ZGESVX(3F) ZGESVX(3F)
ZGESVX - use the LU factorization to compute the solution to a complex
system of linear equations A * X = B,
SUBROUTINE ZGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, EQUED,
R, C, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK,
INFO )
CHARACTER EQUED, FACT, TRANS
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
INTEGER IPIV( * )
DOUBLE PRECISION BERR( * ), C( * ), FERR( * ), R( * ), RWORK(
* )
COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ), WORK( * ), X(
LDX, * )
ZGESVX uses the LU factorization to compute the solution to a complex
system of linear equations
A * X = B, where A is an N-by-N 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 = 'E', real scaling factors are computed to equilibrate
the system:
TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*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(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
or diag(C)*B (if TRANS = 'T' or 'C').
2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
matrix A (after equilibration if FACT = 'E') as
A = P * L * U,
where P is a permutation matrix, L is a unit lower triangular
matrix, and U is upper triangular.
3. The factored form of A is used to estimate the condition number
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ZGESVX(3F) ZGESVX(3F)
of the matrix A. If the reciprocal of the condition number is
less than machine precision, steps 4-6 are skipped.
4. The system of equations is solved for X using the factored form
of A.
5. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.
6. If equilibration was used, the matrix X is premultiplied by
diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
that it solves the original system before equilibration.
FACT (input) 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 R
and C. 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.
TRANS (input) CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A**T * X = B (Transpose)
= 'C': A**H * X = B (Conjugate transpose)
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/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is not
'N', then A must have been equilibrated by the scaling factors in
R and/or C. A is not modified if FACT = 'F' or
On exit, if EQUED .ne. 'N', A is scaled as follows: EQUED = 'R':
A := diag(R) * A
EQUED = 'C': A := A * diag(C)
EQUED = 'B': A := diag(R) * A * diag(C).
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ZGESVX(3F) ZGESVX(3F)
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 factors L and U from the factorization A = P*L*U as computed
by ZGETRF. If EQUED .ne. 'N', then AF is the factored form of
the equilibrated matrix A.
If FACT = 'N', then AF is an output argument and on exit returns
the factors L and U from the factorization A = P*L*U of the
original matrix A.
If FACT = 'E', then AF is an output argument and on exit returns
the factors L and U from the factorization A = P*L*U of the
equilibrated matrix A (see the description of A for the form of
the equilibrated matrix).
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 the pivot indices from the factorization A = P*L*U as
computed by ZGETRF; row i of the matrix was interchanged with row
IPIV(i).
If FACT = 'N', then IPIV is an output argument and on exit
contains the pivot indices from the factorization A = P*L*U of
the original matrix A.
If FACT = 'E', then IPIV is an output argument and on exit
contains the pivot indices from the factorization A = P*L*U of
the equilibrated matrix A.
EQUED (input or output) CHARACTER*1
Specifies the form of equilibration that was done. = 'N': No
equilibration (always true if FACT = 'N').
= 'R': Row equilibration, i.e., A has been premultiplied by
diag(R). = 'C': Column equilibration, i.e., A has been
postmultiplied by diag(C). = 'B': Both row and column
equilibration, i.e., A has been replaced by diag(R) * A *
diag(C). EQUED is an input argument if FACT = 'F'; otherwise, it
is an output argument.
R (input or output) DOUBLE PRECISION array, dimension (N)
The row scale factors for A. If EQUED = 'R' or 'B', A is
multiplied on the left by diag(R); if EQUED = 'N' or 'C', R is
not accessed. R is an input argument if FACT = 'F'; otherwise, R
is an output argument. If FACT = 'F' and EQUED = 'R' or 'B',
each element of R must be positive.
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ZGESVX(3F) ZGESVX(3F)
C (input or output) DOUBLE PRECISION array, dimension (N)
The column scale factors for A. If EQUED = 'C' or 'B', A is
multiplied on the right by diag(C); if EQUED = 'N' or 'R', C is
not accessed. C is an input argument if FACT = 'F'; otherwise, C
is an output argument. If FACT = 'F' and EQUED = 'C' or 'B',
each element of C must be positive.
B (input/output) 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 TRANS = 'N' and EQUED = 'R' or
'B', B is overwritten by diag(R)*B; if TRANS = 'T' or 'C' and
EQUED = 'C' or 'B', B is overwritten by diag(C)*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 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(C))*X if TRANS = 'N' and EQUED = 'C' or or 'B'.
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
after equilibration (if done). 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) COMPLEX*16 array, dimension (2*N)
RWORK (workspace/output) DOUBLE PRECISION array, dimension (2*N)
On exit, RWORK(1) contains the reciprocal pivot growth factor
norm(A)/norm(U). The "max absolute element" norm is used. If
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ZGESVX(3F) ZGESVX(3F)
RWORK(1) 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, condition estimator RCOND, and
forward error bound FERR could be unreliable. If factorization
fails with 0<INFO<=N, then RWORK(1) contains the reciprocal pivot
growth factor for the leading INFO columns of A.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, and i is
<= N: U(i,i) 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. = N+1: RCOND is less
than machine precision. The factorization has been completed,
but the matrix is singular to working precision, and the solution
and error bounds have not been computed.
ZGESVX(3F) ZGESVX(3F)
ZGESVX - use the LU factorization to compute the solution to a complex
system of linear equations A * X = B,
SUBROUTINE ZGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, EQUED,
R, C, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK,
INFO )
CHARACTER EQUED, FACT, TRANS
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
INTEGER IPIV( * )
DOUBLE PRECISION BERR( * ), C( * ), FERR( * ), R( * ), RWORK(
* )
COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ), WORK( * ), X(
LDX, * )
ZGESVX uses the LU factorization to compute the solution to a complex
system of linear equations
A * X = B, where A is an N-by-N 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 = 'E', real scaling factors are computed to equilibrate
the system:
TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*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(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
or diag(C)*B (if TRANS = 'T' or 'C').
2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
matrix A (after equilibration if FACT = 'E') as
A = P * L * U,
where P is a permutation matrix, L is a unit lower triangular
matrix, and U is upper triangular.
3. The factored form of A is used to estimate the condition number
Page 1
ZGESVX(3F) ZGESVX(3F)
of the matrix A. If the reciprocal of the condition number is
less than machine precision, steps 4-6 are skipped.
4. The system of equations is solved for X using the factored form
of A.
5. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.
6. If equilibration was used, the matrix X is premultiplied by
diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
that it solves the original system before equilibration.
FACT (input) 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 R
and C. 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.
TRANS (input) CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A**T * X = B (Transpose)
= 'C': A**H * X = B (Conjugate transpose)
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/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is not
'N', then A must have been equilibrated by the scaling factors in
R and/or C. A is not modified if FACT = 'F' or
On exit, if EQUED .ne. 'N', A is scaled as follows: EQUED = 'R':
A := diag(R) * A
EQUED = 'C': A := A * diag(C)
EQUED = 'B': A := diag(R) * A * diag(C).
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ZGESVX(3F) ZGESVX(3F)
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 factors L and U from the factorization A = P*L*U as computed
by ZGETRF. If EQUED .ne. 'N', then AF is the factored form of
the equilibrated matrix A.
If FACT = 'N', then AF is an output argument and on exit returns
the factors L and U from the factorization A = P*L*U of the
original matrix A.
If FACT = 'E', then AF is an output argument and on exit returns
the factors L and U from the factorization A = P*L*U of the
equilibrated matrix A (see the description of A for the form of
the equilibrated matrix).
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 the pivot indices from the factorization A = P*L*U as
computed by ZGETRF; row i of the matrix was interchanged with row
IPIV(i).
If FACT = 'N', then IPIV is an output argument and on exit
contains the pivot indices from the factorization A = P*L*U of
the original matrix A.
If FACT = 'E', then IPIV is an output argument and on exit
contains the pivot indices from the factorization A = P*L*U of
the equilibrated matrix A.
EQUED (input or output) CHARACTER*1
Specifies the form of equilibration that was done. = 'N': No
equilibration (always true if FACT = 'N').
= 'R': Row equilibration, i.e., A has been premultiplied by
diag(R). = 'C': Column equilibration, i.e., A has been
postmultiplied by diag(C). = 'B': Both row and column
equilibration, i.e., A has been replaced by diag(R) * A *
diag(C). EQUED is an input argument if FACT = 'F'; otherwise, it
is an output argument.
R (input or output) DOUBLE PRECISION array, dimension (N)
The row scale factors for A. If EQUED = 'R' or 'B', A is
multiplied on the left by diag(R); if EQUED = 'N' or 'C', R is
not accessed. R is an input argument if FACT = 'F'; otherwise, R
is an output argument. If FACT = 'F' and EQUED = 'R' or 'B',
each element of R must be positive.
Page 3
ZGESVX(3F) ZGESVX(3F)
C (input or output) DOUBLE PRECISION array, dimension (N)
The column scale factors for A. If EQUED = 'C' or 'B', A is
multiplied on the right by diag(C); if EQUED = 'N' or 'R', C is
not accessed. C is an input argument if FACT = 'F'; otherwise, C
is an output argument. If FACT = 'F' and EQUED = 'C' or 'B',
each element of C must be positive.
B (input/output) 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 TRANS = 'N' and EQUED = 'R' or
'B', B is overwritten by diag(R)*B; if TRANS = 'T' or 'C' and
EQUED = 'C' or 'B', B is overwritten by diag(C)*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 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(C))*X if TRANS = 'N' and EQUED = 'C' or or 'B'.
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
after equilibration (if done). 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) COMPLEX*16 array, dimension (2*N)
RWORK (workspace/output) DOUBLE PRECISION array, dimension (2*N)
On exit, RWORK(1) contains the reciprocal pivot growth factor
norm(A)/norm(U). The "max absolute element" norm is used. If
Page 4
ZGESVX(3F) ZGESVX(3F)
RWORK(1) 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, condition estimator RCOND, and
forward error bound FERR could be unreliable. If factorization
fails with 0<INFO<=N, then RWORK(1) contains the reciprocal pivot
growth factor for the leading INFO columns of A.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, and i is
<= N: U(i,i) 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. = N+1: RCOND is less
than machine precision. The factorization has been completed,
but the matrix is singular to working precision, and the solution
and error bounds have not been computed.
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