table of contents
| hesvxx(3) | LAPACK | hesvxx(3) |
NAME¶
hesvxx - {he,sy}svxx: rook (v1, expert)
SYNOPSIS¶
Functions¶
subroutine chesvxx (fact, uplo, n, nrhs, a, lda, af, ldaf,
ipiv, equed, s, b, ldb, x, ldx, rcond, rpvgrw, berr, n_err_bnds,
err_bnds_norm, err_bnds_comp, nparams, params, work, rwork, info)
CHESVXX computes the solution to system of linear equations A * X = B for
HE matrices subroutine csysvxx (fact, uplo, n, nrhs, a, lda, af,
ldaf, ipiv, equed, s, b, ldb, x, ldx, rcond, rpvgrw, berr, n_err_bnds,
err_bnds_norm, err_bnds_comp, nparams, params, work, rwork, info)
CSYSVXX computes the solution to system of linear equations A * X = B for
SY matrices subroutine dsysvxx (fact, uplo, n, nrhs, a, lda, af,
ldaf, ipiv, equed, s, b, ldb, x, ldx, rcond, rpvgrw, berr, n_err_bnds,
err_bnds_norm, err_bnds_comp, nparams, params, work, iwork, info)
DSYSVXX subroutine ssysvxx (fact, uplo, n, nrhs, a, lda, af,
ldaf, ipiv, equed, s, b, ldb, x, ldx, rcond, rpvgrw, berr, n_err_bnds,
err_bnds_norm, err_bnds_comp, nparams, params, work, iwork, info)
SSYSVXX subroutine zhesvxx (fact, uplo, n, nrhs, a, lda, af,
ldaf, ipiv, equed, s, b, ldb, x, ldx, rcond, rpvgrw, berr, n_err_bnds,
err_bnds_norm, err_bnds_comp, nparams, params, work, rwork, info)
ZHESVXX computes the solution to system of linear equations A * X = B for
HE matrices subroutine zsysvxx (fact, uplo, n, nrhs, a, lda, af,
ldaf, ipiv, equed, s, b, ldb, x, ldx, rcond, rpvgrw, berr, n_err_bnds,
err_bnds_norm, err_bnds_comp, nparams, params, work, rwork, info)
ZSYSVXX computes the solution to system of linear equations A * X = B for
SY matrices
Detailed Description¶
Function Documentation¶
subroutine chesvxx (character fact, character uplo, integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, character equed, real, dimension( * ) s, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldx, * ) x, integer ldx, real rcond, real rpvgrw, real, dimension( * ) berr, integer n_err_bnds, real, dimension( nrhs, * ) err_bnds_norm, real, dimension( nrhs, * ) err_bnds_comp, integer nparams, real, dimension( * ) params, complex, dimension( * ) work, real, dimension( * ) rwork, integer info)¶
CHESVXX computes the solution to system of linear equations A * X = B for HE matrices
Purpose:
CHESVXX uses the diagonal pivoting factorization to compute the
solution to a complex system of linear equations A * X = B, where
A is an N-by-N Hermitian matrix and X and B are N-by-NRHS
matrices.
If requested, both normwise and maximum componentwise error bounds
are returned. CHESVXX will return a solution with a tiny
guaranteed error (O(eps) where eps is the working machine
precision) unless the matrix is very ill-conditioned, in which
case a warning is returned. Relevant condition numbers also are
calculated and returned.
CHESVXX accepts user-provided factorizations and equilibration
factors; see the definitions of the FACT and EQUED options.
Solving with refinement and using a factorization from a previous
CHESVXX call will also produce a solution with either O(eps)
errors or warnings, but we cannot make that claim for general
user-provided factorizations and equilibration factors if they
differ from what CHESVXX would itself produce.
Description:
The following steps are performed:
1. If FACT = 'E', real scaling factors are computed to equilibrate
the system:
diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B
Whether or not the system will be equilibrated depends on the
scaling of the matrix A, but if equilibration is used, A is
overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
2. If FACT = 'N' or 'E', the LU decomposition is used to factor
the matrix A (after equilibration if FACT = 'E') as
A = U * D * U**T, if UPLO = 'U', or
A = L * D * L**T, if UPLO = 'L',
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.
3. If some D(i,i)=0, so that D is exactly singular, then the
routine returns with INFO = i. Otherwise, the factored form of A
is used to estimate the condition number of the matrix A (see
argument RCOND). If the reciprocal of the condition number is
less than machine precision, the routine still goes on to solve
for X and compute error bounds as described below.
4. The system of equations is solved for X using the factored form
of A.
5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
the routine will use iterative refinement to try to get a small
error and error bounds. Refinement calculates the residual to at
least twice the working precision.
6. If equilibration was used, the matrix X is premultiplied by
diag(R) so that it solves the original system before
equilibration.
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.
Parameters
FACT
FACT is CHARACTER*1
Specifies whether or not the factored form of the matrix A is
supplied on entry, and if not, whether the matrix A should be
equilibrated before it is factored.
= 'F': On entry, AF and IPIV contain the factored form of A.
If EQUED is not 'N', the matrix A has been
equilibrated with scaling factors given by S.
A, AF, and IPIV are not modified.
= 'N': The matrix A will be copied to AF and factored.
= 'E': The matrix A will be equilibrated if necessary, then
copied to AF and factored.
UPLO
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 matrices B and X. NRHS >= 0.
A
A is COMPLEX 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.
On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
diag(S)*A*diag(S).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF
AF is COMPLEX array, dimension (LDAF,N)
If FACT = 'F', then AF is an input argument and on entry
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 CHETRF.
If FACT = 'N', then AF is an output argument and on exit
returns 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.
LDAF
LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
If FACT = 'F', then IPIV is an input argument and on entry
contains details of the interchanges and the block
structure of D, as determined by CHETRF. 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.
If FACT = 'N', then IPIV is an output argument and on exit
contains details of the interchanges and the block
structure of D, as determined by CHETRF.
EQUED
EQUED is CHARACTER*1
Specifies the form of equilibration that was done.
= 'N': No equilibration (always true if FACT = 'N').
= 'Y': Both row and column equilibration, i.e., A has been
replaced by diag(S) * A * diag(S).
EQUED is an input argument if FACT = 'F'; otherwise, it is an
output argument.
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.
B
B is COMPLEX array, dimension (LDB,NRHS)
On entry, the N-by-NRHS right hand side matrix B.
On exit,
if EQUED = 'N', B is not modified;
if EQUED = 'Y', B is overwritten by diag(S)*B;
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X
X is COMPLEX array, dimension (LDX,NRHS)
If INFO = 0, the N-by-NRHS solution matrix X to the original
system of equations. Note that A and B are modified on exit if
EQUED .ne. 'N', and the solution to the equilibrated system is
inv(diag(S))*X.
LDX
LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N).
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.
RPVGRW
RPVGRW is REAL
Reciprocal pivot growth. On exit, this contains the reciprocal
pivot growth factor norm(A)/norm(U). The 'max absolute element'
norm is used. If this is much less than 1, then 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. If factorization fails with
0<INFO<=N, then this contains the reciprocal pivot growth factor
for the leading INFO columns of A.
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).
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 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.
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 < 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.
NPARAMS
NPARAMS is INTEGER
Specifies the number of parameters set in PARAMS. If <= 0, the
PARAMS array is never referenced and default values are used.
PARAMS
PARAMS is REAL array, dimension NPARAMS
Specifies algorithm parameters. If an entry is < 0.0, then
that entry will be filled with default value used for that
parameter. Only positions up to NPARAMS are accessed; defaults
are used for higher-numbered parameters.
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)
WORK
WORK is COMPLEX array, dimension (5*N)
RWORK
RWORK is REAL 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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine csysvxx (character fact, character uplo, integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, character equed, real, dimension( * ) s, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldx, * ) x, integer ldx, real rcond, real rpvgrw, real, dimension( * ) berr, integer n_err_bnds, real, dimension( nrhs, * ) err_bnds_norm, real, dimension( nrhs, * ) err_bnds_comp, integer nparams, real, dimension( * ) params, complex, dimension( * ) work, real, dimension( * ) rwork, integer info)¶
CSYSVXX computes the solution to system of linear equations A * X = B for SY matrices
Purpose:
CSYSVXX uses the diagonal pivoting factorization to compute the
solution to a complex system of linear equations A * X = B, where
A is an N-by-N symmetric matrix and X and B are N-by-NRHS
matrices.
If requested, both normwise and maximum componentwise error bounds
are returned. CSYSVXX will return a solution with a tiny
guaranteed error (O(eps) where eps is the working machine
precision) unless the matrix is very ill-conditioned, in which
case a warning is returned. Relevant condition numbers also are
calculated and returned.
CSYSVXX accepts user-provided factorizations and equilibration
factors; see the definitions of the FACT and EQUED options.
Solving with refinement and using a factorization from a previous
CSYSVXX call will also produce a solution with either O(eps)
errors or warnings, but we cannot make that claim for general
user-provided factorizations and equilibration factors if they
differ from what CSYSVXX would itself produce.
Description:
The following steps are performed:
1. If FACT = 'E', real scaling factors are computed to equilibrate
the system:
diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B
Whether or not the system will be equilibrated depends on the
scaling of the matrix A, but if equilibration is used, A is
overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
2. If FACT = 'N' or 'E', the LU decomposition is used to factor
the matrix A (after equilibration if FACT = 'E') as
A = U * D * U**T, if UPLO = 'U', or
A = L * D * L**T, if UPLO = 'L',
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.
3. If some D(i,i)=0, so that D is exactly singular, then the
routine returns with INFO = i. Otherwise, the factored form of A
is used to estimate the condition number of the matrix A (see
argument RCOND). If the reciprocal of the condition number is
less than machine precision, the routine still goes on to solve
for X and compute error bounds as described below.
4. The system of equations is solved for X using the factored form
of A.
5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
the routine will use iterative refinement to try to get a small
error and error bounds. Refinement calculates the residual to at
least twice the working precision.
6. If equilibration was used, the matrix X is premultiplied by
diag(R) so that it solves the original system before
equilibration.
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.
Parameters
FACT
FACT is CHARACTER*1
Specifies whether or not the factored form of the matrix A is
supplied on entry, and if not, whether the matrix A should be
equilibrated before it is factored.
= 'F': On entry, AF and IPIV contain the factored form of A.
If EQUED is not 'N', the matrix A has been
equilibrated with scaling factors given by S.
A, AF, and IPIV are not modified.
= 'N': The matrix A will be copied to AF and factored.
= 'E': The matrix A will be equilibrated if necessary, then
copied to AF and factored.
UPLO
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 matrices B and X. NRHS >= 0.
A
A is COMPLEX 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.
On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
diag(S)*A*diag(S).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF
AF is COMPLEX array, dimension (LDAF,N)
If FACT = 'F', then AF is an input argument and on entry
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 CSYTRF.
If FACT = 'N', then AF is an output argument and on exit
returns 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.
LDAF
LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
If FACT = 'F', then IPIV is an input argument and on entry
contains details of the interchanges and the block
structure of D, as determined by CSYTRF. 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.
If FACT = 'N', then IPIV is an output argument and on exit
contains details of the interchanges and the block
structure of D, as determined by CSYTRF.
EQUED
EQUED is CHARACTER*1
Specifies the form of equilibration that was done.
= 'N': No equilibration (always true if FACT = 'N').
= 'Y': Both row and column equilibration, i.e., A has been
replaced by diag(S) * A * diag(S).
EQUED is an input argument if FACT = 'F'; otherwise, it is an
output argument.
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.
B
B is COMPLEX array, dimension (LDB,NRHS)
On entry, the N-by-NRHS right hand side matrix B.
On exit,
if EQUED = 'N', B is not modified;
if EQUED = 'Y', B is overwritten by diag(S)*B;
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X
X is COMPLEX array, dimension (LDX,NRHS)
If INFO = 0, the N-by-NRHS solution matrix X to the original
system of equations. Note that A and B are modified on exit if
EQUED .ne. 'N', and the solution to the equilibrated system is
inv(diag(S))*X.
LDX
LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N).
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.
RPVGRW
RPVGRW is REAL
Reciprocal pivot growth. On exit, this contains the reciprocal
pivot growth factor norm(A)/norm(U). The 'max absolute element'
norm is used. If this is much less than 1, then 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. If factorization fails with
0<INFO<=N, then this contains the reciprocal pivot growth factor
for the leading INFO columns of A.
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).
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 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.
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 < 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.
NPARAMS
NPARAMS is INTEGER
Specifies the number of parameters set in PARAMS. If <= 0, the
PARAMS array is never referenced and default values are used.
PARAMS
PARAMS is REAL array, dimension NPARAMS
Specifies algorithm parameters. If an entry is < 0.0, then
that entry will be filled with default value used for that
parameter. Only positions up to NPARAMS are accessed; defaults
are used for higher-numbered parameters.
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)
WORK
WORK is COMPLEX array, dimension (2*N)
RWORK
RWORK is REAL 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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine dsysvxx (character fact, character uplo, integer n, integer nrhs, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, character equed, double precision, dimension( * ) s, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldx, * ) x, integer ldx, double precision rcond, double precision rpvgrw, double precision, dimension( * ) berr, integer n_err_bnds, double precision, dimension( nrhs, * ) err_bnds_norm, double precision, dimension( nrhs, * ) err_bnds_comp, integer nparams, double precision, dimension( * ) params, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)¶
DSYSVXX
Purpose:
DSYSVXX uses the diagonal pivoting factorization to compute the
solution to a double precision system of linear equations A * X = B, where A
is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices.
If requested, both normwise and maximum componentwise error bounds
are returned. DSYSVXX will return a solution with a tiny
guaranteed error (O(eps) where eps is the working machine
precision) unless the matrix is very ill-conditioned, in which
case a warning is returned. Relevant condition numbers also are
calculated and returned.
DSYSVXX accepts user-provided factorizations and equilibration
factors; see the definitions of the FACT and EQUED options.
Solving with refinement and using a factorization from a previous
DSYSVXX call will also produce a solution with either O(eps)
errors or warnings, but we cannot make that claim for general
user-provided factorizations and equilibration factors if they
differ from what DSYSVXX would itself produce.
Description:
The following steps are performed:
1. If FACT = 'E', double precision scaling factors are computed to equilibrate
the system:
diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B
Whether or not the system will be equilibrated depends on the
scaling of the matrix A, but if equilibration is used, A is
overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
2. If FACT = 'N' or 'E', the LU decomposition is used to factor
the matrix A (after equilibration if FACT = 'E') as
A = U * D * U**T, if UPLO = 'U', or
A = L * D * L**T, if UPLO = 'L',
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.
3. If some D(i,i)=0, so that D is exactly singular, then the
routine returns with INFO = i. Otherwise, the factored form of A
is used to estimate the condition number of the matrix A (see
argument RCOND). If the reciprocal of the condition number is
less than machine precision, the routine still goes on to solve
for X and compute error bounds as described below.
4. The system of equations is solved for X using the factored form
of A.
5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
the routine will use iterative refinement to try to get a small
error and error bounds. Refinement calculates the residual to at
least twice the working precision.
6. If equilibration was used, the matrix X is premultiplied by
diag(R) so that it solves the original system before
equilibration.
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.
Parameters
FACT
FACT is CHARACTER*1
Specifies whether or not the factored form of the matrix A is
supplied on entry, and if not, whether the matrix A should be
equilibrated before it is factored.
= 'F': On entry, AF and IPIV contain the factored form of A.
If EQUED is not 'N', the matrix A has been
equilibrated with scaling factors given by S.
A, AF, and IPIV are not modified.
= 'N': The matrix A will be copied to AF and factored.
= 'E': The matrix A will be equilibrated if necessary, then
copied to AF and factored.
UPLO
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 matrices B and X. NRHS >= 0.
A
A is DOUBLE PRECISION 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.
On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
diag(S)*A*diag(S).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF
AF is DOUBLE PRECISION array, dimension (LDAF,N)
If FACT = 'F', then AF is an input argument and on entry
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.
If FACT = 'N', then AF is an output argument and on exit
returns 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.
LDAF
LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
If FACT = 'F', then IPIV is an input argument and on entry
contains details of the interchanges and the block
structure of D, as determined by DSYTRF. 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.
If FACT = 'N', then IPIV is an output argument and on exit
contains details of the interchanges and the block
structure of D, as determined by DSYTRF.
EQUED
EQUED is CHARACTER*1
Specifies the form of equilibration that was done.
= 'N': No equilibration (always true if FACT = 'N').
= 'Y': Both row and column equilibration, i.e., A has been
replaced by diag(S) * A * diag(S).
EQUED is an input argument if FACT = 'F'; otherwise, it is an
output argument.
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 DOUBLE PRECISION array, dimension (LDB,NRHS)
On entry, the N-by-NRHS right hand side matrix B.
On exit,
if EQUED = 'N', B is not modified;
if EQUED = 'Y', B is overwritten by diag(S)*B;
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X
X is DOUBLE PRECISION array, dimension (LDX,NRHS)
If INFO = 0, the N-by-NRHS solution matrix X to the original
system of equations. Note that A and B are modified on exit if
EQUED .ne. 'N', and the solution to the equilibrated system is
inv(diag(S))*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.
RPVGRW
RPVGRW is DOUBLE PRECISION
Reciprocal pivot growth. On exit, this contains the reciprocal
pivot growth factor norm(A)/norm(U). The 'max absolute element'
norm is used. If this is much less than 1, then 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. If factorization fails with
0<INFO<=N, then this contains the reciprocal pivot growth factor
for the leading INFO columns of A.
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 < 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 <= 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 < 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 extra-precise refinement algorithm.
(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 DOUBLE PRECISION array, dimension (4*N)
IWORK
IWORK is INTEGER array, dimension (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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine ssysvxx (character fact, character uplo, integer n, integer nrhs, real, dimension( lda, * ) a, integer lda, real, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, character equed, real, dimension( * ) s, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldx, * ) x, integer ldx, real rcond, real rpvgrw, 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)¶
SSYSVXX
Purpose:
SSYSVXX uses the diagonal pivoting factorization to compute the
solution to a real system of linear equations A * X = B, where A
is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices.
If requested, both normwise and maximum componentwise error bounds
are returned. SSYSVXX will return a solution with a tiny
guaranteed error (O(eps) where eps is the working machine
precision) unless the matrix is very ill-conditioned, in which
case a warning is returned. Relevant condition numbers also are
calculated and returned.
SSYSVXX accepts user-provided factorizations and equilibration
factors; see the definitions of the FACT and EQUED options.
Solving with refinement and using a factorization from a previous
SSYSVXX call will also produce a solution with either O(eps)
errors or warnings, but we cannot make that claim for general
user-provided factorizations and equilibration factors if they
differ from what SSYSVXX would itself produce.
Description:
The following steps are performed:
1. If FACT = 'E', real scaling factors are computed to equilibrate
the system:
diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B
Whether or not the system will be equilibrated depends on the
scaling of the matrix A, but if equilibration is used, A is
overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
2. If FACT = 'N' or 'E', the LU decomposition is used to factor
the matrix A (after equilibration if FACT = 'E') as
A = U * D * U**T, if UPLO = 'U', or
A = L * D * L**T, if UPLO = 'L',
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.
3. If some D(i,i)=0, so that D is exactly singular, then the
routine returns with INFO = i. Otherwise, the factored form of A
is used to estimate the condition number of the matrix A (see
argument RCOND). If the reciprocal of the condition number is
less than machine precision, the routine still goes on to solve
for X and compute error bounds as described below.
4. The system of equations is solved for X using the factored form
of A.
5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
the routine will use iterative refinement to try to get a small
error and error bounds. Refinement calculates the residual to at
least twice the working precision.
6. If equilibration was used, the matrix X is premultiplied by
diag(R) so that it solves the original system before
equilibration.
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.
Parameters
FACT
FACT is CHARACTER*1
Specifies whether or not the factored form of the matrix A is
supplied on entry, and if not, whether the matrix A should be
equilibrated before it is factored.
= 'F': On entry, AF and IPIV contain the factored form of A.
If EQUED is not 'N', the matrix A has been
equilibrated with scaling factors given by S.
A, AF, and IPIV are not modified.
= 'N': The matrix A will be copied to AF and factored.
= 'E': The matrix A will be equilibrated if necessary, then
copied to AF and factored.
UPLO
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 matrices B and X. NRHS >= 0.
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.
On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
diag(S)*A*diag(S).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF
AF is REAL array, dimension (LDAF,N)
If FACT = 'F', then AF is an input argument and on entry
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.
If FACT = 'N', then AF is an output argument and on exit
returns 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.
LDAF
LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
If FACT = 'F', then IPIV is an input argument and on entry
contains details of the interchanges and the block
structure of D, as determined by SSYTRF. 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.
If FACT = 'N', then IPIV is an output argument and on exit
contains details of the interchanges and the block
structure of D, as determined by SSYTRF.
EQUED
EQUED is CHARACTER*1
Specifies the form of equilibration that was done.
= 'N': No equilibration (always true if FACT = 'N').
= 'Y': Both row and column equilibration, i.e., A has been
replaced by diag(S) * A * diag(S).
EQUED is an input argument if FACT = 'F'; otherwise, it is an
output argument.
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.
B
B is REAL array, dimension (LDB,NRHS)
On entry, the N-by-NRHS right hand side matrix B.
On exit,
if EQUED = 'N', B is not modified;
if EQUED = 'Y', B is overwritten by diag(S)*B;
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X
X is REAL array, dimension (LDX,NRHS)
If INFO = 0, the N-by-NRHS solution matrix X to the original
system of equations. Note that A and B are modified on exit if
EQUED .ne. 'N', and the solution to the equilibrated system is
inv(diag(S))*X.
LDX
LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N).
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.
RPVGRW
RPVGRW is REAL
Reciprocal pivot growth. On exit, this contains the reciprocal
pivot growth factor norm(A)/norm(U). The 'max absolute element'
norm is used. If this is much less than 1, then 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. If factorization fails with
0<INFO<=N, then this contains the reciprocal pivot growth factor
for the leading INFO columns of A.
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).
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 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.
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 < 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.
NPARAMS
NPARAMS is INTEGER
Specifies the number of parameters set in PARAMS. If <= 0, the
PARAMS array is never referenced and default values are used.
PARAMS
PARAMS is REAL array, dimension NPARAMS
Specifies algorithm parameters. If an entry is < 0.0, then
that entry will be filled with default value used for that
parameter. Only positions up to NPARAMS are accessed; defaults
are used for higher-numbered parameters.
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)
WORK
WORK is REAL array, dimension (4*N)
IWORK
IWORK is INTEGER array, dimension (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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zhesvxx (character fact, character uplo, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, character equed, double precision, dimension( * ) s, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldx, * ) x, integer ldx, double precision rcond, double precision rpvgrw, double precision, dimension( * ) berr, integer n_err_bnds, double precision, dimension( nrhs, * ) err_bnds_norm, double precision, dimension( nrhs, * ) err_bnds_comp, integer nparams, double precision, dimension( * ) params, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)¶
ZHESVXX computes the solution to system of linear equations A * X = B for HE matrices
Purpose:
ZHESVXX uses the diagonal pivoting factorization to compute the
solution to a complex*16 system of linear equations A * X = B, where
A is an N-by-N Hermitian matrix and X and B are N-by-NRHS
matrices.
If requested, both normwise and maximum componentwise error bounds
are returned. ZHESVXX will return a solution with a tiny
guaranteed error (O(eps) where eps is the working machine
precision) unless the matrix is very ill-conditioned, in which
case a warning is returned. Relevant condition numbers also are
calculated and returned.
ZHESVXX accepts user-provided factorizations and equilibration
factors; see the definitions of the FACT and EQUED options.
Solving with refinement and using a factorization from a previous
ZHESVXX call will also produce a solution with either O(eps)
errors or warnings, but we cannot make that claim for general
user-provided factorizations and equilibration factors if they
differ from what ZHESVXX would itself produce.
Description:
The following steps are performed:
1. If FACT = 'E', double precision scaling factors are computed to equilibrate
the system:
diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B
Whether or not the system will be equilibrated depends on the
scaling of the matrix A, but if equilibration is used, A is
overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
2. If FACT = 'N' or 'E', the LU decomposition is used to factor
the matrix A (after equilibration if FACT = 'E') as
A = U * D * U**T, if UPLO = 'U', or
A = L * D * L**T, if UPLO = 'L',
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.
3. If some D(i,i)=0, so that D is exactly singular, then the
routine returns with INFO = i. Otherwise, the factored form of A
is used to estimate the condition number of the matrix A (see
argument RCOND). If the reciprocal of the condition number is
less than machine precision, the routine still goes on to solve
for X and compute error bounds as described below.
4. The system of equations is solved for X using the factored form
of A.
5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
the routine will use iterative refinement to try to get a small
error and error bounds. Refinement calculates the residual to at
least twice the working precision.
6. If equilibration was used, the matrix X is premultiplied by
diag(R) so that it solves the original system before
equilibration.
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.
Parameters
FACT
FACT is CHARACTER*1
Specifies whether or not the factored form of the matrix A is
supplied on entry, and if not, whether the matrix A should be
equilibrated before it is factored.
= 'F': On entry, AF and IPIV contain the factored form of A.
If EQUED is not 'N', the matrix A has been
equilibrated with scaling factors given by S.
A, AF, and IPIV are not modified.
= 'N': The matrix A will be copied to AF and factored.
= 'E': The matrix A will be equilibrated if necessary, then
copied to AF and factored.
UPLO
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 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.
On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
diag(S)*A*diag(S).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF
AF is COMPLEX*16 array, dimension (LDAF,N)
If FACT = 'F', then AF is an input argument and on entry
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.
If FACT = 'N', then AF is an output argument and on exit
returns 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.
LDAF
LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
If FACT = 'F', then IPIV is an input argument and on entry
contains details of the interchanges and the block
structure of D, as determined by ZHETRF. 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.
If FACT = 'N', then IPIV is an output argument and on exit
contains details of the interchanges and the block
structure of D, as determined by ZHETRF.
EQUED
EQUED is CHARACTER*1
Specifies the form of equilibration that was done.
= 'N': No equilibration (always true if FACT = 'N').
= 'Y': Both row and column equilibration, i.e., A has been
replaced by diag(S) * A * diag(S).
EQUED is an input argument if FACT = 'F'; otherwise, it is an
output argument.
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)
On entry, the N-by-NRHS right hand side matrix B.
On exit,
if EQUED = 'N', B is not modified;
if EQUED = 'Y', B is overwritten by diag(S)*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)
If INFO = 0, the N-by-NRHS solution matrix X to the original
system of equations. Note that A and B are modified on exit if
EQUED .ne. 'N', and the solution to the equilibrated system is
inv(diag(S))*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.
RPVGRW
RPVGRW is DOUBLE PRECISION
Reciprocal pivot growth. On exit, this contains the reciprocal
pivot growth factor norm(A)/norm(U). The 'max absolute element'
norm is used. If this is much less than 1, then 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. If factorization fails with
0<INFO<=N, then this contains the reciprocal pivot growth factor
for the leading INFO columns of A.
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 < 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 <= 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 < 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 extra-precise refinement algorithm.
(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 (5*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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zsysvxx (character fact, character uplo, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, character equed, double precision, dimension( * ) s, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldx, * ) x, integer ldx, double precision rcond, double precision rpvgrw, double precision, dimension( * ) berr, integer n_err_bnds, double precision, dimension( nrhs, * ) err_bnds_norm, double precision, dimension( nrhs, * ) err_bnds_comp, integer nparams, double precision, dimension( * ) params, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)¶
ZSYSVXX computes the solution to system of linear equations A * X = B for SY matrices
Purpose:
ZSYSVXX uses the diagonal pivoting factorization to compute the
solution to a complex*16 system of linear equations A * X = B, where
A is an N-by-N symmetric matrix and X and B are N-by-NRHS
matrices.
If requested, both normwise and maximum componentwise error bounds
are returned. ZSYSVXX will return a solution with a tiny
guaranteed error (O(eps) where eps is the working machine
precision) unless the matrix is very ill-conditioned, in which
case a warning is returned. Relevant condition numbers also are
calculated and returned.
ZSYSVXX accepts user-provided factorizations and equilibration
factors; see the definitions of the FACT and EQUED options.
Solving with refinement and using a factorization from a previous
ZSYSVXX call will also produce a solution with either O(eps)
errors or warnings, but we cannot make that claim for general
user-provided factorizations and equilibration factors if they
differ from what ZSYSVXX would itself produce.
Description:
The following steps are performed:
1. If FACT = 'E', double precision scaling factors are computed to equilibrate
the system:
diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B
Whether or not the system will be equilibrated depends on the
scaling of the matrix A, but if equilibration is used, A is
overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
2. If FACT = 'N' or 'E', the LU decomposition is used to factor
the matrix A (after equilibration if FACT = 'E') as
A = U * D * U**T, if UPLO = 'U', or
A = L * D * L**T, if UPLO = 'L',
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.
3. If some D(i,i)=0, so that D is exactly singular, then the
routine returns with INFO = i. Otherwise, the factored form of A
is used to estimate the condition number of the matrix A (see
argument RCOND). If the reciprocal of the condition number is
less than machine precision, the routine still goes on to solve
for X and compute error bounds as described below.
4. The system of equations is solved for X using the factored form
of A.
5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
the routine will use iterative refinement to try to get a small
error and error bounds. Refinement calculates the residual to at
least twice the working precision.
6. If equilibration was used, the matrix X is premultiplied by
diag(R) so that it solves the original system before
equilibration.
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.
Parameters
FACT
FACT is CHARACTER*1
Specifies whether or not the factored form of the matrix A is
supplied on entry, and if not, whether the matrix A should be
equilibrated before it is factored.
= 'F': On entry, AF and IPIV contain the factored form of A.
If EQUED is not 'N', the matrix A has been
equilibrated with scaling factors given by S.
A, AF, and IPIV are not modified.
= 'N': The matrix A will be copied to AF and factored.
= 'E': The matrix A will be equilibrated if necessary, then
copied to AF and factored.
UPLO
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 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.
On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
diag(S)*A*diag(S).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF
AF is COMPLEX*16 array, dimension (LDAF,N)
If FACT = 'F', then AF is an input argument and on entry
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 ZSYTRF.
If FACT = 'N', then AF is an output argument and on exit
returns 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.
LDAF
LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
If FACT = 'F', then IPIV is an input argument and on entry
contains details of the interchanges and the block
structure of D, as determined by ZSYTRF. 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.
If FACT = 'N', then IPIV is an output argument and on exit
contains details of the interchanges and the block
structure of D, as determined by ZSYTRF.
EQUED
EQUED is CHARACTER*1
Specifies the form of equilibration that was done.
= 'N': No equilibration (always true if FACT = 'N').
= 'Y': Both row and column equilibration, i.e., A has been
replaced by diag(S) * A * diag(S).
EQUED is an input argument if FACT = 'F'; otherwise, it is an
output argument.
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)
On entry, the N-by-NRHS right hand side matrix B.
On exit,
if EQUED = 'N', B is not modified;
if EQUED = 'Y', B is overwritten by diag(S)*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)
If INFO = 0, the N-by-NRHS solution matrix X to the original
system of equations. Note that A and B are modified on exit if
EQUED .ne. 'N', and the solution to the equilibrated system is
inv(diag(S))*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.
RPVGRW
RPVGRW is DOUBLE PRECISION
Reciprocal pivot growth. On exit, this contains the reciprocal
pivot growth factor norm(A)/norm(U). The 'max absolute element'
norm is used. If this is much less than 1, then 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. If factorization fails with
0<INFO<=N, then this contains the reciprocal pivot growth factor
for the leading INFO columns of A.
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 < 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 <= 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 < 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 extra-precise refinement algorithm.
(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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Author¶
Generated automatically by Doxygen for LAPACK from the source code.
| Wed Feb 7 2024 11:30:40 | Version 3.12.0 |