On exit, if INFO = 0, the tridiagonal matrix T and the
*> multipliers used to obtain the factor U or L from the
*> factorization A = U*T*U**T or A = L*T*L**T as computed by
*> factorization A = U**T*T*U or A = L*T*L**T as computed by
*> ZSYTRF.
*> \endverbatim
*>
@ -230,7 +230,7 @@
RETURN
END IF
*
* Compute the factorization A = U*T*U**T or A = L*T*L**T.
* Compute the factorization A = U**T*T*U or A = L*T*L**T.
*
CALL ZSYTRF_AA( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
IF( INFO.EQ.0 ) THEN

6
lapack-netlib/SRC/zsysv_aa_2stage.f
View File

@ -43,8 +43,8 @@
*> matrices.
*>
*> Aasen's 2-stage algorithm is used to factor A as
*> A = U * T * U**H, if UPLO = 'U', or
*> A = L * T * L**H, if UPLO = 'L',
*> A = U**T * T * U, if UPLO = 'U', or
*> A = L * T * L**T, if UPLO = 'L',
*> where U (or L) is a product of permutation and unit upper (lower)
*> triangular matrices, and T is symmetric and band. The matrix T is
*> then LU-factored with partial pivoting. The factored form of A
@ -257,7 +257,7 @@
END IF
*
*
* Compute the factorization A = U*T*U**H or A = L*T*L**H.
* Compute the factorization A = U**T*T*U or A = L*T*L**T.
*
CALL ZSYTRF_AA_2STAGE( UPLO, N, A, LDA, TB, LTB, IPIV, IPIV2,
$ WORK, LWORK, INFO )

10
lapack-netlib/SRC/zsysvxx.f
View File

@ -378,7 +378,7 @@
*> information as described below. There currently are up to three
*> pieces of information returned for each right-hand side. If
*> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
*> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
*> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most
*> the first (:,N_ERR_BNDS) entries are returned.
*>
*> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
@ -414,14 +414,14 @@
*> \param[in] NPARAMS
*> \verbatim
*> NPARAMS is INTEGER
*> Specifies the number of parameters set in PARAMS. If .LE. 0, the
*> Specifies the number of parameters set in PARAMS. If <= 0, the
*> PARAMS array is never referenced and default values are used.
*> \endverbatim
*>
*> \param[in,out] PARAMS
*> \verbatim
*> PARAMS is DOUBLE PRECISION array, dimension NPARAMS
*> Specifies algorithm parameters. If an entry is .LT. 0.0, then
*> Specifies algorithm parameters. If an entry is < 0.0, then
*> that entry will be filled with default value used for that
*> parameter. Only positions up to NPARAMS are accessed; defaults
*> are used for higher-numbered parameters.
@ -429,9 +429,9 @@
*> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
*> refinement or not.
*> Default: 1.0D+0
*> = 0.0 : No refinement is performed, and no error bounds are
*> = 0.0: No refinement is performed, and no error bounds are
*> computed.
*> = 1.0 : Use the extra-precise refinement algorithm.
*> = 1.0: Use the extra-precise refinement algorithm.
*> (other values are reserved for future use)
*>
*> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual

4
lapack-netlib/SRC/zsytf2_rk.f
View File

@ -321,7 +321,7 @@
*
* Factorize A as U*D*U**T using the upper triangle of A
*
* Initilize the first entry of array E, where superdiagonal
* Initialize the first entry of array E, where superdiagonal
* elements of D are stored
*
E( 1 ) = CZERO
@ -632,7 +632,7 @@
*
* Factorize A as L*D*L**T using the lower triangle of A
*
* Initilize the unused last entry of the subdiagonal array E.
* Initialize the unused last entry of the subdiagonal array E.
*
E( N ) = CZERO
*

2
lapack-netlib/SRC/zsytrf.f
View File

@ -43,7 +43,7 @@
*>
*> where U (or L) is a product of permutation and unit upper (lower)
*> triangular matrices, and D is symmetric and block diagonal with
*> with 1-by-1 and 2-by-2 diagonal blocks.
*> 1-by-1 and 2-by-2 diagonal blocks.
*>
*> This is the blocked version of the algorithm, calling Level 3 BLAS.
*> \endverbatim

8
lapack-netlib/SRC/zsytrf_aa.f
View File

@ -37,7 +37,7 @@
*> ZSYTRF_AA computes the factorization of a complex symmetric matrix A
*> using the Aasen's algorithm. The form of the factorization is
*>
*> A = U*T*U**T or A = L*T*L**T
*> A = U**T*T*U or A = L*T*L**T
*>
*> where U (or L) is a product of permutation and unit upper (lower)
*> triangular matrices, and T is a complex symmetric tridiagonal matrix.
@ -223,7 +223,7 @@
IF( UPPER ) THEN
*
* .....................................................
* Factorize A as L*D*L**T using the upper triangle of A
* Factorize A as U**T*D*U using the upper triangle of A
* .....................................................
*
* Copy first row A(1, 1:N) into H(1:n) (stored in WORK(1:N))
@ -256,7 +256,7 @@
$ A( MAX(1, J), J+1 ), LDA,
$ IPIV( J+1 ), WORK, N, WORK( N*NB+1 ) )
*
* Ajust IPIV and apply it back (J-th step picks (J+1)-th pivot)
* Adjust IPIV and apply it back (J-th step picks (J+1)-th pivot)
*
DO J2 = J+2, MIN(N, J+JB+1)
IPIV( J2 ) = IPIV( J2 ) + J
@ -375,7 +375,7 @@
$ A( J+1, MAX(1, J) ), LDA,
$ IPIV( J+1 ), WORK, N, WORK( N*NB+1 ) )
*
* Ajust IPIV and apply it back (J-th step picks (J+1)-th pivot)
* Adjust IPIV and apply it back (J-th step picks (J+1)-th pivot)
*
DO J2 = J+2, MIN(N, J+JB+1)
IPIV( J2 ) = IPIV( J2 ) + J

26
lapack-netlib/SRC/zsytrf_aa_2stage.f
View File

@ -38,7 +38,7 @@
*> ZSYTRF_AA_2STAGE computes the factorization of a complex symmetric matrix A
*> using the Aasen's algorithm. The form of the factorization is
*>
*> A = U*T*U**T or A = L*T*L**T
*> A = U**T*T*U or A = L*T*L**T
*>
*> where U (or L) is a product of permutation and unit upper (lower)
*> triangular matrices, and T is a complex symmetric band matrix with the
@ -275,7 +275,7 @@
IF( UPPER ) THEN
*
* .....................................................
* Factorize A as L*D*L**T using the upper triangle of A
* Factorize A as U**T*D*U using the upper triangle of A
* .....................................................
*
DO J = 0, NT-1
@ -448,12 +448,14 @@ c END IF
* > Apply pivots to previous columns of L
CALL ZSWAP( K-1, A( (J+1)*NB+1, I1 ), 1,
$ A( (J+1)*NB+1, I2 ), 1 )
* > Swap A(I1+1:M, I1) with A(I2, I1+1:M)
CALL ZSWAP( I2-I1-1, A( I1, I1+1 ), LDA,
$ A( I1+1, I2 ), 1 )
* > Swap A(I1+1:M, I1) with A(I2, I1+1:M)
IF( I2.GT.(I1+1) )
$ CALL ZSWAP( I2-I1-1, A( I1, I1+1 ), LDA,
$ A( I1+1, I2 ), 1 )
* > Swap A(I2+1:M, I1) with A(I2+1:M, I2)
CALL ZSWAP( N-I2, A( I1, I2+1 ), LDA,
$ A( I2, I2+1 ), LDA )
IF( I2.LT.N )
$ CALL ZSWAP( N-I2, A( I1, I2+1 ), LDA,
$ A( I2, I2+1 ), LDA )
* > Swap A(I1, I1) with A(I2, I2)
PIV = A( I1, I1 )
A( I1, I1 ) = A( I2, I2 )
@ -637,11 +639,13 @@ c END IF
CALL ZSWAP( K-1, A( I1, (J+1)*NB+1 ), LDA,
$ A( I2, (J+1)*NB+1 ), LDA )
* > Swap A(I1+1:M, I1) with A(I2, I1+1:M)
CALL ZSWAP( I2-I1-1, A( I1+1, I1 ), 1,
$ A( I2, I1+1 ), LDA )
IF( I2.GT.(I1+1) )
$ CALL ZSWAP( I2-I1-1, A( I1+1, I1 ), 1,
$ A( I2, I1+1 ), LDA )
* > Swap A(I2+1:M, I1) with A(I2+1:M, I2)
CALL ZSWAP( N-I2, A( I2+1, I1 ), 1,
$ A( I2+1, I2 ), 1 )
IF( I2.LT.N )
$ CALL ZSWAP( N-I2, A( I2+1, I1 ), 1,
$ A( I2+1, I2 ), 1 )
* > Swap A(I1, I1) with A(I2, I2)
PIV = A( I1, I1 )
A( I1, I1 ) = A( I2, I2 )

4
lapack-netlib/SRC/zsytri2.f
View File

@ -62,7 +62,7 @@
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX*16 array, dimension (LDA,N)
*> On entry, the NB diagonal matrix D and the multipliers
*> On entry, the block diagonal matrix D and the multipliers
*> used to obtain the factor U or L as computed by ZSYTRF.
*>
*> On exit, if INFO = 0, the (symmetric) inverse of the original
@ -82,7 +82,7 @@
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> Details of the interchanges and the NB structure of D
*> Details of the interchanges and the block structure of D
*> as determined by ZSYTRF.
*> \endverbatim
*>

2
lapack-netlib/SRC/zsytrs2.f
View File

@ -36,7 +36,7 @@
*>
*> \verbatim
*>
*> ZSYTRS2 solves a system of linear equations A*X = B with a real
*> ZSYTRS2 solves a system of linear equations A*X = B with a complex
*> symmetric matrix A using the factorization A = U*D*U**T or
*> A = L*D*L**T computed by ZSYTRF and converted by ZSYCONV.
*> \endverbatim

112
lapack-netlib/SRC/zsytrs_aa.f
View File

@ -37,7 +37,7 @@
*> \verbatim
*>
*> ZSYTRS_AA solves a system of linear equations A*X = B with a complex
*> symmetric matrix A using the factorization A = U*T*U**T or
*> symmetric matrix A using the factorization A = U**T*T*U or
*> A = L*T*L**T computed by ZSYTRF_AA.
*> \endverbatim
*
@ -49,7 +49,7 @@
*> UPLO is CHARACTER*1
*> Specifies whether the details of the factorization are stored
*> as an upper or lower triangular matrix.
*> = 'U': Upper triangular, form is A = U*T*U**T;
*> = 'U': Upper triangular, form is A = U**T*T*U;
*> = 'L': Lower triangular, form is A = L*T*L**T.
*> \endverbatim
*>
@ -97,14 +97,16 @@
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in] WORK
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE array, dimension (MAX(1,LWORK))
*> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER, LWORK >= MAX(1,3*N-2).
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,3*N-2).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
@ -198,22 +200,29 @@
*
IF( UPPER ) THEN
*
* Solve A*X = B, where A = U*T*U**T.
* Solve A*X = B, where A = U**T*T*U.
*
* Pivot, P**T * B
* 1) Forward substitution with U**T
*
DO K = 1, N
KP = IPIV( K )
IF( KP.NE.K )
$ CALL ZSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
END DO
IF( N.GT.1 ) THEN
*
* Compute (U \P**T * B) -> B [ (U \P**T * B) ]
* Pivot, P**T * B -> B
*
CALL ZTRSM('L', 'U', 'T', 'U', N-1, NRHS, ONE, A( 1, 2 ), LDA,
$ B( 2, 1 ), LDB)
DO K = 1, N
KP = IPIV( K )
IF( KP.NE.K )
$ CALL ZSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
END DO
*
* Compute T \ B -> B [ T \ (U \P**T * B) ]
* Compute U**T \ B -> B [ (U**T \P**T * B) ]
*
CALL ZTRSM( 'L', 'U', 'T', 'U', N-1, NRHS, ONE, A( 1, 2 ),
$ LDA, B( 2, 1 ), LDB)
END IF
*
* 2) Solve with triangular matrix T
*
* Compute T \ B -> B [ T \ (U**T \P**T * B) ]
*
CALL ZLACPY( 'F', 1, N, A( 1, 1 ), LDA+1, WORK( N ), 1)
IF( N.GT.1 ) THEN
@ -223,35 +232,47 @@
CALL ZGTSV( N, NRHS, WORK( 1 ), WORK( N ), WORK( 2*N ), B, LDB,
$ INFO )
*
* Compute (U**T \ B) -> B [ U**T \ (T \ (U \P**T * B) ) ]
* 3) Backward substitution with U
*
CALL ZTRSM( 'L', 'U', 'N', 'U', N-1, NRHS, ONE, A( 1, 2 ), LDA,
$ B( 2, 1 ), LDB)
IF( N.GT.1 ) THEN
*
* Pivot, P * B [ P * (U**T \ (T \ (U \P**T * B) )) ]
* Compute U \ B -> B [ U \ (T \ (U**T \P**T * B) ) ]
*
DO K = N, 1, -1
KP = IPIV( K )
IF( KP.NE.K )
$ CALL ZSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
END DO
CALL ZTRSM( 'L', 'U', 'N', 'U', N-1, NRHS, ONE, A( 1, 2 ),
$ LDA, B( 2, 1 ), LDB)
*
* Pivot, P * B -> B [ P * (U \ (T \ (U**T \P**T * B) )) ]
*
DO K = N, 1, -1
KP = IPIV( K )
IF( KP.NE.K )
$ CALL ZSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
END DO
END IF
*
ELSE
*
* Solve A*X = B, where A = L*T*L**T.
*
* Pivot, P**T * B
* 1) Forward substitution with L
*
DO K = 1, N
KP = IPIV( K )
IF( KP.NE.K )
$ CALL ZSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
END DO
IF( N.GT.1 ) THEN
*
* Compute (L \P**T * B) -> B [ (L \P**T * B) ]
* Pivot, P**T * B -> B
*
CALL ZTRSM( 'L', 'L', 'N', 'U', N-1, NRHS, ONE, A( 2, 1 ), LDA,
$ B( 2, 1 ), LDB)
DO K = 1, N
KP = IPIV( K )
IF( KP.NE.K )
$ CALL ZSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
END DO
*
* Compute L \ B -> B [ (L \P**T * B) ]
*
CALL ZTRSM( 'L', 'L', 'N', 'U', N-1, NRHS, ONE, A( 2, 1 ),
$ LDA, B( 2, 1 ), LDB)
END IF
*
* 2) Solve with triangular matrix T
*
* Compute T \ B -> B [ T \ (L \P**T * B) ]
*
@ -263,18 +284,23 @@
CALL ZGTSV( N, NRHS, WORK( 1 ), WORK(N), WORK( 2*N ), B, LDB,
$ INFO)
*
* Compute (L**T \ B) -> B [ L**T \ (T \ (L \P**T * B) ) ]
* 3) Backward substitution with L**T
*
CALL ZTRSM( 'L', 'L', 'T', 'U', N-1, NRHS, ONE, A( 2, 1 ), LDA,
$ B( 2, 1 ), LDB)
IF( N.GT.1 ) THEN
*
* Pivot, P * B [ P * (L**T \ (T \ (L \P**T * B) )) ]
* Compute (L**T \ B) -> B [ L**T \ (T \ (L \P**T * B) ) ]
*
DO K = N, 1, -1
KP = IPIV( K )
IF( KP.NE.K )
$ CALL ZSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
END DO
CALL ZTRSM( 'L', 'L', 'T', 'U', N-1, NRHS, ONE, A( 2, 1 ),
$ LDA, B( 2, 1 ), LDB)
*
* Pivot, P * B -> B [ P * (L**T \ (T \ (L \P**T * B) )) ]
*
DO K = N, 1, -1
KP = IPIV( K )
IF( KP.NE.K )
$ CALL ZSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
END DO
END IF
*
END IF
*

18
lapack-netlib/SRC/zsytrs_aa_2stage.f
View File

@ -36,7 +36,7 @@
*> \verbatim
*>
*> ZSYTRS_AA_2STAGE solves a system of linear equations A*X = B with a complex
*> symmetric matrix A using the factorization A = U*T*U**T or
*> symmetric matrix A using the factorization A = U**T*T*U or
*> A = L*T*L**T computed by ZSYTRF_AA_2STAGE.
*> \endverbatim
*
@ -48,7 +48,7 @@
*> UPLO is CHARACTER*1
*> Specifies whether the details of the factorization are stored
*> as an upper or lower triangular matrix.
*> = 'U': Upper triangular, form is A = U*T*U**T;
*> = 'U': Upper triangular, form is A = U**T*T*U;
*> = 'L': Lower triangular, form is A = L*T*L**T.
*> \endverbatim
*>
@ -208,15 +208,15 @@
*
IF( UPPER ) THEN
*
* Solve A*X = B, where A = U*T*U**T.
* Solve A*X = B, where A = U**T*T*U.
*
IF( N.GT.NB ) THEN
*
* Pivot, P**T * B
* Pivot, P**T * B -> B
*
CALL ZLASWP( NRHS, B, LDB, NB+1, N, IPIV, 1 )
*
* Compute (U**T \P**T * B) -> B [ (U**T \P**T * B) ]
* Compute (U**T \ B) -> B [ (U**T \P**T * B) ]
*
CALL ZTRSM( 'L', 'U', 'T', 'U', N-NB, NRHS, ONE, A(1, NB+1),
$ LDA, B(NB+1, 1), LDB)
@ -234,7 +234,7 @@
CALL ZTRSM( 'L', 'U', 'N', 'U', N-NB, NRHS, ONE, A(1, NB+1),
$ LDA, B(NB+1, 1), LDB)
*
* Pivot, P * B [ P * (U \ (T \ (U**T \P**T * B) )) ]
* Pivot, P * B -> B [ P * (U \ (T \ (U**T \P**T * B) )) ]
*
CALL ZLASWP( NRHS, B, LDB, NB+1, N, IPIV, -1 )
*
@ -246,11 +246,11 @@
*
IF( N.GT.NB ) THEN
*
* Pivot, P**T * B
* Pivot, P**T * B -> B
*
CALL ZLASWP( NRHS, B, LDB, NB+1, N, IPIV, 1 )
*
* Compute (L \P**T * B) -> B [ (L \P**T * B) ]
* Compute (L \ B) -> B [ (L \P**T * B) ]
*
CALL ZTRSM( 'L', 'L', 'N', 'U', N-NB, NRHS, ONE, A(NB+1, 1),
$ LDA, B(NB+1, 1), LDB)
@ -268,7 +268,7 @@
CALL ZTRSM( 'L', 'L', 'T', 'U', N-NB, NRHS, ONE, A(NB+1, 1),
$ LDA, B(NB+1, 1), LDB)
*
* Pivot, P * B [ P * (L**T \ (T \ (L \P**T * B) )) ]
* Pivot, P * B -> B [ P * (L**T \ (T \ (L \P**T * B) )) ]
*
CALL ZLASWP( NRHS, B, LDB, NB+1, N, IPIV, -1 )
*

4
lapack-netlib/SRC/ztgsy2.f
View File

@ -67,7 +67,7 @@
*> R * B**H + L * E**H = scale * -F
*>
*> This case is used to compute an estimate of Dif[(A, D), (B, E)] =
*> = sigma_min(Z) using reverse communicaton with ZLACON.
*> = sigma_min(Z) using reverse communication with ZLACON.
*>
*> ZTGSY2 also (IJOB >= 1) contributes to the computation in ZTGSYL
*> of an upper bound on the separation between to matrix pairs. Then
@ -81,7 +81,7 @@
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N', solve the generalized Sylvester equation (1).
*> = 'N': solve the generalized Sylvester equation (1).
*> = 'T': solve the 'transposed' system (3).
*> \endverbatim
*>

2
lapack-netlib/SRC/ztpmlqt.f
View File

@ -94,7 +94,7 @@
*>
*> \param[in] V
*> \verbatim
*> V is COMPLEX*16 array, dimension (LDA,K)
*> V is COMPLEX*16 array, dimension (LDV,K)
*> The i-th row must contain the vector which defines the
*> elementary reflector H(i), for i = 1,2,...,k, as returned by
*> DTPLQT in B. See Further Details.

2
lapack-netlib/SRC/ztpmqrt.f
View File

@ -94,7 +94,7 @@
*>
*> \param[in] V
*> \verbatim
*> V is COMPLEX*16 array, dimension (LDA,K)
*> V is COMPLEX*16 array, dimension (LDV,K)
*> The i-th column must contain the vector which defines the
*> elementary reflector H(i), for i = 1,2,...,k, as returned by
*> CTPQRT in B. See Further Details.

4
lapack-netlib/SRC/ztprfb.f
View File

@ -152,8 +152,8 @@
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A.
*> If SIDE = 'L', LDC >= max(1,K);
*> If SIDE = 'R', LDC >= max(1,M).
*> If SIDE = 'L', LDA >= max(1,K);
*> If SIDE = 'R', LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B

307
lapack-netlib/SRC/zungtsqr.f Normal file
View File

@ -0,0 +1,307 @@
*> \brief \b ZUNGTSQR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ZUNGTSQR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zuntsqr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zungtsqr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zungtsqr.f">
*> [TXT]</a>
*>
* Definition:
* ===========
*
* SUBROUTINE ZUNGTSQR( M, N, MB, NB, A, LDA, T, LDT, WORK, LWORK,
* $ INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDT, LWORK, M, N, MB, NB
* ..
* .. Array Arguments ..
* COMPLEX*16 A( LDA, * ), T( LDT, * ), WORK( * )
* ..
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZUNGTSQR generates an M-by-N complex matrix Q_out with orthonormal
*> columns, which are the first N columns of a product of comlpex unitary
*> matrices of order M which are returned by ZLATSQR
*>
*> Q_out = first_N_columns_of( Q(1)_in * Q(2)_in * ... * Q(k)_in ).
*>
*> See the documentation for ZLATSQR.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. M >= N >= 0.
*> \endverbatim
*>
*> \param[in] MB
*> \verbatim
*> MB is INTEGER
*> The row block size used by DLATSQR to return
*> arrays A and T. MB > N.
*> (Note that if MB > M, then M is used instead of MB
*> as the row block size).
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*> NB is INTEGER
*> The column block size used by ZLATSQR to return
*> arrays A and T. NB >= 1.
*> (Note that if NB > N, then N is used instead of NB
*> as the column block size).
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX*16 array, dimension (LDA,N)
*>
*> On entry:
*>
*> The elements on and above the diagonal are not accessed.
*> The elements below the diagonal represent the unit
*> lower-trapezoidal blocked matrix V computed by ZLATSQR
*> that defines the input matrices Q_in(k) (ones on the
*> diagonal are not stored) (same format as the output A
*> below the diagonal in ZLATSQR).
*>
*> On exit:
*>
*> The array A contains an M-by-N orthonormal matrix Q_out,
*> i.e the columns of A are orthogonal unit vectors.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] T
*> \verbatim
*> T is COMPLEX*16 array,
*> dimension (LDT, N * NIRB)
*> where NIRB = Number_of_input_row_blocks
*> = MAX( 1, CEIL((M-N)/(MB-N)) )
*> Let NICB = Number_of_input_col_blocks
*> = CEIL(N/NB)
*>
*> The upper-triangular block reflectors used to define the
*> input matrices Q_in(k), k=(1:NIRB*NICB). The block
*> reflectors are stored in compact form in NIRB block
*> reflector sequences. Each of NIRB block reflector sequences
*> is stored in a larger NB-by-N column block of T and consists
*> of NICB smaller NB-by-NB upper-triangular column blocks.
*> (same format as the output T in ZLATSQR).
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T.
*> LDT >= max(1,min(NB1,N)).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> (workspace) COMPLEX*16 array, dimension (MAX(2,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> The dimension of the array WORK. LWORK >= (M+NB)*N.
*> If LWORK = -1, then a workspace query is assumed.
*> The routine only calculates the optimal size of the WORK
*> array, 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 = -i, the i-th argument had an illegal value
*> \endverbatim
*>
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2019
*
*> \ingroup comlex16OTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> \verbatim
*>
*> November 2019, Igor Kozachenko,
*> Computer Science Division,
*> University of California, Berkeley
*>
*> \endverbatim
*
* =====================================================================
SUBROUTINE ZUNGTSQR( M, N, MB, NB, A, LDA, T, LDT, WORK, LWORK,
$ INFO )
IMPLICIT NONE
*
* -- LAPACK computational routine (version 3.9.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2019
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDT, LWORK, M, N, MB, NB
* ..
* .. Array Arguments ..
COMPLEX*16 A( LDA, * ), T( LDT, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
COMPLEX*16 CONE, CZERO
PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ),
$ CZERO = ( 0.0D+0, 0.0D+0 ) )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER IINFO, LDC, LWORKOPT, LC, LW, NBLOCAL, J
* ..
* .. External Subroutines ..
EXTERNAL ZCOPY, ZLAMTSQR, ZLASET, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC DCMPLX, MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
LQUERY = LWORK.EQ.-1
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 .OR. M.LT.N ) THEN
INFO = -2
ELSE IF( MB.LE.N ) THEN
INFO = -3
ELSE IF( NB.LT.1 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -6
ELSE IF( LDT.LT.MAX( 1, MIN( NB, N ) ) ) THEN
INFO = -8
ELSE
*
* Test the input LWORK for the dimension of the array WORK.
* This workspace is used to store array C(LDC, N) and WORK(LWORK)
* in the call to ZLAMTSQR. See the documentation for ZLAMTSQR.
*
IF( LWORK.LT.2 .AND. (.NOT.LQUERY) ) THEN
INFO = -10
ELSE
*
* Set block size for column blocks
*
NBLOCAL = MIN( NB, N )
*
* LWORK = -1, then set the size for the array C(LDC,N)
* in ZLAMTSQR call and set the optimal size of the work array
* WORK(LWORK) in ZLAMTSQR call.
*
LDC = M
LC = LDC*N
LW = N * NBLOCAL
*
LWORKOPT = LC+LW
*
IF( ( LWORK.LT.MAX( 1, LWORKOPT ) ).AND.(.NOT.LQUERY) ) THEN
INFO = -10
END IF
END IF
*
END IF
*
* Handle error in the input parameters and return workspace query.
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZUNGTSQR', -INFO )
RETURN
ELSE IF ( LQUERY ) THEN
WORK( 1 ) = DCMPLX( LWORKOPT )
RETURN
END IF
*
* Quick return if possible
*
IF( MIN( M, N ).EQ.0 ) THEN
WORK( 1 ) = DCMPLX( LWORKOPT )
RETURN
END IF
*
* (1) Form explicitly the tall-skinny M-by-N left submatrix Q1_in
* of M-by-M orthogonal matrix Q_in, which is implicitly stored in
* the subdiagonal part of input array A and in the input array T.
* Perform by the following operation using the routine ZLAMTSQR.
*
* Q1_in = Q_in * ( I ), where I is a N-by-N identity matrix,
* ( 0 ) 0 is a (M-N)-by-N zero matrix.
*
* (1a) Form M-by-N matrix in the array WORK(1:LDC*N) with ones
* on the diagonal and zeros elsewhere.
*
CALL ZLASET( 'F', M, N, CZERO, CONE, WORK, LDC )
*
* (1b) On input, WORK(1:LDC*N) stores ( I );
* ( 0 )
*
* On output, WORK(1:LDC*N) stores Q1_in.
*
CALL ZLAMTSQR( 'L', 'N', M, N, N, MB, NBLOCAL, A, LDA, T, LDT,
$ WORK, LDC, WORK( LC+1 ), LW, IINFO )
*
* (2) Copy the result from the part of the work array (1:M,1:N)
* with the leading dimension LDC that starts at WORK(1) into
* the output array A(1:M,1:N) column-by-column.
*
DO J = 1, N
CALL ZCOPY( M, WORK( (J-1)*LDC + 1 ), 1, A( 1, J ), 1 )
END DO
*
WORK( 1 ) = DCMPLX( LWORKOPT )
RETURN
*
* End of ZUNGTSQR
*
END

441
lapack-netlib/SRC/zunhr_col.f Normal file
View File

@ -0,0 +1,441 @@
*> \brief \b ZUNHR_COL
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ZUNHR_COL + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zunhr_col.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zunhr_col.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zunhr_col.f">
*> [TXT]</a>
*>
* Definition:
* ===========
*
* SUBROUTINE ZUNHR_COL( M, N, NB, A, LDA, T, LDT, D, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDT, M, N, NB
* ..
* .. Array Arguments ..
* COMPLEX*16 A( LDA, * ), D( * ), T( LDT, * )
* ..
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZUNHR_COL takes an M-by-N complex matrix Q_in with orthonormal columns
*> as input, stored in A, and performs Householder Reconstruction (HR),
*> i.e. reconstructs Householder vectors V(i) implicitly representing
*> another M-by-N matrix Q_out, with the property that Q_in = Q_out*S,
*> where S is an N-by-N diagonal matrix with diagonal entries
*> equal to +1 or -1. The Householder vectors (columns V(i) of V) are
*> stored in A on output, and the diagonal entries of S are stored in D.
*> Block reflectors are also returned in T
*> (same output format as ZGEQRT).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. M >= N >= 0.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*> NB is INTEGER
*> The column block size to be used in the reconstruction
*> of Householder column vector blocks in the array A and
*> corresponding block reflectors in the array T. NB >= 1.
*> (Note that if NB > N, then N is used instead of NB
*> as the column block size.)
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX*16 array, dimension (LDA,N)
*>
*> On entry:
*>
*> The array A contains an M-by-N orthonormal matrix Q_in,
*> i.e the columns of A are orthogonal unit vectors.
*>
*> On exit:
*>
*> The elements below the diagonal of A represent the unit
*> lower-trapezoidal matrix V of Householder column vectors
*> V(i). The unit diagonal entries of V are not stored
*> (same format as the output below the diagonal in A from
*> ZGEQRT). The matrix T and the matrix V stored on output
*> in A implicitly define Q_out.
*>
*> The elements above the diagonal contain the factor U
*> of the "modified" LU-decomposition:
*> Q_in - ( S ) = V * U
*> ( 0 )
*> where 0 is a (M-N)-by-(M-N) zero matrix.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*> T is COMPLEX*16 array,
*> dimension (LDT, N)
*>
*> Let NOCB = Number_of_output_col_blocks
*> = CEIL(N/NB)
*>
*> On exit, T(1:NB, 1:N) contains NOCB upper-triangular
*> block reflectors used to define Q_out stored in compact
*> form as a sequence of upper-triangular NB-by-NB column
*> blocks (same format as the output T in ZGEQRT).
*> The matrix T and the matrix V stored on output in A
*> implicitly define Q_out. NOTE: The lower triangles
*> below the upper-triangular blcoks will be filled with
*> zeros. See Further Details.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T.
*> LDT >= max(1,min(NB,N)).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is COMPLEX*16 array, dimension min(M,N).
*> The elements can be only plus or minus one.
*>
*> D(i) is constructed as D(i) = -SIGN(Q_in_i(i,i)), where
*> 1 <= i <= min(M,N), and Q_in_i is Q_in after performing
*> i-1 steps of “modified” Gaussian elimination.
*> See Further Details.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*>
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The computed M-by-M unitary factor Q_out is defined implicitly as
*> a product of unitary matrices Q_out(i). Each Q_out(i) is stored in
*> the compact WY-representation format in the corresponding blocks of
*> matrices V (stored in A) and T.
*>
*> The M-by-N unit lower-trapezoidal matrix V stored in the M-by-N
*> matrix A contains the column vectors V(i) in NB-size column
*> blocks VB(j). For example, VB(1) contains the columns
*> V(1), V(2), ... V(NB). NOTE: The unit entries on
*> the diagonal of Y are not stored in A.
*>
*> The number of column blocks is
*>
*> NOCB = Number_of_output_col_blocks = CEIL(N/NB)
*>
*> where each block is of order NB except for the last block, which
*> is of order LAST_NB = N - (NOCB-1)*NB.
*>
*> For example, if M=6, N=5 and NB=2, the matrix V is
*>
*>
*> V = ( VB(1), VB(2), VB(3) ) =
*>
*> = ( 1 )
*> ( v21 1 )
*> ( v31 v32 1 )
*> ( v41 v42 v43 1 )
*> ( v51 v52 v53 v54 1 )
*> ( v61 v62 v63 v54 v65 )
*>
*>
*> For each of the column blocks VB(i), an upper-triangular block
*> reflector TB(i) is computed. These blocks are stored as
*> a sequence of upper-triangular column blocks in the NB-by-N
*> matrix T. The size of each TB(i) block is NB-by-NB, except
*> for the last block, whose size is LAST_NB-by-LAST_NB.
*>
*> For example, if M=6, N=5 and NB=2, the matrix T is
*>
*> T = ( TB(1), TB(2), TB(3) ) =
*>
*> = ( t11 t12 t13 t14 t15 )
*> ( t22 t24 )
*>
*>
*> The M-by-M factor Q_out is given as a product of NOCB
*> unitary M-by-M matrices Q_out(i).
*>
*> Q_out = Q_out(1) * Q_out(2) * ... * Q_out(NOCB),
*>
*> where each matrix Q_out(i) is given by the WY-representation
*> using corresponding blocks from the matrices V and T:
*>
*> Q_out(i) = I - VB(i) * TB(i) * (VB(i))**T,
*>
*> where I is the identity matrix. Here is the formula with matrix
*> dimensions:
*>
*> Q(i){M-by-M} = I{M-by-M} -
*> VB(i){M-by-INB} * TB(i){INB-by-INB} * (VB(i))**T {INB-by-M},
*>
*> where INB = NB, except for the last block NOCB
*> for which INB=LAST_NB.
*>
*> =====
*> NOTE:
*> =====
*>
*> If Q_in is the result of doing a QR factorization
*> B = Q_in * R_in, then:
*>
*> B = (Q_out*S) * R_in = Q_out * (S * R_in) = O_out * R_out.
*>
*> So if one wants to interpret Q_out as the result
*> of the QR factorization of B, then corresponding R_out
*> should be obtained by R_out = S * R_in, i.e. some rows of R_in
*> should be multiplied by -1.
*>
*> For the details of the algorithm, see [1].
*>
*> [1] "Reconstructing Householder vectors from tall-skinny QR",
*> G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
*> E. Solomonik, J. Parallel Distrib. Comput.,
*> vol. 85, pp. 3-31, 2015.
*> \endverbatim
*>
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2019
*
*> \ingroup complex16OTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> \verbatim
*>
*> November 2019, Igor Kozachenko,
*> Computer Science Division,
*> University of California, Berkeley
*>
*> \endverbatim
*
* =====================================================================
SUBROUTINE ZUNHR_COL( M, N, NB, A, LDA, T, LDT, D, INFO )
IMPLICIT NONE
*
* -- LAPACK computational routine (version 3.9.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2019
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDT, M, N, NB
* ..
* .. Array Arguments ..
COMPLEX*16 A( LDA, * ), D( * ), T( LDT, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
COMPLEX*16 CONE, CZERO
PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ),
$ CZERO = ( 0.0D+0, 0.0D+0 ) )
* ..
* .. Local Scalars ..
INTEGER I, IINFO, J, JB, JBTEMP1, JBTEMP2, JNB,
$ NPLUSONE
* ..
* .. External Subroutines ..
EXTERNAL ZCOPY, ZLAUNHR_COL_GETRFNP, ZSCAL, ZTRSM,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 .OR. N.GT.M ) THEN
INFO = -2
ELSE IF( NB.LT.1 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
ELSE IF( LDT.LT.MAX( 1, MIN( NB, N ) ) ) THEN
INFO = -7
END IF
*
* Handle error in the input parameters.
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZUNHR_COL', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( MIN( M, N ).EQ.0 ) THEN
RETURN
END IF
*
* On input, the M-by-N matrix A contains the unitary
* M-by-N matrix Q_in.
*
* (1) Compute the unit lower-trapezoidal V (ones on the diagonal
* are not stored) by performing the "modified" LU-decomposition.
*
* Q_in - ( S ) = V * U = ( V1 ) * U,
* ( 0 ) ( V2 )
*
* where 0 is an (M-N)-by-N zero matrix.
*
* (1-1) Factor V1 and U.
CALL ZLAUNHR_COL_GETRFNP( N, N, A, LDA, D, IINFO )
*
* (1-2) Solve for V2.
*
IF( M.GT.N ) THEN
CALL ZTRSM( 'R', 'U', 'N', 'N', M-N, N, CONE, A, LDA,
$ A( N+1, 1 ), LDA )
END IF
*
* (2) Reconstruct the block reflector T stored in T(1:NB, 1:N)
* as a sequence of upper-triangular blocks with NB-size column
* blocking.
*
* Loop over the column blocks of size NB of the array A(1:M,1:N)
* and the array T(1:NB,1:N), JB is the column index of a column
* block, JNB is the column block size at each step JB.
*
NPLUSONE = N + 1
DO JB = 1, N, NB
*
* (2-0) Determine the column block size JNB.
*
JNB = MIN( NPLUSONE-JB, NB )
*
* (2-1) Copy the upper-triangular part of the current JNB-by-JNB
* diagonal block U(JB) (of the N-by-N matrix U) stored
* in A(JB:JB+JNB-1,JB:JB+JNB-1) into the upper-triangular part
* of the current JNB-by-JNB block T(1:JNB,JB:JB+JNB-1)
* column-by-column, total JNB*(JNB+1)/2 elements.
*
JBTEMP1 = JB - 1
DO J = JB, JB+JNB-1
CALL ZCOPY( J-JBTEMP1, A( JB, J ), 1, T( 1, J ), 1 )
END DO
*
* (2-2) Perform on the upper-triangular part of the current
* JNB-by-JNB diagonal block U(JB) (of the N-by-N matrix U) stored
* in T(1:JNB,JB:JB+JNB-1) the following operation in place:
* (-1)*U(JB)*S(JB), i.e the result will be stored in the upper-
* triangular part of T(1:JNB,JB:JB+JNB-1). This multiplication
* of the JNB-by-JNB diagonal block U(JB) by the JNB-by-JNB
* diagonal block S(JB) of the N-by-N sign matrix S from the
* right means changing the sign of each J-th column of the block
* U(JB) according to the sign of the diagonal element of the block
* S(JB), i.e. S(J,J) that is stored in the array element D(J).
*
DO J = JB, JB+JNB-1
IF( D( J ).EQ.CONE ) THEN
CALL ZSCAL( J-JBTEMP1, -CONE, T( 1, J ), 1 )
END IF
END DO
*
* (2-3) Perform the triangular solve for the current block
* matrix X(JB):
*
* X(JB) * (A(JB)**T) = B(JB), where:
*
* A(JB)**T is a JNB-by-JNB unit upper-triangular
* coefficient block, and A(JB)=V1(JB), which
* is a JNB-by-JNB unit lower-triangular block
* stored in A(JB:JB+JNB-1,JB:JB+JNB-1).
* The N-by-N matrix V1 is the upper part
* of the M-by-N lower-trapezoidal matrix V
* stored in A(1:M,1:N);
*
* B(JB) is a JNB-by-JNB upper-triangular right-hand
* side block, B(JB) = (-1)*U(JB)*S(JB), and
* B(JB) is stored in T(1:JNB,JB:JB+JNB-1);
*
* X(JB) is a JNB-by-JNB upper-triangular solution
* block, X(JB) is the upper-triangular block
* reflector T(JB), and X(JB) is stored
* in T(1:JNB,JB:JB+JNB-1).
*
* In other words, we perform the triangular solve for the
* upper-triangular block T(JB):
*
* T(JB) * (V1(JB)**T) = (-1)*U(JB)*S(JB).
*
* Even though the blocks X(JB) and B(JB) are upper-
* triangular, the routine ZTRSM will access all JNB**2
* elements of the square T(1:JNB,JB:JB+JNB-1). Therefore,
* we need to set to zero the elements of the block
* T(1:JNB,JB:JB+JNB-1) below the diagonal before the call
* to ZTRSM.
*
* (2-3a) Set the elements to zero.
*
JBTEMP2 = JB - 2
DO J = JB, JB+JNB-2
DO I = J-JBTEMP2, NB
T( I, J ) = CZERO
END DO
END DO
*
* (2-3b) Perform the triangular solve.
*
CALL ZTRSM( 'R', 'L', 'C', 'U', JNB, JNB, CONE,
$ A( JB, JB ), LDA, T( 1, JB ), LDT )
*
END DO
*
RETURN
*
* End of ZUNHR_COL
*
END