CGTSVX(3F) CGTSVX(3F)
CGTSVX - use the LU factorization to compute the solution to a complex
system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
SUBROUTINE CGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2,
IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK,
INFO )
CHARACTER FACT, TRANS
INTEGER INFO, LDB, LDX, N, NRHS
REAL RCOND
INTEGER IPIV( * )
REAL BERR( * ), FERR( * ), RWORK( * )
COMPLEX B( LDB, * ), D( * ), DF( * ), DL( * ), DLF( * ), DU( *
), DU2( * ), DUF( * ), WORK( * ), X( LDX, * )
CGTSVX uses the LU factorization to compute the solution to a complex
system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
where A is a tridiagonal matrix of order N 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 LU decomposition is used to factor the matrix A
as A = L * U, where L is a product of permutation and unit lower
bidiagonal matrices and U is upper triangular with nonzeros in
only the main diagonal and first two superdiagonals.
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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CGTSVX(3F) CGTSVX(3F)
FACT (input) CHARACTER*1
Specifies whether or not the factored form of A has been supplied
on entry. = 'F': DLF, DF, DUF, DU2, and IPIV contain the
factored form of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV will
not be modified. = 'N': The matrix will be copied to DLF, DF,
and DUF 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 order of the matrix A. N >= 0.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of columns of
the matrix B. NRHS >= 0.
DL (input) COMPLEX array, dimension (N-1)
The (n-1) subdiagonal elements of A.
D (input) COMPLEX array, dimension (N)
The n diagonal elements of A.
DU (input) COMPLEX array, dimension (N-1)
The (n-1) superdiagonal elements of A.
DLF (input or output) COMPLEX array, dimension (N-1)
If FACT = 'F', then DLF is an input argument and on entry
contains the (n-1) multipliers that define the matrix L from the
LU factorization of A as computed by CGTTRF.
If FACT = 'N', then DLF is an output argument and on exit
contains the (n-1) multipliers that define the matrix L from the
LU factorization of A.
DF (input or output) COMPLEX array, dimension (N)
If FACT = 'F', then DF is an input argument and on entry contains
the n diagonal elements of the upper triangular matrix U from the
LU factorization of A.
If FACT = 'N', then DF is an output argument and on exit contains
the n diagonal elements of the upper triangular matrix U from the
LU factorization of A.
DUF (input or output) COMPLEX array, dimension (N-1)
If FACT = 'F', then DUF is an input argument and on entry
contains the (n-1) elements of the first superdiagonal of U.
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CGTSVX(3F) CGTSVX(3F)
If FACT = 'N', then DUF is an output argument and on exit
contains the (n-1) elements of the first superdiagonal of U.
DU2 (input or output) COMPLEX array, dimension (N-2)
If FACT = 'F', then DU2 is an input argument and on entry
contains the (n-2) elements of the second superdiagonal of U.
If FACT = 'N', then DU2 is an output argument and on exit
contains the (n-2) elements of the second superdiagonal of U.
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 LU factorization of A as
computed by CGTTRF.
If FACT = 'N', then IPIV is an output argument and on exit
contains the pivot indices from the LU factorization of A; row i
of the matrix was interchanged with row IPIV(i). IPIV(i) will
always be either i or i+1; IPIV(i) = i indicates a row
interchange was not required.
B (input) COMPLEX 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 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) REAL
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) REAL array, dimension (NRHS)
The estimated forward error bound for each solution vector X(j)
(the j-th column of the solution matrix X). If XTRUE is the true
solution corresponding to X(j), FERR(j) is an estimated upper
bound for the magnitude of the largest element in (X(j) - XTRUE)
divided by the magnitude of the largest element in X(j). The
estimate is as reliable as the estimate for RCOND, and is almost
always a slight overestimate of the true error.
BERR (output) REAL array, dimension (NRHS)
The componentwise relative backward error of each solution vector
X(j) (i.e., the smallest relative change in any element of A or B
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CGTSVX(3F) CGTSVX(3F)
that makes X(j) an exact solution).
WORK (workspace) COMPLEX array, dimension (2*N)
RWORK (workspace) REAL array, dimension (N)
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 not been
completed unless i = N, 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.
CGTSVX(3F) CGTSVX(3F)
CGTSVX - use the LU factorization to compute the solution to a complex
system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
SUBROUTINE CGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2,
IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK,
INFO )
CHARACTER FACT, TRANS
INTEGER INFO, LDB, LDX, N, NRHS
REAL RCOND
INTEGER IPIV( * )
REAL BERR( * ), FERR( * ), RWORK( * )
COMPLEX B( LDB, * ), D( * ), DF( * ), DL( * ), DLF( * ), DU( *
), DU2( * ), DUF( * ), WORK( * ), X( LDX, * )
CGTSVX uses the LU factorization to compute the solution to a complex
system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
where A is a tridiagonal matrix of order N 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 LU decomposition is used to factor the matrix A
as A = L * U, where L is a product of permutation and unit lower
bidiagonal matrices and U is upper triangular with nonzeros in
only the main diagonal and first two superdiagonals.
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
CGTSVX(3F) CGTSVX(3F)
FACT (input) CHARACTER*1
Specifies whether or not the factored form of A has been supplied
on entry. = 'F': DLF, DF, DUF, DU2, and IPIV contain the
factored form of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV will
not be modified. = 'N': The matrix will be copied to DLF, DF,
and DUF 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 order of the matrix A. N >= 0.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of columns of
the matrix B. NRHS >= 0.
DL (input) COMPLEX array, dimension (N-1)
The (n-1) subdiagonal elements of A.
D (input) COMPLEX array, dimension (N)
The n diagonal elements of A.
DU (input) COMPLEX array, dimension (N-1)
The (n-1) superdiagonal elements of A.
DLF (input or output) COMPLEX array, dimension (N-1)
If FACT = 'F', then DLF is an input argument and on entry
contains the (n-1) multipliers that define the matrix L from the
LU factorization of A as computed by CGTTRF.
If FACT = 'N', then DLF is an output argument and on exit
contains the (n-1) multipliers that define the matrix L from the
LU factorization of A.
DF (input or output) COMPLEX array, dimension (N)
If FACT = 'F', then DF is an input argument and on entry contains
the n diagonal elements of the upper triangular matrix U from the
LU factorization of A.
If FACT = 'N', then DF is an output argument and on exit contains
the n diagonal elements of the upper triangular matrix U from the
LU factorization of A.
DUF (input or output) COMPLEX array, dimension (N-1)
If FACT = 'F', then DUF is an input argument and on entry
contains the (n-1) elements of the first superdiagonal of U.
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CGTSVX(3F) CGTSVX(3F)
If FACT = 'N', then DUF is an output argument and on exit
contains the (n-1) elements of the first superdiagonal of U.
DU2 (input or output) COMPLEX array, dimension (N-2)
If FACT = 'F', then DU2 is an input argument and on entry
contains the (n-2) elements of the second superdiagonal of U.
If FACT = 'N', then DU2 is an output argument and on exit
contains the (n-2) elements of the second superdiagonal of U.
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 LU factorization of A as
computed by CGTTRF.
If FACT = 'N', then IPIV is an output argument and on exit
contains the pivot indices from the LU factorization of A; row i
of the matrix was interchanged with row IPIV(i). IPIV(i) will
always be either i or i+1; IPIV(i) = i indicates a row
interchange was not required.
B (input) COMPLEX 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 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) REAL
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) REAL array, dimension (NRHS)
The estimated forward error bound for each solution vector X(j)
(the j-th column of the solution matrix X). If XTRUE is the true
solution corresponding to X(j), FERR(j) is an estimated upper
bound for the magnitude of the largest element in (X(j) - XTRUE)
divided by the magnitude of the largest element in X(j). The
estimate is as reliable as the estimate for RCOND, and is almost
always a slight overestimate of the true error.
BERR (output) REAL array, dimension (NRHS)
The componentwise relative backward error of each solution vector
X(j) (i.e., the smallest relative change in any element of A or B
Page 3
CGTSVX(3F) CGTSVX(3F)
that makes X(j) an exact solution).
WORK (workspace) COMPLEX array, dimension (2*N)
RWORK (workspace) REAL array, dimension (N)
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 not been
completed unless i = N, 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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