table of contents
other versions
- jessie 3.5.0-4
- jessie-backports 3.7.0-1~bpo8+1
- stretch 3.7.0-2
- testing 3.8.0-1
- unstable 3.8.0-1
complex16(3) | LAPACK | complex16(3) |
NAME¶
complex16 -Functions¶
subroutine zhecon (UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, INFO)
Detailed Description¶
This is the group of complex16 computational functions for HE matricesFunction Documentation¶
subroutine zhecon (characterUPLO, integerN, complex*16, dimension( lda, * )A, integerLDA, integer, dimension( * )IPIV, double precisionANORM, double precisionRCOND, complex*16, dimension( * )WORK, integerINFO)¶
ZHECON Purpose:ZHECON estimates the reciprocal of the condition number of a complex Hermitian matrix A using the factorization A = U*D*U**H or A = L*D*L**H computed by ZHETRF. An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
UPLO
Author:
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*D*U**H; = 'L': Lower triangular, form is A = L*D*L**H.N
N is INTEGER The order of the matrix A. N >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZHETRF.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIV
IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by ZHETRF.ANORM
ANORM is DOUBLE PRECISION The 1-norm of the original matrix A.RCOND
RCOND is DOUBLE PRECISION The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an estimate of the 1-norm of inv(A) computed in this routine.WORK
WORK is COMPLEX*16 array, dimension (2*N)INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
subroutine zhecon_3 (characterUPLO, integerN, complex*16, dimension( lda, * )A, integerLDA, complex*16, dimension( * )E, integer, dimension( * )IPIV, double precisionANORM, double precisionRCOND, complex*16, dimension( * )WORK, integerINFO)¶
ZHECON_3 Purpose:ZHECON_3 estimates the reciprocal of the condition number (in the 1-norm) of a complex Hermitian matrix A using the factorization computed by ZHETRF_RK or ZHETRF_BK: A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T), where U (or L) is unit upper (or lower) triangular matrix, U**H (or L**H) is the conjugate of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). This routine uses BLAS3 solver ZHETRS_3.
UPLO
Author:
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 = P*U*D*(U**H)*(P**T); = 'L': Lower triangular, form is A = P*L*D*(L**H)*(P**T).N
N is INTEGER The order of the matrix A. N >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) Diagonal of the block diagonal matrix D and factors U or L as computed by ZHETRF_RK and ZHETRF_BK: a) ONLY diagonal elements of the Hermitian block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D should be provided on entry 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.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).E
E is COMPLEX*16 array, dimension (N) On entry, contains the superdiagonal (or subdiagonal) elements of the Hermitian 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) not referenced; If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced. NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is not referenced in both UPLO = 'U' or UPLO = 'L' cases.IPIV
IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by ZHETRF_RK or ZHETRF_BK.ANORM
ANORM is DOUBLE PRECISION The 1-norm of the original matrix A.RCOND
RCOND is DOUBLE PRECISION The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an estimate of the 1-norm of inv(A) computed in this routine.WORK
WORK is COMPLEX*16 array, dimension (2*N)IWORK
IWORK is INTEGER array, dimension (N)INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Contributors:
subroutine zhecon_rook (characterUPLO, integerN, complex*16, dimension( lda, * )A, integerLDA, integer, dimension( * )IPIV, double precisionANORM, double precisionRCOND, complex*16, dimension( * )WORK, integerINFO)¶
ZHECON_ROOK estimates the reciprocal of the condition number fort HE matrices using factorization obtained with one of the bounded diagonal pivoting methods (max 2 interchanges) Purpose:ZHECON_ROOK estimates the reciprocal of the condition number of a complex Hermitian matrix A using the factorization A = U*D*U**H or A = L*D*L**H computed by CHETRF_ROOK. An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
UPLO
Author:
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*D*U**H; = 'L': Lower triangular, form is A = L*D*L**H.N
N is INTEGER The order of the matrix A. N >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by CHETRF_ROOK.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIV
IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by CHETRF_ROOK.ANORM
ANORM is DOUBLE PRECISION The 1-norm of the original matrix A.RCOND
RCOND is DOUBLE PRECISION The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an estimate of the 1-norm of inv(A) computed in this routine.WORK
WORK is COMPLEX*16 array, dimension (2*N)INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Contributors:
subroutine zheequb (characterUPLO, integerN, complex*16, dimension( lda, * )A, integerLDA, double precision, dimension( * )S, double precisionSCOND, double precisionAMAX, complex*16, dimension( * )WORK, integerINFO)¶
ZHEEQUB Purpose:ZHEEQUB computes row and column scalings intended to equilibrate a Hermitian matrix A (with respect to the Euclidean norm) and reduce its condition number. The scale factors S are computed by the BIN algorithm (see references) so that the scaled matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has a condition number within a factor N of the smallest possible condition number over all possible diagonal scalings.
UPLO
Author:
UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.N
N is INTEGER The order of the matrix A. N >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) The N-by-N Hermitian matrix whose scaling factors are to be computed.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).S
S is DOUBLE PRECISION array, dimension (N) If INFO = 0, S contains the scale factors for A.SCOND
SCOND is DOUBLE PRECISION If INFO = 0, S contains the ratio of the smallest S(i) to the largest S(i). If SCOND >= 0.1 and AMAX is neither too large nor too small, it is not worth scaling by S.AMAX
AMAX is DOUBLE PRECISION Largest absolute value of any matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled.WORK
WORK is COMPLEX*16 array, dimension (2*N)INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the i-th diagonal element is nonpositive.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
April 2012
References:
Livne, O.E. and Golub, G.H., 'Scaling by
Binormalization',
Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004.
DOI 10.1023/B:NUMA.0000016606.32820.69
Tech report version: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.3.1679
Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004.
DOI 10.1023/B:NUMA.0000016606.32820.69
Tech report version: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.3.1679
subroutine zhegs2 (integerITYPE, characterUPLO, integerN, complex*16, dimension( lda, * )A, integerLDA, complex*16, dimension( ldb, * )B, integerLDB, integerINFO)¶
ZHEGS2 reduces a Hermitian definite generalized eigenproblem to standard form, using the factorization results obtained from cpotrf (unblocked algorithm). Purpose:ZHEGS2 reduces a complex Hermitian-definite generalized eigenproblem to standard form. If ITYPE = 1, the problem is A*x = lambda*B*x, and A is overwritten by inv(U**H)*A*inv(U) or inv(L)*A*inv(L**H) If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or B*A*x = lambda*x, and A is overwritten by U*A*U**H or L**H *A*L. B must have been previously factorized as U**H *U or L*L**H by ZPOTRF.
ITYPE
Author:
ITYPE is INTEGER = 1: compute inv(U**H)*A*inv(U) or inv(L)*A*inv(L**H); = 2 or 3: compute U*A*U**H or L**H *A*L.UPLO
UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored, and how B has been factorized. = 'U': Upper triangular = 'L': Lower triangularN
N is INTEGER The order of the matrices A and B. N >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) 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 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, the transformed matrix, stored in the same format as A.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).B
B is COMPLEX*16 array, dimension (LDB,N) The triangular factor from the Cholesky factorization of B, as returned by ZPOTRF.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).INFO
INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
subroutine zhegst (integerITYPE, characterUPLO, integerN, complex*16, dimension( lda, * )A, integerLDA, complex*16, dimension( ldb, * )B, integerLDB, integerINFO)¶
ZHEGST Purpose:ZHEGST reduces a complex Hermitian-definite generalized eigenproblem to standard form. If ITYPE = 1, the problem is A*x = lambda*B*x, and A is overwritten by inv(U**H)*A*inv(U) or inv(L)*A*inv(L**H) If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or B*A*x = lambda*x, and A is overwritten by U*A*U**H or L**H*A*L. B must have been previously factorized as U**H*U or L*L**H by ZPOTRF.
ITYPE
Author:
ITYPE is INTEGER = 1: compute inv(U**H)*A*inv(U) or inv(L)*A*inv(L**H); = 2 or 3: compute U*A*U**H or L**H*A*L.UPLO
UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored and B is factored as U**H*U; = 'L': Lower triangle of A is stored and B is factored as L*L**H.N
N is INTEGER The order of the matrices A and B. N >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) 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 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, the transformed matrix, stored in the same format as A.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).B
B is COMPLEX*16 array, dimension (LDB,N) The triangular factor from the Cholesky factorization of B, as returned by ZPOTRF.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
subroutine zherfs (characterUPLO, integerN, integerNRHS, complex*16, dimension( lda, * )A, integerLDA, complex*16, dimension( ldaf, * )AF, integerLDAF, integer, dimension( * )IPIV, complex*16, dimension( ldb, * )B, integerLDB, complex*16, dimension( ldx, * )X, integerLDX, double precision, dimension( * )FERR, double precision, dimension( * )BERR, complex*16, dimension( * )WORK, double precision, dimension( * )RWORK, integerINFO)¶
ZHERFS Purpose:ZHERFS improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian indefinite, and provides error bounds and backward error estimates for the solution.
UPLO
Internal Parameters:
UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.N
N is INTEGER The order of the matrix A. N >= 0.NRHS
NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.A
A is COMPLEX*16 array, dimension (LDA,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 triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).AF
AF is COMPLEX*16 array, dimension (LDAF,N) The factored form of the matrix A. AF contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**H or A = L*D*L**H as computed by ZHETRF.LDAF
LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).IPIV
IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by ZHETRF.B
B is COMPLEX*16 array, dimension (LDB,NRHS) The right hand side matrix B.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).X
X is COMPLEX*16 array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by ZHETRS. On exit, the improved solution matrix X.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N).FERR
FERR is DOUBLE PRECISION array, dimension (NRHS) The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), FERR(j) is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error.BERR
BERR is DOUBLE PRECISION array, dimension (NRHS) The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution).WORK
WORK is COMPLEX*16 array, dimension (2*N)RWORK
RWORK is DOUBLE PRECISION array, dimension (N)INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value
ITMAX is the maximum number of steps of iterative refinement.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
subroutine zherfsx (characterUPLO, characterEQUED, integerN, integerNRHS, complex*16, dimension( lda, * )A, integerLDA, complex*16, dimension( ldaf, * )AF, integerLDAF, integer, dimension( * )IPIV, double precision, dimension( * )S, complex*16, dimension( ldb, * )B, integerLDB, complex*16, dimension( ldx, * )X, integerLDX, double precisionRCOND, double precision, dimension( * )BERR, integerN_ERR_BNDS, double precision, dimension( nrhs, * )ERR_BNDS_NORM, double precision, dimension( nrhs, * )ERR_BNDS_COMP, integerNPARAMS, double precision, dimension( * )PARAMS, complex*16, dimension( * )WORK, double precision, dimension( * )RWORK, integerINFO)¶
ZHERFSX Purpose:ZHERFSX improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian indefinite, 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 comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the error bounds. The original system of linear equations may have been equilibrated before calling this routine, as described by arguments EQUED and S below. In this case, the solution and error bounds returned are for the original unequilibrated system.
Some optional parameters are bundled in the PARAMS array. These settings determine how refinement is performed, but often the defaults are acceptable. If the defaults are acceptable, users can pass NPARAMS = 0 which prevents the source code from accessing the PARAMS argument.
UPLO
Author:
UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.EQUED
EQUED is CHARACTER*1 Specifies the form of equilibration that was done to A before calling this routine. This is needed to compute the solution and error bounds correctly. = 'N': No equilibration = 'Y': Both row and column equilibration, i.e., A has been replaced by diag(S) * A * diag(S). The right hand side B has been changed accordingly.N
N is INTEGER The order of the matrix A. N >= 0.NRHS
NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) 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.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).AF
AF is COMPLEX*16 array, dimension (LDAF,N) The factored form of the matrix A. AF contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by DSYTRF.LDAF
LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).IPIV
IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by DSYTRF.S
S is DOUBLE PRECISION array, dimension (N) 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 element of S is a power of the radix. If S is input, each element of S should be a power of the radix to ensure a reliable solution and error estimates. Scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows. Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable.B
B is COMPLEX*16 array, dimension (LDB,NRHS) The right hand side matrix B.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).X
X is COMPLEX*16 array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by DGETRS. On exit, the improved solution matrix X.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N).RCOND
RCOND is DOUBLE PRECISION Reciprocal scaled condition number. This is an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if done). If this is less than the machine precision (in particular, if it is zero), the matrix is singular to working precision. Note that the error may still be small even if this number is very small and the matrix appears ill- conditioned.BERR
BERR is DOUBLE PRECISION array, dimension (NRHS) Componentwise relative backward error. This is the componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution).N_ERR_BNDS
N_ERR_BNDS is INTEGER Number of error bounds to return for each right hand side and each type (normwise or componentwise). See ERR_BNDS_NORM and ERR_BNDS_COMP below.ERR_BNDS_NORM
ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the normwise relative error, which is defined as follows: Normwise relative error in the ith solution vector: max_j (abs(XTRUE(j,i) - X(j,i))) ------------------------------ max_j abs(X(j,i)) The array is indexed by the type of error information as described below. There currently are up to three pieces of information returned. The first index in ERR_BNDS_NORM(i,:) corresponds to the ith right-hand side. The second index in ERR_BNDS_NORM(:,err) contains the following three fields: err = 1 "Trust/don't trust" boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * dlamch('Epsilon'). err = 2 "Guaranteed" error bound: The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * dlamch('Epsilon'). This error bound should only be trusted if the previous boolean is true. err = 3 Reciprocal condition number: Estimated normwise reciprocal condition number. Compared with the threshold sqrt(n) * dlamch('Epsilon') to determine if the error estimate is "guaranteed". These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*A, where S scales each row by a power of the radix so all absolute row sums of Z are approximately 1. See Lapack Working Note 165 for further details and extra cautions.ERR_BNDS_COMP
ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the componentwise relative error, which is defined as follows: Componentwise relative error in the ith solution vector: abs(XTRUE(j,i) - X(j,i)) max_j ---------------------- abs(X(j,i)) The array is indexed by the right-hand side i (on which the componentwise relative error depends), and the type of error 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 the first (:,N_ERR_BNDS) entries are returned. The first index in ERR_BNDS_COMP(i,:) corresponds to the ith right-hand side. The second index in ERR_BNDS_COMP(:,err) contains the following three fields: err = 1 "Trust/don't trust" boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * dlamch('Epsilon'). err = 2 "Guaranteed" error bound: The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * dlamch('Epsilon'). This error bound should only be trusted if the previous boolean is true. err = 3 Reciprocal condition number: Estimated componentwise reciprocal condition number. Compared with the threshold sqrt(n) * dlamch('Epsilon') to determine if the error estimate is "guaranteed". These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*(A*diag(x)), where x is the solution for the current right-hand side and S scales each row of A*diag(x) by a power of the radix so all absolute row sums of Z are approximately 1. See Lapack Working Note 165 for further details and extra cautions.NPARAMS
NPARAMS is INTEGER Specifies the number of parameters set in PARAMS. If .LE. 0, the PARAMS array is never referenced and default values are used.PARAMS
PARAMS is DOUBLE PRECISION array, dimension NPARAMS Specifies algorithm parameters. If an entry is .LT. 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. 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 computed. = 1.0 : Use the double-precision refinement algorithm, possibly with doubled-single computations if the compilation environment does not support DOUBLE PRECISION. (other values are reserved for future use) PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual computations allowed for refinement. Default: 10 Aggressive: Set to 100 to permit convergence using approximate factorizations or factorizations other than LU. If the factorization uses a technique other than Gaussian elimination, the guarantees in err_bnds_norm and err_bnds_comp may no longer be trustworthy. PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code will attempt to find a solution with small componentwise relative error in the double-precision algorithm. Positive is true, 0.0 is false. Default: 1.0 (attempt componentwise convergence)WORK
WORK is COMPLEX*16 array, dimension (2*N)RWORK
RWORK is DOUBLE PRECISION array, dimension (2*N)INFO
INFO is INTEGER = 0: Successful exit. The solution to every right-hand side is guaranteed. < 0: If INFO = -i, the i-th argument had an illegal value > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned. = N+J: The solution corresponding to the Jth right-hand side is not guaranteed. The solutions corresponding to other right- hand sides K with K > J may not be guaranteed as well, but only the first such right-hand side is reported. If a small componentwise error is not requested (PARAMS(3) = 0.0) then the Jth right-hand side is the first with a normwise error bound that is not guaranteed (the smallest J such that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) the Jth right-hand side is the first with either a normwise or componentwise error bound that is not guaranteed (the smallest J such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1) = 0.0). See the definition of ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information about all of the right-hand sides check ERR_BNDS_NORM or ERR_BNDS_COMP.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
April 2012
subroutine zhetd2 (characterUPLO, integerN, complex*16, dimension( lda, * )A, integerLDA, double precision, dimension( * )D, double precision, dimension( * )E, complex*16, dimension( * )TAU, integerINFO)¶
ZHETD2 reduces a Hermitian matrix to real symmetric tridiagonal form by an unitary similarity transformation (unblocked algorithm). Purpose:ZHETD2 reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: Q**H * A * Q = T.
UPLO
Author:
UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored: = 'U': Upper triangular = 'L': Lower triangularN
N is INTEGER The order of the matrix A. N >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) 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 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 UPLO = 'U', the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of A are over- written by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors. See Further Details.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).D
D is DOUBLE PRECISION array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).E
E is DOUBLE PRECISION array, dimension (N-1) The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.TAU
TAU is COMPLEX*16 array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Further Details:
If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors Q = H(n-1) . . . H(2) H(1). Each H(i) has the form H(i) = I - tau * v * v**H where tau is a complex scalar, and v is a complex vector with v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in A(1:i-1,i+1), and tau in TAU(i). If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors Q = H(1) H(2) . . . H(n-1). Each H(i) has the form H(i) = I - tau * v * v**H where tau is a complex scalar, and v is a complex vector with v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), and tau in TAU(i). The contents of A on exit are illustrated by the following examples with n = 5: if UPLO = 'U': if UPLO = 'L': ( d e v2 v3 v4 ) ( d ) ( d e v3 v4 ) ( e d ) ( d e v4 ) ( v1 e d ) ( d e ) ( v1 v2 e d ) ( d ) ( v1 v2 v3 e d ) where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i).
subroutine zhetf2 (characterUPLO, integerN, complex*16, dimension( lda, * )A, integerLDA, integer, dimension( * )IPIV, integerINFO)¶
ZHETF2 computes the factorization of a complex Hermitian matrix, using the diagonal pivoting method (unblocked algorithm, calling Level 2 BLAS). Purpose:ZHETF2 computes the factorization of a complex Hermitian matrix A using the Bunch-Kaufman diagonal pivoting method: A = U*D*U**H or A = L*D*L**H where U (or L) is a product of permutation and unit upper (lower) triangular matrices, U**H is the conjugate transpose of U, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. This is the unblocked version of the algorithm, calling Level 2 BLAS.
UPLO
Author:
UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored: = 'U': Upper triangular = 'L': Lower triangularN
N is INTEGER The order of the matrix A. N >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) 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 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, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details).LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIV
IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D. If UPLO = 'U': If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L': If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal value > 0: if INFO = k, D(k,k) is exactly zero. 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.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2013
Further Details:
If UPLO = 'U', then A = U*D*U**H, where U = P(n)*U(n)* ... *P(k)U(k)* ..., i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and U(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then ( I v 0 ) k-s U(k) = ( 0 I 0 ) s ( 0 0 I ) n-k k-s s n-k If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(1:k-2,k-1:k). If UPLO = 'L', then A = L*D*L**H, where L = P(1)*L(1)* ... *P(k)*L(k)* ..., i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then ( I 0 0 ) k-1 L(k) = ( 0 I 0 ) s ( 0 v I ) n-k-s+1 k-1 s n-k-s+1 If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
09-29-06 - patch from Bobby Cheng, MathWorks Replace l.210 and l.393 IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN by IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN 01-01-96 - Based on modifications by J. Lewis, Boeing Computer Services Company A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
subroutine zhetf2_rk (characterUPLO, integerN, complex*16, dimension( lda, * )A, integerLDA, complex*16, dimension( * )E, integer, dimension( * )IPIV, integerINFO)¶
ZHETF2_RK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm). Purpose:ZHETF2_RK computes the factorization of a complex Hermitian matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method: A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T), where U (or L) is unit upper (or lower) triangular matrix, U**H (or L**H) is the conjugate of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. This is the unblocked version of the algorithm, calling Level 2 BLAS. For more information see Further Details section.
UPLO
Author:
UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored: = 'U': Upper triangular = 'L': Lower triangularN
N is INTEGER The order of the matrix A. N >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) 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 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 Hermitian 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.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).E
E is COMPLEX*16 array, dimension (N) On exit, contains the superdiagonal (or subdiagonal) elements of the Hermitian 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.IPIV
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 Hermitian 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. For more info see Further Details section. 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 matrix A(1:N,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,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 matrix A(1:N,1:N). If -IPIV(k-1) = k-1, no interchange occurred. c) In both cases a) and b), 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 matrix A(1:N,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: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 matrix A(1:N,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 matrix A(1:N,1:N). If -IPIV(k+1) = k+1, no interchange occurred. c) In both cases a) and b), always ABS( IPIV(k) ) >= k. d) NOTE: Any entry IPIV(k) is always NONZERO on output.INFO
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.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Further Details:
TODO: put further details
December 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 01-01-96 - Based on modifications by J. Lewis, Boeing Computer Services Company A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville abd , USA
subroutine zhetf2_rook (characterUPLO, integerN, complex*16, dimension( lda, * )A, integerLDA, integer, dimension( * )IPIV, integerINFO)¶
ZHETF2_ROOK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (unblocked algorithm). Purpose:ZHETF2_ROOK computes the factorization of a complex Hermitian matrix A using the bounded Bunch-Kaufman ("rook") diagonal pivoting method: A = U*D*U**H or A = L*D*L**H where U (or L) is a product of permutation and unit upper (lower) triangular matrices, U**H is the conjugate transpose of U, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. This is the unblocked version of the algorithm, calling Level 2 BLAS.
UPLO
Author:
UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored: = 'U': Upper triangular = 'L': Lower triangularN
N is INTEGER The order of the matrix A. N >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) 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 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, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details).LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIV
IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D. If UPLO = 'U': If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k-1 and -IPIV(k-1) were inerchaged, D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L': If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k+1 and -IPIV(k+1) were inerchaged, D(k:k+1,k:k+1) is a 2-by-2 diagonal block.INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal value > 0: if INFO = k, D(k,k) is exactly zero. 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.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2013
Further Details:
If UPLO = 'U', then A = U*D*U**H, where U = P(n)*U(n)* ... *P(k)U(k)* ..., i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and U(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then ( I v 0 ) k-s U(k) = ( 0 I 0 ) s ( 0 0 I ) n-k k-s s n-k If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(1:k-2,k-1:k). If UPLO = 'L', then A = L*D*L**H, where L = P(1)*L(1)* ... *P(k)*L(k)* ..., i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then ( I 0 0 ) k-1 L(k) = ( 0 I 0 ) s ( 0 v I ) n-k-s+1 k-1 s n-k-s+1 If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
November 2013, Igor Kozachenko, Computer Science Division, University of California, Berkeley September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester 01-01-96 - Based on modifications by J. Lewis, Boeing Computer Services Company A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
subroutine zhetrd (characterUPLO, integerN, complex*16, dimension( lda, * )A, integerLDA, double precision, dimension( * )D, double precision, dimension( * )E, complex*16, dimension( * )TAU, complex*16, dimension( * )WORK, integerLWORK, integerINFO)¶
ZHETRD Purpose:ZHETRD reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: Q**H * A * Q = T.
UPLO
Author:
UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.N
N is INTEGER The order of the matrix A. N >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) 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 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 UPLO = 'U', the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of A are over- written by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors. See Further Details.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).D
D is DOUBLE PRECISION array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).E
E is DOUBLE PRECISION array, dimension (N-1) The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.TAU
TAU is COMPLEX*16 array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).WORK
WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.LWORK
LWORK is INTEGER The dimension of the array WORK. LWORK >= 1. For optimum performance LWORK >= N*NB, where NB is the optimal blocksize. 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.INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Further Details:
If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors Q = H(n-1) . . . H(2) H(1). Each H(i) has the form H(i) = I - tau * v * v**H where tau is a complex scalar, and v is a complex vector with v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in A(1:i-1,i+1), and tau in TAU(i). If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors Q = H(1) H(2) . . . H(n-1). Each H(i) has the form H(i) = I - tau * v * v**H where tau is a complex scalar, and v is a complex vector with v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), and tau in TAU(i). The contents of A on exit are illustrated by the following examples with n = 5: if UPLO = 'U': if UPLO = 'L': ( d e v2 v3 v4 ) ( d ) ( d e v3 v4 ) ( e d ) ( d e v4 ) ( v1 e d ) ( d e ) ( v1 v2 e d ) ( d ) ( v1 v2 v3 e d ) where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i).
subroutine zhetrd_2stage (characterVECT, characterUPLO, integerN, complex*16, dimension( lda, * )A, integerLDA, double precision, dimension( * )D, double precision, dimension( * )E, complex*16, dimension( * )TAU, complex*16, dimension( * )HOUS2, integerLHOUS2, complex*16, dimension( * )WORK, integerLWORK, integerINFO)¶
ZHETRD_2STAGE Purpose:ZHETRD_2STAGE reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: Q1**H Q2**H* A * Q2 * Q1 = T.
VECT
Author:
VECT is CHARACTER*1 = 'N': No need for the Housholder representation, in particular for the second stage (Band to tridiagonal) and thus LHOUS2 is of size max(1, 4*N); = 'V': the Householder representation is needed to either generate Q1 Q2 or to apply Q1 Q2, then LHOUS2 is to be queried and computed. (NOT AVAILABLE IN THIS RELEASE).UPLO
UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.N
N is INTEGER The order of the matrix A. N >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) 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 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 UPLO = 'U', the band superdiagonal of A are overwritten by the corresponding elements of the internal band-diagonal matrix AB, and the elements above the KD superdiagonal, with the array TAU, represent the unitary matrix Q1 as a product of elementary reflectors; if UPLO = 'L', the diagonal and band subdiagonal of A are over- written by the corresponding elements of the internal band-diagonal matrix AB, and the elements below the KD subdiagonal, with the array TAU, represent the unitary matrix Q1 as a product of elementary reflectors. See Further Details.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).D
D is DOUBLE PRECISION array, dimension (N) The diagonal elements of the tridiagonal matrix T.E
E is DOUBLE PRECISION array, dimension (N-1) The off-diagonal elements of the tridiagonal matrix T.TAU
TAU is COMPLEX*16 array, dimension (N-KD) The scalar factors of the elementary reflectors of the first stage (see Further Details).HOUS2
HOUS2 is COMPLEX*16 array, dimension LHOUS2, that store the Householder representation of the stage2 band to tridiagonal.LHOUS2
LHOUS2 is INTEGER The dimension of the array HOUS2. LHOUS2 = MAX(1, dimension) If LWORK = -1, or LHOUS2=-1, then a query is assumed; the routine only calculates the optimal size of the HOUS2 array, returns this value as the first entry of the HOUS2 array, and no error message related to LHOUS2 is issued by XERBLA. LHOUS2 = MAX(1, dimension) where dimension = 4*N if VECT='N' not available now if VECT='H'WORK
WORK is COMPLEX*16 array, dimension LWORK.LWORK
LWORK is INTEGER The dimension of the array WORK. LWORK = MAX(1, dimension) If LWORK = -1, or LHOUS2=-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. LWORK = MAX(1, dimension) where dimension = max(stage1,stage2) + (KD+1)*N = N*KD + N*max(KD+1,FACTOPTNB) + max(2*KD*KD, KD*NTHREADS) + (KD+1)*N where KD is the blocking size of the reduction, FACTOPTNB is the blocking used by the QR or LQ algorithm, usually FACTOPTNB=128 is a good choice NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1.INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Further Details:
Implemented by Azzam Haidar. All details are available on technical report, SC11, SC13 papers. Azzam Haidar, Hatem Ltaief, and Jack Dongarra. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '11), New York, NY, USA, Article 8 , 11 pages. http://doi.acm.org/10.1145/2063384.2063394 A. Haidar, J. Kurzak, P. Luszczek, 2013. An improved parallel singular value algorithm and its implementation for multicore hardware, In Proceedings of 2013 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '13). Denver, Colorado, USA, 2013. Article 90, 12 pages. http://doi.acm.org/10.1145/2503210.2503292 A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. A novel hybrid CPU-GPU generalized eigensolver for electronic structure calculations based on fine-grained memory aware tasks. International Journal of High Performance Computing Applications. Volume 28 Issue 2, Pages 196-209, May 2014. http://hpc.sagepub.com/content/28/2/196
subroutine zhetrd_he2hb (characterUPLO, integerN, integerKD, complex*16, dimension( lda, * )A, integerLDA, complex*16, dimension( ldab, * )AB, integerLDAB, complex*16, dimension( * )TAU, complex*16, dimension( * )WORK, integerLWORK, integerINFO)¶
ZHETRD_HE2HB Purpose:ZHETRD_HE2HB reduces a complex Hermitian matrix A to complex Hermitian band-diagonal form AB by a unitary similarity transformation: Q**H * A * Q = AB.
UPLO
Author:
UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.N
N is INTEGER The order of the matrix A. N >= 0.KD
KD is INTEGER The number of superdiagonals of the reduced matrix if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0. The reduced matrix is stored in the array AB.A
A is COMPLEX*16 array, dimension (LDA,N) 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 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 UPLO = 'U', the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of A are over- written by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors. See Further Details.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).AB
AB is COMPLEX*16 array, dimension (LDAB,N) On exit, the upper or lower triangle of the Hermitian band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).LDAB
LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1.TAU
TAU is COMPLEX*16 array, dimension (N-KD) The scalar factors of the elementary reflectors (see Further Details).WORK
WORK is COMPLEX*16 array, dimension LWORK. On exit, if INFO = 0, or if LWORK=-1, WORK(1) returns the size of LWORK.LWORK
LWORK is INTEGER The dimension of the array WORK which should be calculated 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. LWORK_QUERY = N*KD + N*max(KD,FACTOPTNB) + 2*KD*KD where FACTOPTNB is the blocking used by the QR or LQ algorithm, usually FACTOPTNB=128 is a good choice otherwise putting LWORK=-1 will provide the size of WORK.INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Further Details:
Implemented by Azzam Haidar. All details are available on technical report, SC11, SC13 papers. Azzam Haidar, Hatem Ltaief, and Jack Dongarra. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '11), New York, NY, USA, Article 8 , 11 pages. http://doi.acm.org/10.1145/2063384.2063394 A. Haidar, J. Kurzak, P. Luszczek, 2013. An improved parallel singular value algorithm and its implementation for multicore hardware, In Proceedings of 2013 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '13). Denver, Colorado, USA, 2013. Article 90, 12 pages. http://doi.acm.org/10.1145/2503210.2503292 A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. A novel hybrid CPU-GPU generalized eigensolver for electronic structure calculations based on fine-grained memory aware tasks. International Journal of High Performance Computing Applications. Volume 28 Issue 2, Pages 196-209, May 2014. http://hpc.sagepub.com/content/28/2/196
If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors Q = H(k)**H . . . H(2)**H H(1)**H, where k = n-kd. Each H(i) has the form H(i) = I - tau * v * v**H where tau is a complex scalar, and v is a complex vector with v(1:i+kd-1) = 0 and v(i+kd) = 1; conjg(v(i+kd+1:n)) is stored on exit in A(i,i+kd+1:n), and tau in TAU(i). If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors Q = H(1) H(2) . . . H(k), where k = n-kd. Each H(i) has the form H(i) = I - tau * v * v**H where tau is a complex scalar, and v is a complex vector with v(kd+1:i) = 0 and v(i+kd+1) = 1; v(i+kd+2:n) is stored on exit in The contents of A on exit are illustrated by the following examples with n = 5: if UPLO = 'U': if UPLO = 'L': ( ab ab/v1 v1 v1 v1 ) ( ab ) ( ab ab/v2 v2 v2 ) ( ab/v1 ab ) ( ab ab/v3 v3 ) ( v1 ab/v2 ab ) ( ab ab/v4 ) ( v1 v2 ab/v3 ab ) ( ab ) ( v1 v2 v3 ab/v4 ab ) where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i)..fi
subroutine zhetrf (characterUPLO, integerN, complex*16, dimension( lda, * )A, integerLDA, integer, dimension( * )IPIV, complex*16, dimension( * )WORK, integerLWORK, integerINFO)¶
ZHETRF Purpose:ZHETRF computes the factorization of a complex Hermitian matrix A using the Bunch-Kaufman diagonal pivoting method. The form of the factorization is A = U*D*U**H or A = L*D*L**H where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. This is the blocked version of the algorithm, calling Level 3 BLAS.
UPLO
Author:
UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.N
N is INTEGER The order of the matrix A. N >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) 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 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, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details).LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIV
IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D. If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.WORK
WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.LWORK
LWORK is INTEGER The length of WORK. LWORK >=1. For best performance LWORK >= N*NB, where NB is the block size returned by ILAENV.INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, D(i,i) is exactly zero. 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.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Further Details:
If UPLO = 'U', then A = U*D*U**H, where U = P(n)*U(n)* ... *P(k)U(k)* ..., i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and U(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then ( I v 0 ) k-s U(k) = ( 0 I 0 ) s ( 0 0 I ) n-k k-s s n-k If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(1:k-2,k-1:k). If UPLO = 'L', then A = L*D*L**H, where L = P(1)*L(1)* ... *P(k)*L(k)* ..., i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then ( I 0 0 ) k-1 L(k) = ( 0 I 0 ) s ( 0 v I ) n-k-s+1 k-1 s n-k-s+1 If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
subroutine zhetrf_aa (characterUPLO, integerN, complex*16, dimension( lda, * )A, integerLDA, integer, dimension( * )IPIV, complex*16, dimension( * )WORK, integerLWORK, integerINFO)¶
ZHETRF_AA Purpose:ZHETRF_AA computes the factorization of a complex hermitian matrix A using the Aasen's algorithm. The form of the factorization is A = U*T*U**H or A = L*T*L**H where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and T is a hermitian tridiagonal matrix. This is the blocked version of the algorithm, calling Level 3 BLAS.
UPLO
Author:
UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.N
N is INTEGER The order of the matrix A. N >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) 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 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, the tridiagonal matrix is stored in the diagonals and the subdiagonals of A just below (or above) the diagonals, and L is stored below (or above) the subdiaonals, when UPLO is 'L' (or 'U').LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIV
IPIV is INTEGER array, dimension (N) On exit, it contains the details of the interchanges, i.e., the row and column k of A were interchanged with the row and column IPIV(k).WORK
WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.LWORK
LWORK is INTEGER The length of WORK. LWORK >= MAX(1,2*N). For optimum performance LWORK >= N*(1+NB), where NB is the optimal blocksize. 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.INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, D(i,i) is exactly zero. 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.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
subroutine zhetrf_rk (characterUPLO, integerN, complex*16, dimension( lda, * )A, integerLDA, complex*16, dimension( * )E, integer, dimension( * )IPIV, complex*16, dimension( * )WORK, integerLWORK, integerINFO)¶
ZHETRF_RK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm). Purpose:ZHETRF_RK computes the factorization of a complex Hermitian matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method: A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T), where U (or L) is unit upper (or lower) triangular matrix, U**H (or L**H) is the conjugate of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. This is the blocked version of the algorithm, calling Level 3 BLAS. For more information see Further Details section.
UPLO
Author:
UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored: = 'U': Upper triangular = 'L': Lower triangularN
N is INTEGER The order of the matrix A. N >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) 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 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 Hermitian 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.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).E
E is COMPLEX*16 array, dimension (N) On exit, contains the superdiagonal (or subdiagonal) elements of the Hermitian 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.IPIV
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 Hermitian 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. For more info see Further Details section. 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 matrix A(1:N,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,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 matrix A(1:N,1:N). If -IPIV(k-1) = k-1, no interchange occurred. c) In both cases a) and b), 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 matrix A(1:N,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: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 matrix A(1:N,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 matrix A(1:N,1:N). If -IPIV(k+1) = k+1, no interchange occurred. c) In both cases a) and b), always ABS( IPIV(k) ) >= k. d) NOTE: Any entry IPIV(k) is always NONZERO on output.WORK
WORK is COMPLEX*16 array, dimension ( MAX(1,LWORK) ). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.LWORK
LWORK is INTEGER The length of WORK. LWORK >=1. For best performance LWORK >= N*NB, where NB is the block size returned by ILAENV. 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.INFO
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.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Further Details:
TODO: put correct description
December 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
subroutine zhetrf_rook (characterUPLO, integerN, complex*16, dimension( lda, * )A, integerLDA, integer, dimension( * )IPIV, complex*16, dimension( * )WORK, integerLWORK, integerINFO)¶
ZHETRF_ROOK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (blocked algorithm, calling Level 3 BLAS). Purpose:ZHETRF_ROOK computes the factorization of a complex Hermitian matrix A using the bounded Bunch-Kaufman ("rook") diagonal pivoting method. The form of the factorization is A = U*D*U**T or A = L*D*L**T where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. This is the blocked version of the algorithm, calling Level 3 BLAS.
UPLO
Author:
UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.N
N is INTEGER The order of the matrix A. N >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) 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 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, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details).LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIV
IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D. If UPLO = 'U': Only the last KB elements of IPIV are set. If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k-1 and -IPIV(k-1) were inerchaged, D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L': Only the first KB elements of IPIV are set. If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k+1 and -IPIV(k+1) were inerchaged, D(k:k+1,k:k+1) is a 2-by-2 diagonal block.WORK
WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.LWORK
LWORK is INTEGER The length of WORK. LWORK >=1. For best performance LWORK >= N*NB, where NB is the block size returned by ILAENV. 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.INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, D(i,i) is exactly zero. 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.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
June 2016
Further Details:
If UPLO = 'U', then A = U*D*U**T, where U = P(n)*U(n)* ... *P(k)U(k)* ..., i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and U(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then ( I v 0 ) k-s U(k) = ( 0 I 0 ) s ( 0 0 I ) n-k k-s s n-k If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(1:k-2,k-1:k). If UPLO = 'L', then A = L*D*L**T, where L = P(1)*L(1)* ... *P(k)*L(k)* ..., i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then ( I 0 0 ) k-1 L(k) = ( 0 I 0 ) s ( 0 v I ) n-k-s+1 k-1 s n-k-s+1 If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
June 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
subroutine zhetri (characterUPLO, integerN, complex*16, dimension( lda, * )A, integerLDA, integer, dimension( * )IPIV, complex*16, dimension( * )WORK, integerINFO)¶
ZHETRI Purpose:ZHETRI computes the inverse of a complex Hermitian indefinite matrix A using the factorization A = U*D*U**H or A = L*D*L**H computed by ZHETRF.
UPLO
Author:
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*D*U**H; = 'L': Lower triangular, form is A = L*D*L**H.N
N is INTEGER The order of the matrix A. N >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) On entry, the block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZHETRF. On exit, if INFO = 0, the (Hermitian) inverse of the original matrix. If UPLO = 'U', the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced; if UPLO = 'L' the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIV
IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by ZHETRF.WORK
WORK is COMPLEX*16 array, dimension (N)INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
subroutine zhetri2 (characterUPLO, integerN, complex*16, dimension( lda, * )A, integerLDA, integer, dimension( * )IPIV, complex*16, dimension( * )WORK, integerLWORK, integerINFO)¶
ZHETRI2 Purpose:ZHETRI2 computes the inverse of a COMPLEX*16 hermitian indefinite matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by ZHETRF. ZHETRI2 set the LEADING DIMENSION of the workspace before calling ZHETRI2X that actually computes the inverse.
UPLO
Author:
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*D*U**T; = 'L': Lower triangular, form is A = L*D*L**T.N
N is INTEGER The order of the matrix A. N >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) On entry, the NB diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZHETRF. On exit, if INFO = 0, the (symmetric) inverse of the original matrix. If UPLO = 'U', the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced; if UPLO = 'L' the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIV
IPIV is INTEGER array, dimension (N) Details of the interchanges and the NB structure of D as determined by ZHETRF.WORK
WORK is COMPLEX*16 array, dimension (N+NB+1)*(NB+3)LWORK
LWORK is INTEGER The dimension of the array WORK. WORK is size >= (N+NB+1)*(NB+3) If LWORK = -1, then a workspace query is assumed; the routine 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.INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
subroutine zhetri2x (characterUPLO, integerN, complex*16, dimension( lda, * )A, integerLDA, integer, dimension( * )IPIV, complex*16, dimension( n+nb+1,* )WORK, integerNB, integerINFO)¶
ZHETRI2X Purpose:ZHETRI2X computes the inverse of a COMPLEX*16 Hermitian indefinite matrix A using the factorization A = U*D*U**H or A = L*D*L**H computed by ZHETRF.
UPLO
Author:
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*D*U**H; = 'L': Lower triangular, form is A = L*D*L**H.N
N is INTEGER The order of the matrix A. N >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) On entry, the NNB diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZHETRF. On exit, if INFO = 0, the (symmetric) inverse of the original matrix. If UPLO = 'U', the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced; if UPLO = 'L' the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIV
IPIV is INTEGER array, dimension (N) Details of the interchanges and the NNB structure of D as determined by ZHETRF.WORK
WORK is COMPLEX*16 array, dimension (N+NB+1,NB+3)NB
NB is INTEGER Block sizeINFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
subroutine zhetri_3 (characterUPLO, integerN, complex*16, dimension( lda, * )A, integerLDA, complex*16, dimension( * )E, integer, dimension( * )IPIV, complex*16, dimension( * )WORK, integerLWORK, integerINFO)¶
ZHETRI_3 Purpose:ZHETRI_3 computes the inverse of a complex Hermitian indefinite matrix A using the factorization computed by ZHETRF_RK or ZHETRF_BK: A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T), where U (or L) is unit upper (or lower) triangular matrix, U**H (or L**H) is the conjugate of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. ZHETRI_3 sets the leading dimension of the workspace before calling ZHETRI_3X that actually computes the inverse. This is the blocked version of the algorithm, calling Level 3 BLAS.
UPLO
Author:
UPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix. = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.N
N is INTEGER The order of the matrix A. N >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) On entry, diagonal of the block diagonal matrix D and factors U or L as computed by ZHETRF_RK and ZHETRF_BK: a) ONLY diagonal elements of the Hermitian block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D should be provided on entry 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. On exit, if INFO = 0, the Hermitian inverse of the original matrix. If UPLO = 'U': the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced; If UPLO = 'L': the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).E
E is COMPLEX*16 array, dimension (N) On entry, contains the superdiagonal (or subdiagonal) elements of the Hermitian 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) not referenced; If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced. NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is not referenced in both UPLO = 'U' or UPLO = 'L' cases.IPIV
IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by ZHETRF_RK or ZHETRF_BK.WORK
WORK is COMPLEX*16 array, dimension (N+NB+1)*(NB+3). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.LWORK
LWORK is INTEGER The length of WORK. LWORK >= (N+NB+1)*(NB+3). If LDWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of 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.INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Contributors:
subroutine zhetri_3x (characterUPLO, integerN, complex*16, dimension( lda, * )A, integerLDA, complex*16, dimension( * )E, integer, dimension( * )IPIV, complex*16, dimension( n+nb+1, * )WORK, integerNB, integerINFO)¶
ZHETRI_3X Purpose:ZHETRI_3X computes the inverse of a complex Hermitian indefinite matrix A using the factorization computed by ZHETRF_RK or ZHETRF_BK: A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T), where U (or L) is unit upper (or lower) triangular matrix, U**H (or L**H) is the conjugate of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. This is the blocked version of the algorithm, calling Level 3 BLAS.
UPLO
Author:
UPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix. = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.N
N is INTEGER The order of the matrix A. N >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) On entry, diagonal of the block diagonal matrix D and factors U or L as computed by ZHETRF_RK and ZHETRF_BK: a) ONLY diagonal elements of the Hermitian block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D should be provided on entry 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. On exit, if INFO = 0, the Hermitian inverse of the original matrix. If UPLO = 'U': the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced; If UPLO = 'L': the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).E
E is COMPLEX*16 array, dimension (N) On entry, contains the superdiagonal (or subdiagonal) elements of the Hermitian 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) not referenced; If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) not referenced. NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is not referenced in both UPLO = 'U' or UPLO = 'L' cases.IPIV
IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by ZHETRF_RK or ZHETRF_BK.WORK
WORK is COMPLEX*16 array, dimension (N+NB+1,NB+3).NB
NB is INTEGER Block size.INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Contributors:
subroutine zhetri_rook (characterUPLO, integerN, complex*16, dimension( lda, * )A, integerLDA, integer, dimension( * )IPIV, complex*16, dimension( * )WORK, integerINFO)¶
ZHETRI_ROOK computes the inverse of HE matrix using the factorization obtained with the bounded Bunch-Kaufman ('rook') diagonal pivoting method. Purpose:ZHETRI_ROOK computes the inverse of a complex Hermitian indefinite matrix A using the factorization A = U*D*U**H or A = L*D*L**H computed by ZHETRF_ROOK.
UPLO
Author:
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*D*U**H; = 'L': Lower triangular, form is A = L*D*L**H.N
N is INTEGER The order of the matrix A. N >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) On entry, the block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZHETRF_ROOK. On exit, if INFO = 0, the (Hermitian) inverse of the original matrix. If UPLO = 'U', the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced; if UPLO = 'L' the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIV
IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by ZHETRF_ROOK.WORK
WORK is COMPLEX*16 array, dimension (N)INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2013
Contributors:
November 2013, Igor Kozachenko, Computer Science Division, University of California, Berkeley September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester
subroutine zhetrs (characterUPLO, integerN, integerNRHS, complex*16, dimension( lda, * )A, integerLDA, integer, dimension( * )IPIV, complex*16, dimension( ldb, * )B, integerLDB, integerINFO)¶
ZHETRS Purpose:ZHETRS solves a system of linear equations A*X = B with a complex Hermitian matrix A using the factorization A = U*D*U**H or A = L*D*L**H computed by ZHETRF.
UPLO
Author:
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*D*U**H; = 'L': Lower triangular, form is A = L*D*L**H.N
N is INTEGER The order of the matrix A. N >= 0.NRHS
NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZHETRF.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIV
IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by ZHETRF.B
B is COMPLEX*16 array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, the solution matrix X.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
subroutine zhetrs2 (characterUPLO, integerN, integerNRHS, complex*16, dimension( lda, * )A, integerLDA, integer, dimension( * )IPIV, complex*16, dimension( ldb, * )B, integerLDB, complex*16, dimension( * )WORK, integerINFO)¶
ZHETRS2 Purpose:ZHETRS2 solves a system of linear equations A*X = B with a complex Hermitian matrix A using the factorization A = U*D*U**H or A = L*D*L**H computed by ZHETRF and converted by ZSYCONV.
UPLO
Author:
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*D*U**H; = 'L': Lower triangular, form is A = L*D*L**H.N
N is INTEGER The order of the matrix A. N >= 0.NRHS
NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZHETRF.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIV
IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by ZHETRF.B
B is COMPLEX*16 array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, the solution matrix X.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).WORK
WORK is COMPLEX*16 array, dimension (N)INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
June 2016
subroutine zhetrs_3 (characterUPLO, integerN, integerNRHS, complex*16, dimension( lda, * )A, integerLDA, complex*16, dimension( * )E, integer, dimension( * )IPIV, complex*16, dimension( ldb, * )B, integerLDB, integerINFO)¶
ZHETRS_3 Purpose:ZHETRS_3 solves a system of linear equations A * X = B with a complex Hermitian matrix A using the factorization computed by ZHETRF_RK or ZHETRF_BK: A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T), where U (or L) is unit upper (or lower) triangular matrix, U**H (or L**H) is the conjugate of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. This algorithm is using Level 3 BLAS.
UPLO
Author:
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 = P*U*D*(U**H)*(P**T); = 'L': Lower triangular, form is A = P*L*D*(L**H)*(P**T).N
N is INTEGER The order of the matrix A. N >= 0.NRHS
NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) Diagonal of the block diagonal matrix D and factors U or L as computed by ZHETRF_RK and ZHETRF_BK: a) ONLY diagonal elements of the Hermitian block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D should be provided on entry 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.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).E
E is COMPLEX*16 array, dimension (N) On entry, contains the superdiagonal (or subdiagonal) elements of the Hermitian 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) not referenced; If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced. NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is not referenced in both UPLO = 'U' or UPLO = 'L' cases.IPIV
IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by ZHETRF_RK or ZHETRF_BK.B
B is COMPLEX*16 array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, the solution matrix X.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Contributors:
December 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
subroutine zhetrs_aa (characterUPLO, integerN, integerNRHS, complex*16, dimension( lda, * )A, integerLDA, integer, dimension( * )IPIV, complex*16, dimension( ldb, * )B, integerLDB, complex*16, dimension( * )WORK, integerLWORK, integerINFO)¶
ZHETRS_AA Purpose:ZHETRS_AA solves a system of linear equations A*X = B with a complex hermitian matrix A using the factorization A = U*T*U**H or A = L*T*L**T computed by ZHETRF_AA.
UPLO
Author:
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**H; = 'L': Lower triangular, form is A = L*T*L**H.N
N is INTEGER The order of the matrix A. N >= 0.NRHS
NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) Details of factors computed by ZHETRF_AA.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIV
IPIV is INTEGER array, dimension (N) Details of the interchanges as computed by ZHETRF_AA.B
B is COMPLEX*16 array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, the solution matrix X.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).WORK
WORK is DOUBLE array, dimension (MAX(1,LWORK))LWORK
LWORK is INTEGER, LWORK >= MAX(1,3*N-2). aram[out]INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
subroutine zhetrs_rook (characterUPLO, integerN, integerNRHS, complex*16, dimension( lda, * )A, integerLDA, integer, dimension( * )IPIV, complex*16, dimension( ldb, * )B, integerLDB, integerINFO)¶
ZHETRS_ROOK computes the solution to a system of linear equations A * X = B for HE matrices using factorization obtained with one of the bounded diagonal pivoting methods (max 2 interchanges) Purpose:ZHETRS_ROOK solves a system of linear equations A*X = B with a complex Hermitian matrix A using the factorization A = U*D*U**H or A = L*D*L**H computed by ZHETRF_ROOK.
UPLO
Author:
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*D*U**H; = 'L': Lower triangular, form is A = L*D*L**H.N
N is INTEGER The order of the matrix A. N >= 0.NRHS
NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZHETRF_ROOK.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIV
IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by ZHETRF_ROOK.B
B is COMPLEX*16 array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, the solution matrix X.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2013
Contributors:
November 2013, Igor Kozachenko, Computer Science Division, University of California, Berkeley September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester
subroutine zla_heamv (integerUPLO, integerN, double precisionALPHA, complex*16, dimension( lda, * )A, integerLDA, complex*16, dimension( * )X, integerINCX, double precisionBETA, double precision, dimension( * )Y, integerINCY)¶
ZLA_HEAMV computes a matrix-vector product using a Hermitian indefinite matrix to calculate error bounds. Purpose:ZLA_SYAMV performs the matrix-vector operation y := alpha*abs(A)*abs(x) + beta*abs(y), where alpha and beta are scalars, x and y are vectors and A is an n by n symmetric matrix. This function is primarily used in calculating error bounds. To protect against underflow during evaluation, components in the resulting vector are perturbed away from zero by (N+1) times the underflow threshold. To prevent unnecessarily large errors for block-structure embedded in general matrices, "symbolically" zero components are not perturbed. A zero entry is considered "symbolic" if all multiplications involved in computing that entry have at least one zero multiplicand.
UPLO
Author:
UPLO is INTEGER On entry, UPLO specifies whether the upper or lower triangular part of the array A is to be referenced as follows: UPLO = BLAS_UPPER Only the upper triangular part of A is to be referenced. UPLO = BLAS_LOWER Only the lower triangular part of A is to be referenced. Unchanged on exit.N
N is INTEGER On entry, N specifies the number of columns of the matrix A. N must be at least zero. Unchanged on exit.ALPHA
ALPHA is DOUBLE PRECISION . On entry, ALPHA specifies the scalar alpha. Unchanged on exit.A
A is COMPLEX*16 array, DIMENSION ( LDA, n ). Before entry, the leading m by n part of the array A must contain the matrix of coefficients. Unchanged on exit.LDA
LDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, n ). Unchanged on exit.X
X is COMPLEX*16 array, DIMENSION at least ( 1 + ( n - 1 )*abs( INCX ) ) Before entry, the incremented array X must contain the vector x. Unchanged on exit.INCX
INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. Unchanged on exit.BETA
BETA is DOUBLE PRECISION . On entry, BETA specifies the scalar beta. When BETA is supplied as zero then Y need not be set on input. Unchanged on exit.Y
Y is DOUBLE PRECISION array, dimension ( 1 + ( n - 1 )*abs( INCY ) ) Before entry with BETA non-zero, the incremented array Y must contain the vector y. On exit, Y is overwritten by the updated vector y.INCY
INCY is INTEGER On entry, INCY specifies the increment for the elements of Y. INCY must not be zero. Unchanged on exit.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Further Details:
Level 2 Blas routine. -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. -- Modified for the absolute-value product, April 2006 Jason Riedy, UC Berkeley
double precision function zla_hercond_c (characterUPLO, integerN, complex*16, dimension( lda, * )A, integerLDA, complex*16, dimension( ldaf, * )AF, integerLDAF, integer, dimension( * )IPIV, double precision, dimension ( * )C, logicalCAPPLY, integerINFO, complex*16, dimension( * )WORK, double precision, dimension( * )RWORK)¶
ZLA_HERCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for Hermitian indefinite matrices. Purpose:ZLA_HERCOND_C computes the infinity norm condition number of op(A) * inv(diag(C)) where C is a DOUBLE PRECISION vector.
UPLO
Author:
UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.N
N is INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) On entry, the N-by-N matrix ALDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).AF
AF is COMPLEX*16 array, dimension (LDAF,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZHETRF.LDAF
LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).IPIV
IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by CHETRF.C
C is DOUBLE PRECISION array, dimension (N) The vector C in the formula op(A) * inv(diag(C)).CAPPLY
CAPPLY is LOGICAL If .TRUE. then access the vector C in the formula above.INFO
INFO is INTEGER = 0: Successful exit. i > 0: The ith argument is invalid.WORK
WORK is COMPLEX*16 array, dimension (2*N). Workspace.RWORK
RWORK is DOUBLE PRECISION array, dimension (N). Workspace.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
double precision function zla_hercond_x (characterUPLO, integerN, complex*16, dimension( lda, * )A, integerLDA, complex*16, dimension( ldaf, * )AF, integerLDAF, integer, dimension( * )IPIV, complex*16, dimension( * )X, integerINFO, complex*16, dimension( * )WORK, double precision, dimension( * )RWORK)¶
ZLA_HERCOND_X computes the infinity norm condition number of op(A)*diag(x) for Hermitian indefinite matrices. Purpose:ZLA_HERCOND_X computes the infinity norm condition number of op(A) * diag(X) where X is a COMPLEX*16 vector.
UPLO
Author:
UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.N
N is INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0.A
A is COMPLEX*16 array, dimension (LDA,N) On entry, the N-by-N matrix A.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).AF
AF is COMPLEX*16 array, dimension (LDAF,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZHETRF.LDAF
LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).IPIV
IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by CHETRF.X
X is COMPLEX*16 array, dimension (N) The vector X in the formula op(A) * diag(X).INFO
INFO is INTEGER = 0: Successful exit. i > 0: The ith argument is invalid.WORK
WORK is COMPLEX*16 array, dimension (2*N). Workspace.RWORK
RWORK is DOUBLE PRECISION array, dimension (N). Workspace.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
subroutine zla_herfsx_extended (integerPREC_TYPE, characterUPLO, integerN, integerNRHS, complex*16, dimension( lda, * )A, integerLDA, complex*16, dimension( ldaf, * )AF, integerLDAF, integer, dimension( * )IPIV, logicalCOLEQU, double precision, dimension( * )C, complex*16, dimension( ldb, * )B, integerLDB, complex*16, dimension( ldy, * )Y, integerLDY, double precision, dimension( * )BERR_OUT, integerN_NORMS, double precision, dimension( nrhs, * )ERR_BNDS_NORM, double precision, dimension( nrhs, * )ERR_BNDS_COMP, complex*16, dimension( * )RES, double precision, dimension( * )AYB, complex*16, dimension( * )DY, complex*16, dimension( * )Y_TAIL, double precisionRCOND, integerITHRESH, double precisionRTHRESH, double precisionDZ_UB, logicalIGNORE_CWISE, integerINFO)¶
ZLA_HERFSX_EXTENDED improves the computed solution to a system of linear equations for Hermitian indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. Purpose:ZLA_HERFSX_EXTENDED improves the computed solution to a system of linear equations by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. This subroutine is called by ZHERFSX to perform iterative refinement. In addition to normwise error bound, the code provides maximum componentwise error bound if possible. See comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the error bounds. Note that this subroutine is only resonsible for setting the second fields of ERR_BNDS_NORM and ERR_BNDS_COMP.
PREC_TYPE
Author:
PREC_TYPE is INTEGER Specifies the intermediate precision to be used in refinement. The value is defined by ILAPREC(P) where P is a CHARACTER and P = 'S': Single = 'D': Double = 'I': Indigenous = 'X', 'E': ExtraUPLO
UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.N
N is INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0.NRHS
NRHS is INTEGER The number of right-hand-sides, i.e., the number of columns of the matrix B.A
A is COMPLEX*16 array, dimension (LDA,N) On entry, the N-by-N matrix A.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).AF
AF is COMPLEX*16 array, dimension (LDAF,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZHETRF.LDAF
LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).IPIV
IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by ZHETRF.COLEQU
COLEQU is LOGICAL If .TRUE. then column equilibration was done to A before calling this routine. This is needed to compute the solution and error bounds correctly.C
C is DOUBLE PRECISION array, dimension (N) The column scale factors for A. If COLEQU = .FALSE., C is not accessed. If C is input, each element of C should be a power of the radix to ensure a reliable solution and error estimates. Scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows. Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable.B
B is COMPLEX*16 array, dimension (LDB,NRHS) The right-hand-side matrix B.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).Y
Y is COMPLEX*16 array, dimension (LDY,NRHS) On entry, the solution matrix X, as computed by ZHETRS. On exit, the improved solution matrix Y.LDY
LDY is INTEGER The leading dimension of the array Y. LDY >= max(1,N).BERR_OUT
BERR_OUT is DOUBLE PRECISION array, dimension (NRHS) On exit, BERR_OUT(j) contains the componentwise relative backward error for right-hand-side j from the formula max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) where abs(Z) is the componentwise absolute value of the matrix or vector Z. This is computed by ZLA_LIN_BERR.N_NORMS
N_NORMS is INTEGER Determines which error bounds to return (see ERR_BNDS_NORM and ERR_BNDS_COMP). If N_NORMS >= 1 return normwise error bounds. If N_NORMS >= 2 return componentwise error bounds.ERR_BNDS_NORM
ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the normwise relative error, which is defined as follows: Normwise relative error in the ith solution vector: max_j (abs(XTRUE(j,i) - X(j,i))) ------------------------------ max_j abs(X(j,i)) The array is indexed by the type of error information as described below. There currently are up to three pieces of information returned. The first index in ERR_BNDS_NORM(i,:) corresponds to the ith right-hand side. The second index in ERR_BNDS_NORM(:,err) contains the following three fields: err = 1 "Trust/don't trust" boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon'). err = 2 "Guaranteed" error bound: The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true. err = 3 Reciprocal condition number: Estimated normwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is "guaranteed". These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*A, where S scales each row by a power of the radix so all absolute row sums of Z are approximately 1. This subroutine is only responsible for setting the second field above. See Lapack Working Note 165 for further details and extra cautions.ERR_BNDS_COMP
ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the componentwise relative error, which is defined as follows: Componentwise relative error in the ith solution vector: abs(XTRUE(j,i) - X(j,i)) max_j ---------------------- abs(X(j,i)) The array is indexed by the right-hand side i (on which the componentwise relative error depends), and the type of error 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 the first (:,N_ERR_BNDS) entries are returned. The first index in ERR_BNDS_COMP(i,:) corresponds to the ith right-hand side. The second index in ERR_BNDS_COMP(:,err) contains the following three fields: err = 1 "Trust/don't trust" boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon'). err = 2 "Guaranteed" error bound: The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true. err = 3 Reciprocal condition number: Estimated componentwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is "guaranteed". These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*(A*diag(x)), where x is the solution for the current right-hand side and S scales each row of A*diag(x) by a power of the radix so all absolute row sums of Z are approximately 1. This subroutine is only responsible for setting the second field above. See Lapack Working Note 165 for further details and extra cautions.RES
RES is COMPLEX*16 array, dimension (N) Workspace to hold the intermediate residual.AYB
AYB is DOUBLE PRECISION array, dimension (N) Workspace.DY
DY is COMPLEX*16 array, dimension (N) Workspace to hold the intermediate solution.Y_TAIL
Y_TAIL is COMPLEX*16 array, dimension (N) Workspace to hold the trailing bits of the intermediate solution.RCOND
RCOND is DOUBLE PRECISION Reciprocal scaled condition number. This is an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if done). If this is less than the machine precision (in particular, if it is zero), the matrix is singular to working precision. Note that the error may still be small even if this number is very small and the matrix appears ill- conditioned.ITHRESH
ITHRESH is INTEGER The maximum number of residual computations allowed for refinement. The default is 10. For 'aggressive' set to 100 to permit convergence using approximate factorizations or factorizations other than LU. If the factorization uses a technique other than Gaussian elimination, the guarantees in ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy.RTHRESH
RTHRESH is DOUBLE PRECISION Determines when to stop refinement if the error estimate stops decreasing. Refinement will stop when the next solution no longer satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The default value is 0.5. For 'aggressive' set to 0.9 to permit convergence on extremely ill-conditioned matrices. See LAWN 165 for more details.DZ_UB
DZ_UB is DOUBLE PRECISION Determines when to start considering componentwise convergence. Componentwise convergence is only considered after each component of the solution Y is stable, which we definte as the relative change in each component being less than DZ_UB. The default value is 0.25, requiring the first bit to be stable. See LAWN 165 for more details.IGNORE_CWISE
IGNORE_CWISE is LOGICAL If .TRUE. then ignore componentwise convergence. Default value is .FALSE..INFO
INFO is INTEGER = 0: Successful exit. < 0: if INFO = -i, the ith argument to ZLA_HERFSX_EXTENDED had an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
double precision function zla_herpvgrw (character*1UPLO, integerN, integerINFO, complex*16, dimension( lda, * )A, integerLDA, complex*16, dimension( ldaf, * )AF, integerLDAF, integer, dimension( * )IPIV, double precision, dimension( * )WORK)¶
ZLA_HERPVGRW Purpose:ZLA_HERPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U). The "max absolute element" norm is used. If this is much less than 1, the stability of the LU factorization of the (equilibrated) matrix A could be poor. This also means that the solution X, estimated condition numbers, and error bounds could be unreliable.
UPLO
Author:
UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.N
N is INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0.INFO
INFO is INTEGER The value of INFO returned from ZHETRF, .i.e., the pivot in column INFO is exactly 0.A
A is COMPLEX*16 array, dimension (LDA,N) On entry, the N-by-N matrix A.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).AF
AF is COMPLEX*16 array, dimension (LDAF,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZHETRF.LDAF
LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).IPIV
IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by ZHETRF.WORK
WORK is DOUBLE PRECISION array, dimension (2*N)
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
June 2016
subroutine zlahef (characterUPLO, integerN, integerNB, integerKB, complex*16, dimension( lda, * )A, integerLDA, integer, dimension( * )IPIV, complex*16, dimension( ldw, * )W, integerLDW, integerINFO)¶
ZLAHEF computes a partial factorization of a complex Hermitian indefinite matrix using the Bunch-Kaufman diagonal pivoting method (blocked algorithm, calling Level 3 BLAS). Purpose:ZLAHEF computes a partial factorization of a complex Hermitian matrix A using the Bunch-Kaufman 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**H U22**H ) A = ( L11 0 ) ( D 0 ) ( L11**H L21**H ) 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. Note that U**H denotes the conjugate transpose of U. ZLAHEF is an auxiliary routine called by ZHETRF. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L').
UPLO
Author:
UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored: = 'U': Upper triangular = 'L': Lower triangularN
N is INTEGER The order of the matrix A. N >= 0.NB
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.KB
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.A
A is COMPLEX*16 array, dimension (LDA,N) 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 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, A contains details of the partial factorization.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIV
IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D. If UPLO = 'U': Only the last KB elements of IPIV are set. If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L': Only the first KB elements of IPIV are set. If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.W
W is COMPLEX*16 array, dimension (LDW,NB)LDW
LDW is INTEGER The leading dimension of the array W. LDW >= max(1,N).INFO
INFO is INTEGER = 0: successful exit > 0: if INFO = k, D(k,k) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Contributors:
December 2016, Igor Kozachenko, Computer Science Division, University of California, Berkeley
subroutine zlahef_aa (characterUPLO, integerJ1, integerM, integerNB, complex*16, dimension( lda, * )A, integerLDA, integer, dimension( * )IPIV, complex*16, dimension( ldh, * )H, integerLDH, complex*16, dimension( * )WORK, integerINFO)¶
ZLAHEF_AA Purpose:DLAHEF_AA factorizes a panel of a complex hermitian matrix A using the Aasen's algorithm. The panel consists of a set of NB rows of A when UPLO is U, or a set of NB columns when UPLO is L. In order to factorize the panel, the Aasen's algorithm requires the last row, or column, of the previous panel. The first row, or column, of A is set to be the first row, or column, of an identity matrix, which is used to factorize the first panel. The resulting J-th row of U, or J-th column of L, is stored in the (J-1)-th row, or column, of A (without the unit diagonals), while the diagonal and subdiagonal of A are overwritten by those of T.
UPLO
Author:
UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.J1
J1 is INTEGER The location of the first row, or column, of the panel within the submatrix of A, passed to this routine, e.g., when called by ZHETRF_AA, for the first panel, J1 is 1, while for the remaining panels, J1 is 2.M
M is INTEGER The dimension of the submatrix. M >= 0.NB
NB is INTEGER The dimension of the panel to be facotorized.A
A is COMPLEX*16 array, dimension (LDA,M) for the first panel, while dimension (LDA,M+1) for the remaining panels. On entry, A contains the last row, or column, of the previous panel, and the trailing submatrix of A to be factorized, except for the first panel, only the panel is passed. On exit, the leading panel is factorized.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIV
IPIV is INTEGER array, dimension (N) Details of the row and column interchanges, the row and column k were interchanged with the row and column IPIV(k).H
H is COMPLEX*16 workspace, dimension (LDH,NB).LDH
LDH is INTEGER The leading dimension of the workspace H. LDH >= max(1,M).WORK
WORK is COMPLEX*16 workspace, dimension (M).INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, D(i,i) is exactly zero. 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.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
subroutine zlahef_rk (characterUPLO, integerN, integerNB, integerKB, complex*16, dimension( lda, * )A, integerLDA, complex*16, dimension( * )E, integer, dimension( * )IPIV, complex*16, dimension( ldw, * )W, integerLDW, integerINFO)¶
ZLAHEF_RK computes a partial factorization of a complex Hermitian indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method. Purpose:ZLAHEF_RK computes a partial factorization of a complex Hermitian 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**H U22**H ) A = ( L11 0 ) ( D 0 ) ( L11**H L21**H ) 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. ZLAHEF_RK is an auxiliary routine called by ZHETRF_RK. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L').
UPLO
Author:
UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored: = 'U': Upper triangular = 'L': Lower triangularN
N is INTEGER The order of the matrix A. N >= 0.NB
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.KB
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.A
A is COMPLEX*16 array, dimension (LDA,N) 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 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 Hermitian 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.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).E
E is COMPLEX*16 array, dimension (N) On exit, contains the superdiagonal (or subdiagonal) elements of the Hermitian 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.IPIV
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 Hermitian 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.W
W is COMPLEX*16 array, dimension (LDW,NB)LDW
LDW is INTEGER The leading dimension of the array W. LDW >= max(1,N).INFO
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.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Contributors:
December 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
subroutine zlahef_rook (characterUPLO, integerN, integerNB, integerKB, complex*16, dimension( lda, * )A, integerLDA, integer, dimension( * )IPIV, complex*16, dimension( ldw, * )W, integerLDW, integerINFO)¶
Purpose:ZLAHEF_ROOK computes a partial factorization of a complex Hermitian 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**H U22**H ) A = ( L11 0 ) ( D 0 ) ( L11**H L21**H ) 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. Note that U**H denotes the conjugate transpose of U. ZLAHEF_ROOK is an auxiliary routine called by ZHETRF_ROOK. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L').
UPLO
Author:
UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored: = 'U': Upper triangular = 'L': Lower triangularN
N is INTEGER The order of the matrix A. N >= 0.NB
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.KB
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.A
A is COMPLEX*16 array, dimension (LDA,N) 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 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, A contains details of the partial factorization.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIV
IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D. If UPLO = 'U': Only the last KB elements of IPIV are set. If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k-1 and -IPIV(k-1) were inerchaged, D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L': Only the first KB elements of IPIV are set. If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k+1 and -IPIV(k+1) were inerchaged, D(k:k+1,k:k+1) is a 2-by-2 diagonal block.W
W is COMPLEX*16 array, dimension (LDW,NB)LDW
LDW is INTEGER The leading dimension of the array W. LDW >= max(1,N).INFO
INFO is INTEGER = 0: successful exit > 0: if INFO = k, D(k,k) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2013
Contributors:
November 2013, Igor Kozachenko, Computer Science Division, University of California, Berkeley September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester
Author¶
Generated automatically by Doxygen for LAPACK from the source code.Wed Mar 8 2017 | Version 3.7.0 |