LAPACK  3.4.2 LAPACK: Linear Algebra PACKage
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## Functions/Subroutines

subroutine sgtcon (NORM, N, DL, D, DU, DU2, IPIV, ANORM, RCOND, WORK, IWORK, INFO)
SGTCON
subroutine sgtrfs (TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
SGTRFS
subroutine sgttrf (N, DL, D, DU, DU2, IPIV, INFO)
SGTTRF
subroutine sgttrs (TRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB, INFO)
SGTTRS
subroutine sgtts2 (ITRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB)
SGTTS2 solves a system of linear equations with a tridiagonal matrix using the LU factorization computed by sgttrf.

## Detailed Description

This is the group of real computational functions for GT matrices

## Function/Subroutine Documentation

 subroutine sgtcon ( character NORM, integer N, real, dimension( * ) DL, real, dimension( * ) D, real, dimension( * ) DU, real, dimension( * ) DU2, integer, dimension( * ) IPIV, real ANORM, real RCOND, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO )

SGTCON

Purpose:
``` SGTCON estimates the reciprocal of the condition number of a real
tridiagonal matrix A using the LU factorization as computed by
SGTTRF.

An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).```
Parameters:
 [in] NORM ``` NORM is CHARACTER*1 Specifies whether the 1-norm condition number or the infinity-norm condition number is required: = '1' or 'O': 1-norm; = 'I': Infinity-norm.``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in] DL ``` DL is REAL array, dimension (N-1) The (n-1) multipliers that define the matrix L from the LU factorization of A as computed by SGTTRF.``` [in] D ``` D is REAL array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A.``` [in] DU ``` DU is REAL array, dimension (N-1) The (n-1) elements of the first superdiagonal of U.``` [in] DU2 ``` DU2 is REAL array, dimension (N-2) The (n-2) elements of the second superdiagonal of U.``` [in] IPIV ``` IPIV is INTEGER array, dimension (N) The pivot indices; for 1 <= i <= n, 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.``` [in] ANORM ``` ANORM is REAL If NORM = '1' or 'O', the 1-norm of the original matrix A. If NORM = 'I', the infinity-norm of the original matrix A.``` [out] RCOND ``` RCOND is REAL The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an estimate of the 1-norm of inv(A) computed in this routine.``` [out] WORK ` WORK is REAL array, dimension (2*N)` [out] IWORK ` IWORK is INTEGER array, dimension (N)` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
Date:
September 2012

Definition at line 146 of file sgtcon.f.

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 subroutine sgtrfs ( character TRANS, integer N, integer NRHS, real, dimension( * ) DL, real, dimension( * ) D, real, dimension( * ) DU, real, dimension( * ) DLF, real, dimension( * ) DF, real, dimension( * ) DUF, real, dimension( * ) DU2, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO )

SGTRFS

Purpose:
``` SGTRFS improves the computed solution to a system of linear
equations when the coefficient matrix is tridiagonal, and provides
error bounds and backward error estimates for the solution.```
Parameters:
 [in] TRANS ``` TRANS is 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)``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in] NRHS ``` NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.``` [in] DL ``` DL is REAL array, dimension (N-1) The (n-1) subdiagonal elements of A.``` [in] D ``` D is REAL array, dimension (N) The diagonal elements of A.``` [in] DU ``` DU is REAL array, dimension (N-1) The (n-1) superdiagonal elements of A.``` [in] DLF ``` DLF is REAL array, dimension (N-1) The (n-1) multipliers that define the matrix L from the LU factorization of A as computed by SGTTRF.``` [in] DF ``` DF is REAL array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A.``` [in] DUF ``` DUF is REAL array, dimension (N-1) The (n-1) elements of the first superdiagonal of U.``` [in] DU2 ``` DU2 is REAL array, dimension (N-2) The (n-2) elements of the second superdiagonal of U.``` [in] IPIV ``` IPIV is INTEGER array, dimension (N) The pivot indices; for 1 <= i <= n, 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.``` [in] B ``` B is REAL array, dimension (LDB,NRHS) The right hand side matrix B.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).``` [in,out] X ``` X is REAL array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by SGTTRS. On exit, the improved solution matrix X.``` [in] LDX ``` LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N).``` [out] FERR ``` FERR is 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.``` [out] BERR ``` BERR is 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).``` [out] WORK ` WORK is REAL array, dimension (3*N)` [out] IWORK ` IWORK is INTEGER array, dimension (N)` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
Internal Parameters:
`  ITMAX is the maximum number of steps of iterative refinement.`
Date:
September 2012

Definition at line 208 of file sgtrfs.f.

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 subroutine sgttrf ( integer N, real, dimension( * ) DL, real, dimension( * ) D, real, dimension( * ) DU, real, dimension( * ) DU2, integer, dimension( * ) IPIV, integer INFO )

SGTTRF

Purpose:
``` SGTTRF computes an LU factorization of a real tridiagonal matrix A
using elimination with partial pivoting and row interchanges.

The factorization has the form
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.```
Parameters:
 [in] N ``` N is INTEGER The order of the matrix A.``` [in,out] DL ``` DL is REAL array, dimension (N-1) On entry, DL must contain the (n-1) sub-diagonal elements of A. On exit, DL is overwritten by the (n-1) multipliers that define the matrix L from the LU factorization of A.``` [in,out] D ``` D is REAL array, dimension (N) On entry, D must contain the diagonal elements of A. On exit, D is overwritten by the n diagonal elements of the upper triangular matrix U from the LU factorization of A.``` [in,out] DU ``` DU is REAL array, dimension (N-1) On entry, DU must contain the (n-1) super-diagonal elements of A. On exit, DU is overwritten by the (n-1) elements of the first super-diagonal of U.``` [out] DU2 ``` DU2 is REAL array, dimension (N-2) On exit, DU2 is overwritten by the (n-2) elements of the second super-diagonal of U.``` [out] IPIV ``` IPIV is INTEGER array, dimension (N) The pivot indices; for 1 <= i <= n, 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.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal value > 0: if INFO = k, U(k,k) is exactly zero. The factorization has been completed, but the factor U is exactly singular, and division by zero will occur if it is used to solve a system of equations.```
Date:
September 2012

Definition at line 125 of file sgttrf.f.

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 subroutine sgttrs ( character TRANS, integer N, integer NRHS, real, dimension( * ) DL, real, dimension( * ) D, real, dimension( * ) DU, real, dimension( * ) DU2, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB, integer INFO )

SGTTRS

Purpose:
``` SGTTRS solves one of the systems of equations
A*X = B  or  A**T*X = B,
with a tridiagonal matrix A using the LU factorization computed
by SGTTRF.```
Parameters:
 [in] TRANS ``` TRANS is CHARACTER*1 Specifies the form of the system of equations. = 'N': A * X = B (No transpose) = 'T': A**T* X = B (Transpose) = 'C': A**T* X = B (Conjugate transpose = Transpose)``` [in] N ``` N is INTEGER The order of the matrix A.``` [in] NRHS ``` NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.``` [in] DL ``` DL is REAL array, dimension (N-1) The (n-1) multipliers that define the matrix L from the LU factorization of A.``` [in] D ``` D is REAL array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A.``` [in] DU ``` DU is REAL array, dimension (N-1) The (n-1) elements of the first super-diagonal of U.``` [in] DU2 ``` DU2 is REAL array, dimension (N-2) The (n-2) elements of the second super-diagonal of U.``` [in] IPIV ``` IPIV is INTEGER array, dimension (N) The pivot indices; for 1 <= i <= n, 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.``` [in,out] B ``` B is REAL array, dimension (LDB,NRHS) On entry, the matrix of right hand side vectors B. On exit, B is overwritten by the solution vectors X.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
Date:
September 2012

Definition at line 138 of file sgttrs.f.

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 subroutine sgtts2 ( integer ITRANS, integer N, integer NRHS, real, dimension( * ) DL, real, dimension( * ) D, real, dimension( * ) DU, real, dimension( * ) DU2, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB )

SGTTS2 solves a system of linear equations with a tridiagonal matrix using the LU factorization computed by sgttrf.

``` SGTTS2 solves one of the systems of equations
 [in] ITRANS ``` ITRANS is INTEGER Specifies the form of the system of equations. = 0: A * X = B (No transpose) = 1: A**T* X = B (Transpose) = 2: A**T* X = B (Conjugate transpose = Transpose)``` [in] N ``` N is INTEGER The order of the matrix A.``` [in] NRHS ``` NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.``` [in] DL ``` DL is REAL array, dimension (N-1) The (n-1) multipliers that define the matrix L from the LU factorization of A.``` [in] D ``` D is REAL array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A.``` [in] DU ``` DU is REAL array, dimension (N-1) The (n-1) elements of the first super-diagonal of U.``` [in] DU2 ``` DU2 is REAL array, dimension (N-2) The (n-2) elements of the second super-diagonal of U.``` [in] IPIV ``` IPIV is INTEGER array, dimension (N) The pivot indices; for 1 <= i <= n, 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.``` [in,out] B ``` B is REAL array, dimension (LDB,NRHS) On entry, the matrix of right hand side vectors B. On exit, B is overwritten by the solution vectors X.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).```