realGTcomputational(3) LAPACK realGTcomputational(3)

# NAME¶

realGTcomputational

# SYNOPSIS¶

## Functions¶

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 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

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.```

N

```
N is INTEGER

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

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.```

D

```
D is REAL array, dimension (N)

The n diagonal elements of the upper triangular matrix U from

the LU factorization of A.```

DU

```
DU is REAL array, dimension (N-1)

The (n-1) elements of the first superdiagonal of U.```

DU2

```
DU2 is REAL array, dimension (N-2)

The (n-2) elements of the second superdiagonal of U.```

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.```

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.```

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.```

WORK

```
WORK is REAL array, dimension (2*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```

Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date

December 2016

## 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

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 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)

The (n-1) multipliers that define the matrix L from the

LU factorization of A as computed by SGTTRF.```

DF

```
DF is REAL array, dimension (N)

The n diagonal elements of the upper triangular matrix U from

the LU factorization of A.```

DUF

```
DUF is REAL array, dimension (N-1)

The (n-1) elements of the first superdiagonal of U.```

DU2

```
DU2 is REAL array, dimension (N-2)

The (n-2) elements of the second superdiagonal of U.```

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.```

B

```
B is REAL array, dimension (LDB,NRHS)

The 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)

On entry, the solution matrix X, as computed by SGTTRS.

On exit, the improved solution matrix X.```

LDX

```
LDX is INTEGER

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

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```

Internal Parameters:

```
ITMAX is the maximum number of steps of iterative refinement.```

Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date

December 2016

## 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

N

```
N is INTEGER

The order of the matrix A.```

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.```

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.```

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.```

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.```

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.```

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.```

Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date

December 2016

## 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

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)```

N

```
N is INTEGER

The order of the matrix A.```

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) multipliers that define the matrix L from the

LU factorization of A.```

D

```
D is REAL array, dimension (N)

The n diagonal elements of the upper triangular matrix U from

the LU factorization of A.```

DU

```
DU is REAL array, dimension (N-1)

The (n-1) elements of the first super-diagonal of U.```

DU2

```
DU2 is REAL array, dimension (N-2)

The (n-2) elements of the second super-diagonal of U.```

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.```

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.```

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```

Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date

December 2016

## 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.

Purpose:

```
SGTTS2 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

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)```

N

```
N is INTEGER

The order of the matrix A.```

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) multipliers that define the matrix L from the

LU factorization of A.```

D

```
D is REAL array, dimension (N)

The n diagonal elements of the upper triangular matrix U from

the LU factorization of A.```

DU

```
DU is REAL array, dimension (N-1)

The (n-1) elements of the first super-diagonal of U.```

DU2

```
DU2 is REAL array, dimension (N-2)

The (n-2) elements of the second super-diagonal of U.```

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.```

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.```

LDB

```
LDB is INTEGER

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

Author

Univ. of Tennessee

Univ. of California Berkeley