ZSYSVX(3F) ZSYSVX(3F)
ZSYSVX - use the diagonal pivoting factorization to compute the solution
to a complex system of linear equations A * X = B,
SUBROUTINE ZSYSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
X, LDX, RCOND, FERR, BERR, WORK, LWORK, RWORK, INFO )
CHARACTER FACT, UPLO
INTEGER INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS
DOUBLE PRECISION RCOND
INTEGER IPIV( * )
DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ), WORK( * ), X(
LDX, * )
ZSYSVX 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.
Error bounds on the solution and a condition estimate are also provided.
The following steps are performed:
1. If FACT = 'N', the diagonal pivoting method is used to factor A.
The form of the factorization is
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.
2. The factored form of A is used to estimate the condition number
of the matrix A. If the reciprocal of the condition number is
less than machine precision, steps 3 and 4 are skipped.
3. The system of equations is solved for X using the factored form
of A.
4. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.
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ZSYSVX(3F) ZSYSVX(3F)
FACT (input) CHARACTER*1
Specifies whether or not the factored form of A has been supplied
on entry. = 'F': On entry, AF and IPIV contain the factored
form of A. A, AF and IPIV will not be modified. = 'N': The
matrix A will be copied to AF and factored.
UPLO (input) CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N (input) INTEGER
The number of linear equations, i.e., the order of the matrix A.
N >= 0.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of columns of
the matrices B and X. NRHS >= 0.
A (input) COMPLEX*16 array, dimension (LDA,N)
The symmetric matrix A. If UPLO = 'U', the leading N-by-N upper
triangular part of A contains the upper triangular part of the
matrix A, and the strictly lower triangular part of A is not
referenced. If UPLO = 'L', the leading N-by-N lower triangular
part of A contains the lower triangular part of the matrix A, and
the strictly upper triangular part of A is not referenced.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF (input or output) 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 (input) INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV (input or output) 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(k1:k,k-1:k)
is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k)
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ZSYSVX(3F) ZSYSVX(3F)
= 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.
B (input) COMPLEX*16 array, dimension (LDB,NRHS)
The N-by-NRHS right hand side matrix B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X (output) COMPLEX*16 array, dimension (LDX,NRHS)
If INFO = 0, the N-by-NRHS solution matrix X.
LDX (input) INTEGER
The leading dimension of the array X. LDX >= max(1,N).
RCOND (output) DOUBLE PRECISION
The estimate of the reciprocal condition number of the matrix A.
If RCOND is less than the machine precision (in particular, if
RCOND = 0), the matrix is singular to working precision. This
condition is indicated by a return code of INFO > 0, and the
solution and error bounds are not computed.
FERR (output) DOUBLE PRECISION array, dimension (NRHS)
The estimated forward error bound for each solution vector X(j)
(the j-th column of the solution matrix X). If XTRUE is the true
solution corresponding to X(j), FERR(j) is an estimated upper
bound for the magnitude of the largest element in (X(j) - XTRUE)
divided by the magnitude of the largest element in X(j). The
estimate is as reliable as the estimate for RCOND, and is almost
always a slight overestimate of the true error.
BERR (output) DOUBLE PRECISION array, dimension (NRHS)
The componentwise relative backward error of each solution vector
X(j) (i.e., the smallest relative change in any element of A or B
that makes X(j) an exact solution).
WORK (workspace/output) COMPLEX*16 array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The length of WORK. LWORK >= 2*N, and for best performance LWORK
>= N*NB, where NB is the optimal blocksize for ZSYTRF.
RWORK (workspace) DOUBLE PRECISION array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Page 3
ZSYSVX(3F) ZSYSVX(3F)
> 0: if INFO = i, and i is
<= N: D(i,i) is exactly zero. The factorization has been
completed, but the block diagonal matrix D is exactly singular,
so the solution and error bounds could not be computed. = N+1:
the block diagonal matrix D is nonsingular, but RCOND is less
than machine precision. The factorization has been completed,
but the matrix is singular to working precision, so the solution
and error bounds have not been computed.
ZSYSVX(3F) ZSYSVX(3F)
ZSYSVX - use the diagonal pivoting factorization to compute the solution
to a complex system of linear equations A * X = B,
SUBROUTINE ZSYSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
X, LDX, RCOND, FERR, BERR, WORK, LWORK, RWORK, INFO )
CHARACTER FACT, UPLO
INTEGER INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS
DOUBLE PRECISION RCOND
INTEGER IPIV( * )
DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ), WORK( * ), X(
LDX, * )
ZSYSVX 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.
Error bounds on the solution and a condition estimate are also provided.
The following steps are performed:
1. If FACT = 'N', the diagonal pivoting method is used to factor A.
The form of the factorization is
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.
2. The factored form of A is used to estimate the condition number
of the matrix A. If the reciprocal of the condition number is
less than machine precision, steps 3 and 4 are skipped.
3. The system of equations is solved for X using the factored form
of A.
4. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.
Page 1
ZSYSVX(3F) ZSYSVX(3F)
FACT (input) CHARACTER*1
Specifies whether or not the factored form of A has been supplied
on entry. = 'F': On entry, AF and IPIV contain the factored
form of A. A, AF and IPIV will not be modified. = 'N': The
matrix A will be copied to AF and factored.
UPLO (input) CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N (input) INTEGER
The number of linear equations, i.e., the order of the matrix A.
N >= 0.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of columns of
the matrices B and X. NRHS >= 0.
A (input) COMPLEX*16 array, dimension (LDA,N)
The symmetric matrix A. If UPLO = 'U', the leading N-by-N upper
triangular part of A contains the upper triangular part of the
matrix A, and the strictly lower triangular part of A is not
referenced. If UPLO = 'L', the leading N-by-N lower triangular
part of A contains the lower triangular part of the matrix A, and
the strictly upper triangular part of A is not referenced.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF (input or output) 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 (input) INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV (input or output) 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(k1:k,k-1:k)
is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k)
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ZSYSVX(3F) ZSYSVX(3F)
= 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.
B (input) COMPLEX*16 array, dimension (LDB,NRHS)
The N-by-NRHS right hand side matrix B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X (output) COMPLEX*16 array, dimension (LDX,NRHS)
If INFO = 0, the N-by-NRHS solution matrix X.
LDX (input) INTEGER
The leading dimension of the array X. LDX >= max(1,N).
RCOND (output) DOUBLE PRECISION
The estimate of the reciprocal condition number of the matrix A.
If RCOND is less than the machine precision (in particular, if
RCOND = 0), the matrix is singular to working precision. This
condition is indicated by a return code of INFO > 0, and the
solution and error bounds are not computed.
FERR (output) DOUBLE PRECISION array, dimension (NRHS)
The estimated forward error bound for each solution vector X(j)
(the j-th column of the solution matrix X). If XTRUE is the true
solution corresponding to X(j), FERR(j) is an estimated upper
bound for the magnitude of the largest element in (X(j) - XTRUE)
divided by the magnitude of the largest element in X(j). The
estimate is as reliable as the estimate for RCOND, and is almost
always a slight overestimate of the true error.
BERR (output) DOUBLE PRECISION array, dimension (NRHS)
The componentwise relative backward error of each solution vector
X(j) (i.e., the smallest relative change in any element of A or B
that makes X(j) an exact solution).
WORK (workspace/output) COMPLEX*16 array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The length of WORK. LWORK >= 2*N, and for best performance LWORK
>= N*NB, where NB is the optimal blocksize for ZSYTRF.
RWORK (workspace) DOUBLE PRECISION array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Page 3
ZSYSVX(3F) ZSYSVX(3F)
> 0: if INFO = i, and i is
<= N: D(i,i) is exactly zero. The factorization has been
completed, but the block diagonal matrix D is exactly singular,
so the solution and error bounds could not be computed. = N+1:
the block diagonal matrix D is nonsingular, but RCOND is less
than machine precision. The factorization has been completed,
but the matrix is singular to working precision, so the solution
and error bounds have not been computed.
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