---
name: "sgtsvx"
section: "3lapack"
platform: "linux"
package: "liblapack-doc"
version: "3.2.1-8"
release: "Debian 6.0 (Squeeze)"
platforms: ["linux"]
url: "https://new.carta.tech/sgtsvx/3lapack"
---

# sgtsvx(3lapack)

uses the LU factorization to compute the solution to a real system of linear equations A \* X = B or A\*\*T \* X = B,

## NAME

SGTSVX - uses the LU factorization to compute the solution to a real system of linear equations A \* X = B or A\*\*T \* X = B,

## SYNOPSIS

- SUBROUTINE SGTSVX(

  FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO )

- CHARACTER

  FACT, TRANS

- INTEGER

  INFO, LDB, LDX, N, NRHS

- REAL

  RCOND

- INTEGER

  IPIV( \* ), IWORK( \* )

- REAL

  B( LDB, \* ), BERR( \* ), D( \* ), DF( \* ), DL( \* ), DLF( \* ), DU( \* ), DU2( \* ), DUF( \* ), FERR( \* ), WORK( \* ), X( LDX, \* )

## PURPOSE

SGTSVX 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.\
 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 = 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) REAL array, dimension (N-1)

  The (n-1) subdiagonal elements of A.

- D (input) REAL array, dimension (N)

  The n diagonal elements of A.

- DU (input) REAL array, dimension (N-1)

  The (n-1) superdiagonal elements of A.

- DLF (input or output) REAL 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 SGTTRF. 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) REAL 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) REAL 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) REAL 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 SGTTRF. 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) REAL 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) REAL 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) REAL array, dimension (3\*N)

  - IWORK (workspace) INTEGER 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.