 |
LAPACK
3.4.2
LAPACK: Linear Algebra PACKage
|
Functions/Subroutines | |
| subroutine | sla_syamv (UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY) |
| SLA_SYAMV computes a matrix-vector product using a symmetric indefinite matrix to calculate error bounds. | |
| REAL function | sla_syrcond (UPLO, N, A, LDA, AF, LDAF, IPIV, CMODE, C, INFO, WORK, IWORK) |
| SLA_SYRCOND estimates the Skeel condition number for a symmetric indefinite matrix. | |
| subroutine | sla_syrfsx_extended (PREC_TYPE, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, COLEQU, C, B, LDB, Y, LDY, BERR_OUT, N_NORMS, ERR_BNDS_NORM, ERR_BNDS_COMP, RES, AYB, DY, Y_TAIL, RCOND, ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE, INFO) |
| SLA_SYRFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. | |
| REAL function | sla_syrpvgrw (UPLO, N, INFO, A, LDA, AF, LDAF, IPIV, WORK) |
| SLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite matrix. | |
| subroutine | slasyf (UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO) |
| SLASYF computes a partial factorization of a real symmetric matrix, using the diagonal pivoting method. | |
| subroutine | ssycon (UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, IWORK, INFO) |
| SSYCON | |
| subroutine | ssyconv (UPLO, WAY, N, A, LDA, IPIV, WORK, INFO) |
| SSYCONV | |
| subroutine | ssyequb (UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO) |
| SSYEQUB | |
| subroutine | ssygs2 (ITYPE, UPLO, N, A, LDA, B, LDB, INFO) |
| SSYGS2 reduces a symmetric definite generalized eigenproblem to standard form, using the factorization results obtained from spotrf (unblocked algorithm). | |
| subroutine | ssygst (ITYPE, UPLO, N, A, LDA, B, LDB, INFO) |
| SSYGST | |
| subroutine | ssyrfs (UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO) |
| SSYRFS | |
| subroutine | ssyrfsx (UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO) |
| SSYRFSX | |
| subroutine | ssytd2 (UPLO, N, A, LDA, D, E, TAU, INFO) |
| SSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm). | |
| subroutine | ssytf2 (UPLO, N, A, LDA, IPIV, INFO) |
| SSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm). | |
| subroutine | ssytrd (UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO) |
| SSYTRD | |
| subroutine | ssytrf (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO) |
| SSYTRF | |
| subroutine | ssytri (UPLO, N, A, LDA, IPIV, WORK, INFO) |
| SSYTRI | |
| subroutine | ssytri2 (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO) |
| SSYTRI2 | |
| subroutine | ssytri2x (UPLO, N, A, LDA, IPIV, WORK, NB, INFO) |
| SSYTRI2X | |
| subroutine | ssytrs (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO) |
| SSYTRS | |
| subroutine | ssytrs2 (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK, INFO) |
| SSYTRS2 | |
| subroutine | stgsyl (TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO) |
| STGSYL | |
| subroutine | strsyl (TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C, LDC, SCALE, INFO) |
| STRSYL | |
This is the group of real computational functions for SY matrices
| subroutine sla_syamv | ( | integer | UPLO, |
| integer | N, | ||
| real | ALPHA, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( * ) | X, | ||
| integer | INCX, | ||
| real | BETA, | ||
| real, dimension( * ) | Y, | ||
| integer | INCY | ||
| ) |
SLA_SYAMV computes a matrix-vector product using a symmetric indefinite matrix to calculate error bounds.
Download SLA_SYAMV + dependencies [TGZ] [ZIP] [TXT]
SLA_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. | [in] | UPLO | 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. |
| [in] | 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. |
| [in] | ALPHA | ALPHA is REAL .
On entry, ALPHA specifies the scalar alpha.
Unchanged on exit. |
| [in] | A | A is REAL array of DIMENSION ( LDA, n ).
Before entry, the leading m by n part of the array A must
contain the matrix of coefficients.
Unchanged on exit. |
| [in] | 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. |
| [in] | X | X is REAL array, dimension
( 1 + ( n - 1 )*abs( INCX ) )
Before entry, the incremented array X must contain the
vector x.
Unchanged on exit. |
| [in] | INCX | INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X. INCX must not be zero.
Unchanged on exit. |
| [in] | BETA | BETA is REAL .
On entry, BETA specifies the scalar beta. When BETA is
supplied as zero then Y need not be set on input.
Unchanged on exit. |
| [in,out] | Y | Y is REAL 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. |
| [in] | INCY | INCY is INTEGER
On entry, INCY specifies the increment for the elements of
Y. INCY must not be zero.
Unchanged on exit. |
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 Definition at line 177 of file sla_syamv.f.
| REAL function sla_syrcond | ( | character | UPLO, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( ldaf, * ) | AF, | ||
| integer | LDAF, | ||
| integer, dimension( * ) | IPIV, | ||
| integer | CMODE, | ||
| real, dimension( * ) | C, | ||
| integer | INFO, | ||
| real, dimension( * ) | WORK, | ||
| integer, dimension( * ) | IWORK | ||
| ) |
SLA_SYRCOND estimates the Skeel condition number for a symmetric indefinite matrix.
Download SLA_SYRCOND + dependencies [TGZ] [ZIP] [TXT] SLA_SYRCOND estimates the Skeel condition number of op(A) * op2(C)
where op2 is determined by CMODE as follows
CMODE = 1 op2(C) = C
CMODE = 0 op2(C) = I
CMODE = -1 op2(C) = inv(C)
The Skeel condition number cond(A) = norminf( |inv(A)||A| )
is computed by computing scaling factors R such that
diag(R)*A*op2(C) is row equilibrated and computing the standard
infinity-norm condition number. | [in] | UPLO | UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored. |
| [in] | N | N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0. |
| [in] | A | A is REAL array, dimension (LDA,N)
On entry, the N-by-N matrix A. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [in] | AF | AF is REAL array, dimension (LDAF,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by SSYTRF. |
| [in] | LDAF | LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N). |
| [in] | IPIV | IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by SSYTRF. |
| [in] | CMODE | CMODE is INTEGER
Determines op2(C) in the formula op(A) * op2(C) as follows:
CMODE = 1 op2(C) = C
CMODE = 0 op2(C) = I
CMODE = -1 op2(C) = inv(C) |
| [in] | C | C is REAL array, dimension (N)
The vector C in the formula op(A) * op2(C). |
| [out] | INFO | INFO is INTEGER
= 0: Successful exit.
i > 0: The ith argument is invalid. |
| [in] | WORK | WORK is REAL array, dimension (3*N).
Workspace. |
| [in] | IWORK | IWORK is INTEGER array, dimension (N).
Workspace. |
Definition at line 146 of file sla_syrcond.f.
| subroutine sla_syrfsx_extended | ( | integer | PREC_TYPE, |
| character | UPLO, | ||
| integer | N, | ||
| integer | NRHS, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( ldaf, * ) | AF, | ||
| integer | LDAF, | ||
| integer, dimension( * ) | IPIV, | ||
| logical | COLEQU, | ||
| real, dimension( * ) | C, | ||
| real, dimension( ldb, * ) | B, | ||
| integer | LDB, | ||
| real, dimension( ldy, * ) | Y, | ||
| integer | LDY, | ||
| real, dimension( * ) | BERR_OUT, | ||
| integer | N_NORMS, | ||
| real, dimension( nrhs, * ) | ERR_BNDS_NORM, | ||
| real, dimension( nrhs, * ) | ERR_BNDS_COMP, | ||
| real, dimension( * ) | RES, | ||
| real, dimension( * ) | AYB, | ||
| real, dimension( * ) | DY, | ||
| real, dimension( * ) | Y_TAIL, | ||
| real | RCOND, | ||
| integer | ITHRESH, | ||
| real | RTHRESH, | ||
| real | DZ_UB, | ||
| logical | IGNORE_CWISE, | ||
| integer | INFO | ||
| ) |
SLA_SYRFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
Download SLA_SYRFSX_EXTENDED + dependencies [TGZ] [ZIP] [TXT]SLA_SYRFSX_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 SSYRFSX 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.
| [in] | PREC_TYPE | 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': Extra |
| [in] | UPLO | UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored. |
| [in] | N | N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0. |
| [in] | NRHS | NRHS is INTEGER
The number of right-hand-sides, i.e., the number of columns of the
matrix B. |
| [in] | A | A is REAL array, dimension (LDA,N)
On entry, the N-by-N matrix A. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [in] | AF | AF is REAL array, dimension (LDAF,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by SSYTRF. |
| [in] | LDAF | LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N). |
| [in] | IPIV | IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by SSYTRF. |
| [in] | 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. |
| [in] | C | C is REAL 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. |
| [in] | B | B is REAL array, dimension (LDB,NRHS)
The right-hand-side matrix B. |
| [in] | LDB | LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N). |
| [in,out] | Y | Y is REAL array, dimension (LDY,NRHS)
On entry, the solution matrix X, as computed by SSYTRS.
On exit, the improved solution matrix Y. |
| [in] | LDY | LDY is INTEGER
The leading dimension of the array Y. LDY >= max(1,N). |
| [out] | BERR_OUT | BERR_OUT is REAL 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 SLA_LIN_BERR. |
| [in] | 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. |
| [in,out] | ERR_BNDS_NORM | ERR_BNDS_NORM is REAL 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. |
| [in,out] | ERR_BNDS_COMP | ERR_BNDS_COMP is REAL 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. |
| [in] | RES | RES is REAL array, dimension (N)
Workspace to hold the intermediate residual. |
| [in] | AYB | AYB is REAL array, dimension (N)
Workspace. This can be the same workspace passed for Y_TAIL. |
| [in] | DY | DY is REAL array, dimension (N)
Workspace to hold the intermediate solution. |
| [in] | Y_TAIL | Y_TAIL is REAL array, dimension (N)
Workspace to hold the trailing bits of the intermediate solution. |
| [in] | RCOND | RCOND is REAL
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. |
| [in] | 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. |
| [in] | RTHRESH | RTHRESH is REAL
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. |
| [in] | DZ_UB | DZ_UB is REAL
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. |
| [in] | IGNORE_CWISE | IGNORE_CWISE is LOGICAL
If .TRUE. then ignore componentwise convergence. Default value
is .FALSE.. |
| [out] | INFO | INFO is INTEGER
= 0: Successful exit.
< 0: if INFO = -i, the ith argument to SLA_SYRFSX_EXTENDED had an illegal
value |
Definition at line 391 of file sla_syrfsx_extended.f.
| REAL function sla_syrpvgrw | ( | character*1 | UPLO, |
| integer | N, | ||
| integer | INFO, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( ldaf, * ) | AF, | ||
| integer | LDAF, | ||
| integer, dimension( * ) | IPIV, | ||
| real, dimension( * ) | WORK | ||
| ) |
SLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite matrix.
Download SLA_SYRPVGRW + dependencies [TGZ] [ZIP] [TXT]SLA_SYRPVGRW 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.
| [in] | UPLO | UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored. |
| [in] | N | N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0. |
| [in] | INFO | INFO is INTEGER
The value of INFO returned from SSYTRF, .i.e., the pivot in
column INFO is exactly 0. |
| [in] | A | A is REAL array, dimension (LDA,N)
On entry, the N-by-N matrix A. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [in] | AF | AF is REAL array, dimension (LDAF,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by SSYTRF. |
| [in] | LDAF | LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N). |
| [in] | IPIV | IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by SSYTRF. |
| [in] | WORK | WORK is REAL array, dimension (2*N) |
Definition at line 122 of file sla_syrpvgrw.f.
| subroutine slasyf | ( | character | UPLO, |
| integer | N, | ||
| integer | NB, | ||
| integer | KB, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| integer, dimension( * ) | IPIV, | ||
| real, dimension( ldw, * ) | W, | ||
| integer | LDW, | ||
| integer | INFO | ||
| ) |
SLASYF computes a partial factorization of a real symmetric matrix, using the diagonal pivoting method.
Download SLASYF + dependencies [TGZ] [ZIP] [TXT] SLASYF computes a partial factorization of a real symmetric 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**T U22**T )
A = ( L11 0 ) ( D 0 ) ( L11**T L21**T ) if UPLO = 'L'
( L21 I ) ( 0 A22 ) ( 0 I )
where the order of D is at most NB. The actual order is returned in
the argument KB, and is either NB or NB-1, or N if N <= NB.
SLASYF is an auxiliary routine called by SSYTRF. It uses blocked code
(calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or
A22 (if UPLO = 'L'). | [in] | UPLO | UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular |
| [in] | N | N is INTEGER
The order of the matrix A. N >= 0. |
| [in] | 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. |
| [out] | 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. |
| [in,out] | A | A is REAL array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading
n-by-n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading n-by-n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, A contains details of the partial factorization. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [out] | 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 UPLO = 'L', only the first KB elements 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 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. |
| [out] | W | W is REAL array, dimension (LDW,NB) |
| [in] | LDW | LDW is INTEGER
The leading dimension of the array W. LDW >= max(1,N). |
| [out] | 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. |
Definition at line 157 of file slasyf.f.
| subroutine ssycon | ( | character | UPLO, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| integer, dimension( * ) | IPIV, | ||
| real | ANORM, | ||
| real | RCOND, | ||
| real, dimension( * ) | WORK, | ||
| integer, dimension( * ) | IWORK, | ||
| integer | INFO | ||
| ) |
SSYCON
Download SSYCON + dependencies [TGZ] [ZIP] [TXT]SSYCON estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by SSYTRF. An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
| [in] | UPLO | 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. |
| [in] | N | N is INTEGER
The order of the matrix A. N >= 0. |
| [in] | A | A is REAL array, dimension (LDA,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by SSYTRF. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [in] | IPIV | IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by SSYTRF. |
| [in] | ANORM | ANORM is REAL
The 1-norm of the original matrix A. |
| [out] | RCOND | RCOND is REAL
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. |
| [out] | WORK | WORK is REAL array, dimension (2*N) |
| [out] | IWORK | IWORK is INTEGER array, dimension (N) |
| [out] | INFO | INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value |
Definition at line 130 of file ssycon.f.
| subroutine ssyconv | ( | character | UPLO, |
| character | WAY, | ||
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| integer, dimension( * ) | IPIV, | ||
| real, dimension( * ) | WORK, | ||
| integer | INFO | ||
| ) |
SSYCONV
Download SSYCONV + dependencies [TGZ] [ZIP] [TXT]SSYCONV convert A given by TRF into L and D and vice-versa. Get Non-diag elements of D (returned in workspace) and apply or reverse permutation done in TRF.
| [in] | UPLO | 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. |
| [in] | WAY | WAY is CHARACTER*1
= 'C': Convert
= 'R': Revert |
| [in] | N | N is INTEGER
The order of the matrix A. N >= 0. |
| [in] | A | A is REAL array, dimension (LDA,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by SSYTRF. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [in] | IPIV | IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by SSYTRF. |
| [out] | WORK | WORK is REAL array, dimension (N) |
| [out] | INFO | INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value |
Definition at line 113 of file ssyconv.f.
| subroutine ssyequb | ( | character | UPLO, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( * ) | S, | ||
| real | SCOND, | ||
| real | AMAX, | ||
| real, dimension( * ) | WORK, | ||
| integer | INFO | ||
| ) |
SSYEQUB
Download SSYEQUB + dependencies [TGZ] [ZIP] [TXT]SSYEQUB computes row and column scalings intended to equilibrate a symmetric matrix A and reduce its condition number (with respect to the two-norm). S contains the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings.
| [in] | UPLO | 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. |
| [in] | N | N is INTEGER
The order of the matrix A. N >= 0. |
| [in] | A | A is REAL array, dimension (LDA,N)
The N-by-N symmetric matrix whose scaling
factors are to be computed. Only the diagonal elements of A
are referenced. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [out] | S | S is REAL array, dimension (N)
If INFO = 0, S contains the scale factors for A. |
| [out] | SCOND | SCOND is REAL
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. |
| [out] | AMAX | AMAX is REAL
Absolute value of largest matrix element. If AMAX is very
close to overflow or very close to underflow, the matrix
should be scaled. |
| [out] | WORK | WORK is REAL array, dimension (3*N) |
| [out] | 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. |
Definition at line 136 of file ssyequb.f.
| subroutine ssygs2 | ( | integer | ITYPE, |
| character | UPLO, | ||
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( ldb, * ) | B, | ||
| integer | LDB, | ||
| integer | INFO | ||
| ) |
SSYGS2 reduces a symmetric definite generalized eigenproblem to standard form, using the factorization results obtained from spotrf (unblocked algorithm).
Download SSYGS2 + dependencies [TGZ] [ZIP] [TXT]SSYGS2 reduces a real symmetric-definite generalized eigenproblem to standard form. If ITYPE = 1, the problem is A*x = lambda*B*x, and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T) 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**T or L**T *A*L. B must have been previously factorized as U**T *U or L*L**T by SPOTRF.
| [in] | ITYPE | ITYPE is INTEGER
= 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
= 2 or 3: compute U*A*U**T or L**T *A*L. |
| [in] | UPLO | UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored, and how B has been factorized.
= 'U': Upper triangular
= 'L': Lower triangular |
| [in] | N | N is INTEGER
The order of the matrices A and B. N >= 0. |
| [in,out] | A | A is REAL array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading
n by n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading n by n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if INFO = 0, the transformed matrix, stored in the
same format as A. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [in] | B | B is REAL array, dimension (LDB,N)
The triangular factor from the Cholesky factorization of B,
as returned by SPOTRF. |
| [in] | LDB | LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N). |
| [out] | INFO | INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value. |
Definition at line 128 of file ssygs2.f.
| subroutine ssygst | ( | integer | ITYPE, |
| character | UPLO, | ||
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( ldb, * ) | B, | ||
| integer | LDB, | ||
| integer | INFO | ||
| ) |
SSYGST
Download SSYGST + dependencies [TGZ] [ZIP] [TXT]SSYGST reduces a real symmetric-definite generalized eigenproblem to standard form. If ITYPE = 1, the problem is A*x = lambda*B*x, and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T) 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**T or L**T*A*L. B must have been previously factorized as U**T*U or L*L**T by SPOTRF.
| [in] | ITYPE | ITYPE is INTEGER
= 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
= 2 or 3: compute U*A*U**T or L**T*A*L. |
| [in] | UPLO | UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored and B is factored as
U**T*U;
= 'L': Lower triangle of A is stored and B is factored as
L*L**T. |
| [in] | N | N is INTEGER
The order of the matrices A and B. N >= 0. |
| [in,out] | A | A is REAL array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if INFO = 0, the transformed matrix, stored in the
same format as A. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [in] | B | B is REAL array, dimension (LDB,N)
The triangular factor from the Cholesky factorization of B,
as returned by SPOTRF. |
| [in] | LDB | LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N). |
| [out] | INFO | INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value |
Definition at line 128 of file ssygst.f.
| subroutine ssyrfs | ( | character | UPLO, |
| integer | N, | ||
| integer | NRHS, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( ldaf, * ) | AF, | ||
| integer | LDAF, | ||
| integer, dimension( * ) | IPIV, | ||
| real, dimension( ldb, * ) | B, | ||
| integer | LDB, | ||
| real, dimension( ldx, * ) | X, | ||
| integer | LDX, | ||
| real, dimension( * ) | FERR, | ||
| real, dimension( * ) | BERR, | ||
| real, dimension( * ) | WORK, | ||
| integer, dimension( * ) | IWORK, | ||
| integer | INFO | ||
| ) |
SSYRFS
Download SSYRFS + dependencies [TGZ] [ZIP] [TXT]SSYRFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric indefinite, and provides error bounds and backward error estimates for the solution.
| [in] | UPLO | UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored. |
| [in] | N | N is INTEGER
The order of the matrix A. N >= 0. |
| [in] | NRHS | NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0. |
| [in] | A | A is REAL 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. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [in] | AF | AF is REAL 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 SSYTRF. |
| [in] | LDAF | LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N). |
| [in] | IPIV | IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by SSYTRF. |
| [in] | B | B is REAL array, dimension (LDB,NRHS)
The right hand side matrix B. |
| [in] | LDB | LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N). |
| [in,out] | X | X is REAL array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by SSYTRS.
On exit, the improved solution matrix X. |
| [in] | LDX | LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N). |
| [out] | FERR | FERR is REAL 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. |
| [out] | BERR | BERR is REAL 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). |
| [out] | WORK | WORK is REAL array, dimension (3*N) |
| [out] | IWORK | IWORK is INTEGER array, dimension (N) |
| [out] | 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.
Definition at line 191 of file ssyrfs.f.
| subroutine ssyrfsx | ( | character | UPLO, |
| character | EQUED, | ||
| integer | N, | ||
| integer | NRHS, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( ldaf, * ) | AF, | ||
| integer | LDAF, | ||
| integer, dimension( * ) | IPIV, | ||
| real, dimension( * ) | S, | ||
| real, dimension( ldb, * ) | B, | ||
| integer | LDB, | ||
| real, dimension( ldx, * ) | X, | ||
| integer | LDX, | ||
| real | RCOND, | ||
| real, dimension( * ) | BERR, | ||
| integer | N_ERR_BNDS, | ||
| real, dimension( nrhs, * ) | ERR_BNDS_NORM, | ||
| real, dimension( nrhs, * ) | ERR_BNDS_COMP, | ||
| integer | NPARAMS, | ||
| real, dimension( * ) | PARAMS, | ||
| real, dimension( * ) | WORK, | ||
| integer, dimension( * ) | IWORK, | ||
| integer | INFO | ||
| ) |
SSYRFSX
Download SSYRFSX + dependencies [TGZ] [ZIP] [TXT] SSYRFSX improves the computed solution to a system of linear
equations when the coefficient matrix is symmetric 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.| [in] | UPLO | UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored. |
| [in] | 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. |
| [in] | N | N is INTEGER
The order of the matrix A. N >= 0. |
| [in] | NRHS | NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0. |
| [in] | A | A is REAL 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. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [in] | AF | AF is REAL 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 SSYTRF. |
| [in] | LDAF | LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N). |
| [in] | IPIV | IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by SSYTRF. |
| [in,out] | S | S is REAL 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. |
| [in] | B | B is REAL array, dimension (LDB,NRHS)
The right hand side matrix B. |
| [in] | LDB | LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N). |
| [in,out] | X | X is REAL array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by SGETRS.
On exit, the improved solution matrix X. |
| [in] | LDX | LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N). |
| [out] | RCOND | RCOND is REAL
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. |
| [out] | BERR | BERR is REAL 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). |
| [in] | 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. |
| [out] | ERR_BNDS_NORM | ERR_BNDS_NORM is REAL 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.
See Lapack Working Note 165 for further details and extra
cautions. |
| [out] | ERR_BNDS_COMP | ERR_BNDS_COMP is REAL 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.
See Lapack Working Note 165 for further details and extra
cautions. |
| [in] | 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. |
| [in,out] | PARAMS | PARAMS is REAL 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: 1.0
= 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) |
| [out] | WORK | WORK is REAL array, dimension (4*N) |
| [out] | IWORK | IWORK is INTEGER array, dimension (N) |
| [out] | 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. |
Definition at line 400 of file ssyrfsx.f.
| subroutine ssytd2 | ( | character | UPLO, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( * ) | D, | ||
| real, dimension( * ) | E, | ||
| real, dimension( * ) | TAU, | ||
| integer | INFO | ||
| ) |
SSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm).
Download SSYTD2 + dependencies [TGZ] [ZIP] [TXT]SSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: Q**T * A * Q = T.
| [in] | UPLO | UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular |
| [in] | N | N is INTEGER
The order of the matrix A. N >= 0. |
| [in,out] | A | A is REAL array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading
n-by-n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading n-by-n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if 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 orthogonal
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 orthogonal matrix Q as a product
of elementary reflectors. See Further Details. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [out] | D | D is REAL array, dimension (N)
The diagonal elements of the tridiagonal matrix T:
D(i) = A(i,i). |
| [out] | E | E is REAL 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'. |
| [out] | TAU | TAU is REAL array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details). |
| [out] | INFO | INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value. |
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**T
where tau is a real scalar, and v is a real 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**T
where tau is a real scalar, and v is a real 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). Definition at line 174 of file ssytd2.f.
| subroutine ssytf2 | ( | character | UPLO, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| integer, dimension( * ) | IPIV, | ||
| integer | INFO | ||
| ) |
SSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).
Download SSYTF2 + dependencies [TGZ] [ZIP] [TXT] SSYTF2 computes the factorization of a real symmetric matrix A using
the Bunch-Kaufman diagonal pivoting method:
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, U**T is the transpose of U, and D is symmetric 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. | [in] | UPLO | UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular |
| [in] | N | N is INTEGER
The order of the matrix A. N >= 0. |
| [in,out] | A | A is REAL array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading
n-by-n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading n-by-n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, the block diagonal matrix D and the multipliers used
to obtain the factor U or L (see below for further details). |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [out] | 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. |
| [out] | 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. |
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). 09-29-06 - patch from
Bobby Cheng, MathWorks
Replace l.204 and l.372
IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
by
IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. SISNAN(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
1-96 - Based on modifications by J. Lewis, Boeing Computer Services
Company Definition at line 187 of file ssytf2.f.
| subroutine ssytrd | ( | character | UPLO, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( * ) | D, | ||
| real, dimension( * ) | E, | ||
| real, dimension( * ) | TAU, | ||
| real, dimension( * ) | WORK, | ||
| integer | LWORK, | ||
| integer | INFO | ||
| ) |
SSYTRD
Download SSYTRD + dependencies [TGZ] [ZIP] [TXT]SSYTRD reduces a real symmetric matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**T * A * Q = T.
| [in] | UPLO | UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored. |
| [in] | N | N is INTEGER
The order of the matrix A. N >= 0. |
| [in,out] | A | A is REAL array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if 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 orthogonal
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 orthogonal matrix Q as a product
of elementary reflectors. See Further Details. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [out] | D | D is REAL array, dimension (N)
The diagonal elements of the tridiagonal matrix T:
D(i) = A(i,i). |
| [out] | E | E is REAL 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'. |
| [out] | TAU | TAU is REAL array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details). |
| [out] | WORK | WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
| [in] | 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. |
| [out] | INFO | INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value |
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**T
where tau is a real scalar, and v is a real 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**T
where tau is a real scalar, and v is a real 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). Definition at line 193 of file ssytrd.f.
| subroutine ssytrf | ( | character | UPLO, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| integer, dimension( * ) | IPIV, | ||
| real, dimension( * ) | WORK, | ||
| integer | LWORK, | ||
| integer | INFO | ||
| ) |
SSYTRF
Download SSYTRF + dependencies [TGZ] [ZIP] [TXT] SSYTRF computes the factorization of a real symmetric matrix A using
the Bunch-Kaufman 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 symmetric 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. | [in] | UPLO | UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored. |
| [in] | N | N is INTEGER
The order of the matrix A. N >= 0. |
| [in,out] | A | A is REAL array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, the block diagonal matrix D and the multipliers used
to obtain the factor U or L (see below for further details). |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [out] | 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. |
| [out] | WORK | WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
| [in] | 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. |
| [out] | 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. |
If UPLO = 'U', then A = U*D*U**T, where
U = P(n)*U(n)* ... <em>P(k)U(k)</em> ...,
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)* ... <em>P(k)*L(k)</em> ...,
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). Definition at line 183 of file ssytrf.f.
| subroutine ssytri | ( | character | UPLO, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| integer, dimension( * ) | IPIV, | ||
| real, dimension( * ) | WORK, | ||
| integer | INFO | ||
| ) |
SSYTRI
Download SSYTRI + dependencies [TGZ] [ZIP] [TXT]SSYTRI computes the inverse of a real symmetric indefinite matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by SSYTRF.
| [in] | UPLO | 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. |
| [in] | N | N is INTEGER
The order of the matrix A. N >= 0. |
| [in,out] | A | A is REAL 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 SSYTRF.
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. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [in] | IPIV | IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by SSYTRF. |
| [out] | WORK | WORK is REAL array, dimension (N) |
| [out] | 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. |
Definition at line 115 of file ssytri.f.
| subroutine ssytri2 | ( | character | UPLO, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| integer, dimension( * ) | IPIV, | ||
| real, dimension( * ) | WORK, | ||
| integer | LWORK, | ||
| integer | INFO | ||
| ) |
SSYTRI2
Download SSYTRI2 + dependencies [TGZ] [ZIP] [TXT]SSYTRI2 computes the inverse of a REAL symmetric indefinite matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by SSYTRF. SSYTRI2 sets the LEADING DIMENSION of the workspace before calling SSYTRI2X that actually computes the inverse.
| [in] | UPLO | 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. |
| [in] | N | N is INTEGER
The order of the matrix A. N >= 0. |
| [in,out] | A | A is REAL 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 SSYTRF.
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. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [in] | IPIV | IPIV is INTEGER array, dimension (N)
Details of the interchanges and the NB structure of D
as determined by SSYTRF. |
| [out] | WORK | WORK is REAL array, dimension (N+NB+1)*(NB+3) |
| [in] | LWORK | LWORK is INTEGER
The dimension of the array WORK.
WORK is size >= (N+NB+1)*(NB+3)
If LDWORK = -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 LDWORK is issued by XERBLA. |
| [out] | 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. |
Definition at line 128 of file ssytri2.f.
| subroutine ssytri2x | ( | character | UPLO, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| integer, dimension( * ) | IPIV, | ||
| real, dimension( n+nb+1,* ) | WORK, | ||
| integer | NB, | ||
| integer | INFO | ||
| ) |
SSYTRI2X
Download SSYTRI2X + dependencies [TGZ] [ZIP] [TXT]SSYTRI2X computes the inverse of a real symmetric indefinite matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by SSYTRF.
| [in] | UPLO | 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. |
| [in] | N | N is INTEGER
The order of the matrix A. N >= 0. |
| [in,out] | A | A is REAL 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 SSYTRF.
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. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [in] | IPIV | IPIV is INTEGER array, dimension (N)
Details of the interchanges and the NNB structure of D
as determined by SSYTRF. |
| [out] | WORK | WORK is REAL array, dimension (N+NNB+1,NNB+3) |
| [in] | NB | NB is INTEGER
Block size |
| [out] | 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. |
Definition at line 121 of file ssytri2x.f.
| subroutine ssytrs | ( | character | UPLO, |
| integer | N, | ||
| integer | NRHS, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| integer, dimension( * ) | IPIV, | ||
| real, dimension( ldb, * ) | B, | ||
| integer | LDB, | ||
| integer | INFO | ||
| ) |
SSYTRS
Download SSYTRS + dependencies [TGZ] [ZIP] [TXT]SSYTRS solves a system of linear equations A*X = B with a real symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by SSYTRF.
| [in] | UPLO | 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. |
| [in] | N | N is INTEGER
The order of the matrix A. N >= 0. |
| [in] | NRHS | NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0. |
| [in] | A | A is REAL array, dimension (LDA,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by SSYTRF. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [in] | IPIV | IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by SSYTRF. |
| [in,out] | B | B is REAL array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X. |
| [in] | LDB | LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N). |
| [out] | INFO | INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value |
Definition at line 121 of file ssytrs.f.
| subroutine ssytrs2 | ( | character | UPLO, |
| integer | N, | ||
| integer | NRHS, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| integer, dimension( * ) | IPIV, | ||
| real, dimension( ldb, * ) | B, | ||
| integer | LDB, | ||
| real, dimension( * ) | WORK, | ||
| integer | INFO | ||
| ) |
SSYTRS2
Download SSYTRS2 + dependencies [TGZ] [ZIP] [TXT]SSYTRS2 solves a system of linear equations A*X = B with a real symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by SSYTRF and converted by SSYCONV.
| [in] | UPLO | 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. |
| [in] | N | N is INTEGER
The order of the matrix A. N >= 0. |
| [in] | NRHS | NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0. |
| [in] | A | A is REAL array, dimension (LDA,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by SSYTRF. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [in] | IPIV | IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by SSYTRF. |
| [in,out] | B | B is REAL array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X. |
| [in] | LDB | LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N). |
| [out] | WORK | WORK is REAL array, dimension (N) |
| [out] | INFO | INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value |
Definition at line 127 of file ssytrs2.f.
| subroutine stgsyl | ( | character | TRANS, |
| integer | IJOB, | ||
| integer | M, | ||
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( ldb, * ) | B, | ||
| integer | LDB, | ||
| real, dimension( ldc, * ) | C, | ||
| integer | LDC, | ||
| real, dimension( ldd, * ) | D, | ||
| integer | LDD, | ||
| real, dimension( lde, * ) | E, | ||
| integer | LDE, | ||
| real, dimension( ldf, * ) | F, | ||
| integer | LDF, | ||
| real | SCALE, | ||
| real | DIF, | ||
| real, dimension( * ) | WORK, | ||
| integer | LWORK, | ||
| integer, dimension( * ) | IWORK, | ||
| integer | INFO | ||
| ) |
STGSYL
Download STGSYL + dependencies [TGZ] [ZIP] [TXT] STGSYL solves the generalized Sylvester equation:
A * R - L * B = scale * C (1)
D * R - L * E = scale * F
where R and L are unknown m-by-n matrices, (A, D), (B, E) and
(C, F) are given matrix pairs of size m-by-m, n-by-n and m-by-n,
respectively, with real entries. (A, D) and (B, E) must be in
generalized (real) Schur canonical form, i.e. A, B are upper quasi
triangular and D, E are upper triangular.
The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output
scaling factor chosen to avoid overflow.
In matrix notation (1) is equivalent to solve Zx = scale b, where
Z is defined as
Z = [ kron(In, A) -kron(B**T, Im) ] (2)
[ kron(In, D) -kron(E**T, Im) ].
Here Ik is the identity matrix of size k and X**T is the transpose of
X. kron(X, Y) is the Kronecker product between the matrices X and Y.
If TRANS = 'T', STGSYL solves the transposed system Z**T*y = scale*b,
which is equivalent to solve for R and L in
A**T * R + D**T * L = scale * C (3)
R * B**T + L * E**T = scale * -F
This case (TRANS = 'T') is used to compute an one-norm-based estimate
of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D)
and (B,E), using SLACON.
If IJOB >= 1, STGSYL computes a Frobenius norm-based estimate
of Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the
reciprocal of the smallest singular value of Z. See [1-2] for more
information.
This is a level 3 BLAS algorithm. | [in] | TRANS | TRANS is CHARACTER*1
= 'N', solve the generalized Sylvester equation (1).
= 'T', solve the 'transposed' system (3). |
| [in] | IJOB | IJOB is INTEGER
Specifies what kind of functionality to be performed.
=0: solve (1) only.
=1: The functionality of 0 and 3.
=2: The functionality of 0 and 4.
=3: Only an estimate of Dif[(A,D), (B,E)] is computed.
(look ahead strategy IJOB = 1 is used).
=4: Only an estimate of Dif[(A,D), (B,E)] is computed.
( SGECON on sub-systems is used ).
Not referenced if TRANS = 'T'. |
| [in] | M | M is INTEGER
The order of the matrices A and D, and the row dimension of
the matrices C, F, R and L. |
| [in] | N | N is INTEGER
The order of the matrices B and E, and the column dimension
of the matrices C, F, R and L. |
| [in] | A | A is REAL array, dimension (LDA, M)
The upper quasi triangular matrix A. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1, M). |
| [in] | B | B is REAL array, dimension (LDB, N)
The upper quasi triangular matrix B. |
| [in] | LDB | LDB is INTEGER
The leading dimension of the array B. LDB >= max(1, N). |
| [in,out] | C | C is REAL array, dimension (LDC, N)
On entry, C contains the right-hand-side of the first matrix
equation in (1) or (3).
On exit, if IJOB = 0, 1 or 2, C has been overwritten by
the solution R. If IJOB = 3 or 4 and TRANS = 'N', C holds R,
the solution achieved during the computation of the
Dif-estimate. |
| [in] | LDC | LDC is INTEGER
The leading dimension of the array C. LDC >= max(1, M). |
| [in] | D | D is REAL array, dimension (LDD, M)
The upper triangular matrix D. |
| [in] | LDD | LDD is INTEGER
The leading dimension of the array D. LDD >= max(1, M). |
| [in] | E | E is REAL array, dimension (LDE, N)
The upper triangular matrix E. |
| [in] | LDE | LDE is INTEGER
The leading dimension of the array E. LDE >= max(1, N). |
| [in,out] | F | F is REAL array, dimension (LDF, N)
On entry, F contains the right-hand-side of the second matrix
equation in (1) or (3).
On exit, if IJOB = 0, 1 or 2, F has been overwritten by
the solution L. If IJOB = 3 or 4 and TRANS = 'N', F holds L,
the solution achieved during the computation of the
Dif-estimate. |
| [in] | LDF | LDF is INTEGER
The leading dimension of the array F. LDF >= max(1, M). |
| [out] | DIF | DIF is REAL
On exit DIF is the reciprocal of a lower bound of the
reciprocal of the Dif-function, i.e. DIF is an upper bound of
Dif[(A,D), (B,E)] = sigma_min(Z), where Z as in (2).
IF IJOB = 0 or TRANS = 'T', DIF is not touched. |
| [out] | SCALE | SCALE is REAL
On exit SCALE is the scaling factor in (1) or (3).
If 0 < SCALE < 1, C and F hold the solutions R and L, resp.,
to a slightly perturbed system but the input matrices A, B, D
and E have not been changed. If SCALE = 0, C and F hold the
solutions R and L, respectively, to the homogeneous system
with C = F = 0. Normally, SCALE = 1. |
| [out] | WORK | WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
| [in] | LWORK | LWORK is INTEGER
The dimension of the array WORK. LWORK > = 1.
If IJOB = 1 or 2 and TRANS = 'N', LWORK >= max(1,2*M*N).
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA. |
| [out] | IWORK | IWORK is INTEGER array, dimension (M+N+6) |
| [out] | INFO | INFO is INTEGER
=0: successful exit
<0: If INFO = -i, the i-th argument had an illegal value.
>0: (A, D) and (B, E) have common or close eigenvalues. |
[1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
for Solving the Generalized Sylvester Equation and Estimating the
Separation between Regular Matrix Pairs, Report UMINF - 93.23,
Department of Computing Science, Umea University, S-901 87 Umea,
Sweden, December 1993, Revised April 1994, Also as LAPACK Working
Note 75. To appear in ACM Trans. on Math. Software, Vol 22,
No 1, 1996.
[2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester
Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal.
Appl., 15(4):1045-1060, 1994
[3] B. Kagstrom and L. Westin, Generalized Schur Methods with
Condition Estimators for Solving the Generalized Sylvester
Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7,
July 1989, pp 745-751. Definition at line 298 of file stgsyl.f.
| subroutine strsyl | ( | character | TRANA, |
| character | TRANB, | ||
| integer | ISGN, | ||
| integer | M, | ||
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( ldb, * ) | B, | ||
| integer | LDB, | ||
| real, dimension( ldc, * ) | C, | ||
| integer | LDC, | ||
| real | SCALE, | ||
| integer | INFO | ||
| ) |
STRSYL
Download STRSYL + dependencies [TGZ] [ZIP] [TXT] STRSYL solves the real Sylvester matrix equation:
op(A)*X + X*op(B) = scale*C or
op(A)*X - X*op(B) = scale*C,
where op(A) = A or A**T, and A and B are both upper quasi-
triangular. A is M-by-M and B is N-by-N; the right hand side C and
the solution X are M-by-N; and scale is an output scale factor, set
<= 1 to avoid overflow in X.
A and B must be in Schur canonical form (as returned by SHSEQR), that
is, block upper triangular with 1-by-1 and 2-by-2 diagonal blocks;
each 2-by-2 diagonal block has its diagonal elements equal and its
off-diagonal elements of opposite sign. | [in] | TRANA | TRANA is CHARACTER*1
Specifies the option op(A):
= 'N': op(A) = A (No transpose)
= 'T': op(A) = A**T (Transpose)
= 'C': op(A) = A**H (Conjugate transpose = Transpose) |
| [in] | TRANB | TRANB is CHARACTER*1
Specifies the option op(B):
= 'N': op(B) = B (No transpose)
= 'T': op(B) = B**T (Transpose)
= 'C': op(B) = B**H (Conjugate transpose = Transpose) |
| [in] | ISGN | ISGN is INTEGER
Specifies the sign in the equation:
= +1: solve op(A)*X + X*op(B) = scale*C
= -1: solve op(A)*X - X*op(B) = scale*C |
| [in] | M | M is INTEGER
The order of the matrix A, and the number of rows in the
matrices X and C. M >= 0. |
| [in] | N | N is INTEGER
The order of the matrix B, and the number of columns in the
matrices X and C. N >= 0. |
| [in] | A | A is REAL array, dimension (LDA,M)
The upper quasi-triangular matrix A, in Schur canonical form. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M). |
| [in] | B | B is REAL array, dimension (LDB,N)
The upper quasi-triangular matrix B, in Schur canonical form. |
| [in] | LDB | LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N). |
| [in,out] | C | C is REAL array, dimension (LDC,N)
On entry, the M-by-N right hand side matrix C.
On exit, C is overwritten by the solution matrix X. |
| [in] | LDC | LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M) |
| [out] | SCALE | SCALE is REAL
The scale factor, scale, set <= 1 to avoid overflow in X. |
| [out] | INFO | INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
= 1: A and B have common or very close eigenvalues; perturbed
values were used to solve the equation (but the matrices
A and B are unchanged). |
Definition at line 164 of file strsyl.f.
1.8.1.1