.TH DGTSVX 1 "November 2006" " LAPACK routine (version 3.1) " " LAPACK routine (version 3.1) "
.SH NAME
DGTSVX - the LU factorization to compute the solution to a real system of linear equations A * X = B or A**T * X = B,
.SH SYNOPSIS
.TP 19
SUBROUTINE DGTSVX(
FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF,
DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR,
WORK, IWORK, INFO )
.TP 19
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CHARACTER
FACT, TRANS
.TP 19
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INTEGER
INFO, LDB, LDX, N, NRHS
.TP 19
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DOUBLE
PRECISION RCOND
.TP 19
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INTEGER
IPIV( * ), IWORK( * )
.TP 19
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DOUBLE
PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ),
FERR( * ), WORK( * ), X( LDX, * )
.SH PURPOSE
DGTSVX uses the LU factorization to compute the solution to a real
system of linear equations A * X = B or A**T * X = B,
where A is a tridiagonal matrix of order N and X and B are N-by-NRHS
matrices.
.br
Error bounds on the solution and a condition estimate are also
provided.
.br
.SH DESCRIPTION
The following steps are performed:
.br
1. If FACT = \(aqN\(aq, 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.
.br
2. If some U(i,i)=0, so that U 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. If the
reciprocal of the condition number is less than machine precision,
INFO = N+1 is returned as a warning, but the routine still goes on
to solve for X and compute error bounds as described below.
3. The system of equations is solved for X using the factored form
of A.
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4. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.
.br
.SH ARGUMENTS
.TP 8
FACT (input) CHARACTER*1
Specifies whether or not the factored form of A has been
supplied on entry.
= \(aqF\(aq: 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.
= \(aqN\(aq: The matrix will be copied to DLF, DF, and DUF
and factored.
.TP 8
TRANS (input) CHARACTER*1
Specifies the form of the system of equations:
.br
= \(aqN\(aq: A * X = B (No transpose)
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= \(aqT\(aq: A**T * X = B (Transpose)
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= \(aqC\(aq: A**H * X = B (Conjugate transpose = Transpose)
.TP 8
N (input) INTEGER
The order of the matrix A. N >= 0.
.TP 8
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
.TP 8
DL (input) DOUBLE PRECISION array, dimension (N-1)
The (n-1) subdiagonal elements of A.
.TP 8
D (input) DOUBLE PRECISION array, dimension (N)
The n diagonal elements of A.
.TP 8
DU (input) DOUBLE PRECISION array, dimension (N-1)
The (n-1) superdiagonal elements of A.
.TP 8
DLF (input or output) DOUBLE PRECISION array, dimension (N-1)
If FACT = \(aqF\(aq, 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 DGTTRF.
If FACT = \(aqN\(aq, 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.
.TP 8
DF (input or output) DOUBLE PRECISION array, dimension (N)
If FACT = \(aqF\(aq, 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 = \(aqN\(aq, 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.
.TP 8
DUF (input or output) DOUBLE PRECISION array, dimension (N-1)
If FACT = \(aqF\(aq, then DUF is an input argument and on entry
contains the (n-1) elements of the first superdiagonal of U.
If FACT = \(aqN\(aq, then DUF is an output argument and on exit
contains the (n-1) elements of the first superdiagonal of U.
.TP 8
DU2 (input or output) DOUBLE PRECISION array, dimension (N-2)
If FACT = \(aqF\(aq, then DU2 is an input argument and on entry
contains the (n-2) elements of the second superdiagonal of
U.
If FACT = \(aqN\(aq, then DU2 is an output argument and on exit
contains the (n-2) elements of the second superdiagonal of
U.
.TP 8
IPIV (input or output) INTEGER array, dimension (N)
If FACT = \(aqF\(aq, then IPIV is an input argument and on entry
contains the pivot indices from the LU factorization of A as
computed by DGTTRF.
If FACT = \(aqN\(aq, 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.
.TP 8
B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
The N-by-NRHS right hand side matrix B.
.TP 8
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
.TP 8
X (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
.TP 8
LDX (input) INTEGER
The leading dimension of the array X. LDX >= max(1,N).
.TP 8
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.
.TP 8
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.
.TP 8
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).
.TP 8
WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
.TP 8
IWORK (workspace) INTEGER array, dimension (N)
.TP 8
INFO (output) INTEGER
= 0: successful exit
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< 0: if INFO = -i, the i-th argument had an illegal value
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> 0: if INFO = i, and i is
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<= 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.
RCOND = 0 is returned.
= N+1: U is nonsingular, but RCOND is less than machine
precision, meaning that the matrix is singular
to working precision. Nevertheless, the
solution and error bounds are computed because
there are a number of situations where the
computed solution can be more accurate than the
value of RCOND would suggest.