# man cgtsvx (Fonctions bibliothèques) - 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,

## NAME

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,

## SYNOPSIS

- 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, * )

## PURPOSE

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.

## DESCRIPTION

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. 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.

4. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.

## ARGUMENTS

- 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.

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 or INFO = N+1, 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.
- 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 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. 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.