## table of contents

realGTsolve(3) | LAPACK | realGTsolve(3) |

# NAME¶

realGTsolve

# SYNOPSIS¶

## Functions¶

subroutine **sgtsv** (N, NRHS, DL, D, DU, B, LDB, INFO)

** SGTSV computes the solution to system of linear equations A * X = B for GT
matrices ** subroutine **sgtsvx** (FACT, TRANS, N, NRHS, DL, D, DU,
DLF, DF, DUF, DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK,
INFO)

** SGTSVX computes the solution to system of linear equations A * X = B for
GT matrices **

# Detailed Description¶

This is the group of real solve driver functions for GT matrices

# Function Documentation¶

## subroutine sgtsv (integer N, integer NRHS, real, dimension( * ) DL, real, dimension( * ) D, real, dimension( * ) DU, real, dimension( ldb, * ) B, integer LDB, integer INFO)¶

** SGTSV computes the solution to system of linear equations A *
X = B for GT matrices **

**Purpose:**

SGTSV solves the equation

A*X = B,

where A is an n by n tridiagonal matrix, by Gaussian elimination with

partial pivoting.

Note that the equation A**T*X = B may be solved by interchanging the

order of the arguments DU and DL.

**Parameters**

*N*

N is INTEGER

The order of the matrix A. N >= 0.

*NRHS*

NRHS is INTEGER

The number of right hand sides, i.e., the number of columns

of the matrix B. NRHS >= 0.

*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-2) elements of the

second super-diagonal of the upper triangular matrix U from

the LU factorization of A, in DL(1), ..., DL(n-2).

*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 U.

*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.

*B*

B is REAL array, dimension (LDB,NRHS)

On entry, the N by NRHS matrix of right hand side matrix B.

On exit, if INFO = 0, the N by NRHS solution matrix X.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, U(i,i) is exactly zero, and the solution

has not been computed. The factorization has not been

completed unless i = N.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine sgtsvx (character FACT, 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 RCOND, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶

** SGTSVX computes the solution to system of linear equations A *
X = B for GT matrices **

**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.

**Parameters**

*FACT*

FACT is 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*

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)

*N*

N is INTEGER

The order of the matrix A. N >= 0.

*NRHS*

NRHS is INTEGER

The number of right hand sides, i.e., the number of columns

of the matrix B. NRHS >= 0.

*DL*

DL is REAL array, dimension (N-1)

The (n-1) subdiagonal elements of A.

*D*

D is REAL array, dimension (N)

The n diagonal elements of A.

*DU*

DU is REAL array, dimension (N-1)

The (n-1) superdiagonal elements of A.

*DLF*

DLF is 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*

DF is 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*

DUF is 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*

DU2 is 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*

IPIV is 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*

B is REAL array, dimension (LDB,NRHS)

The N-by-NRHS right hand side matrix B.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

*X*

X is REAL array, dimension (LDX,NRHS)

If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.

*LDX*

LDX is INTEGER

The leading dimension of the array X. LDX >= max(1,N).

*RCOND*

RCOND is 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*

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.

*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).

*WORK*

WORK is REAL array, dimension (3*N)

*IWORK*

IWORK is INTEGER array, dimension (N)

*INFO*

INFO is 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.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

# Author¶

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