realPTcomputational(3) LAPACK realPTcomputational(3)

# NAME¶

realPTcomputational

# SYNOPSIS¶

## Functions¶

subroutine sptcon (N, D, E, ANORM, RCOND, WORK, INFO)
SPTCON subroutine spteqr (COMPZ, N, D, E, Z, LDZ, WORK, INFO)
SPTEQR subroutine sptrfs (N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR, WORK, INFO)
SPTRFS subroutine spttrs (N, NRHS, D, E, B, LDB, INFO)
SPTTRS subroutine sptts2 (N, NRHS, D, E, B, LDB)
SPTTS2 solves a tridiagonal system of the form AX=B using the L D LH factorization computed by spttrf.

# Detailed Description¶

This is the group of real computational functions for PT matrices

# Function Documentation¶

## subroutine sptcon (integer N, real, dimension( * ) D, real, dimension( * ) E, real ANORM, real RCOND, real, dimension( * ) WORK, integer INFO)¶

SPTCON

Purpose:

```
SPTCON computes the reciprocal of the condition number (in the

1-norm) of a real symmetric positive definite tridiagonal matrix

using the factorization A = L*D*L**T or A = U**T*D*U computed by

SPTTRF.

Norm(inv(A)) is computed by a direct method, and the reciprocal of

the condition number is computed as

RCOND = 1 / (ANORM * norm(inv(A))).```

Parameters

N

```
N is INTEGER

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

D

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

The n diagonal elements of the diagonal matrix D from the

factorization of A, as computed by SPTTRF.```

E

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

The (n-1) off-diagonal elements of the unit bidiagonal factor

U or L from the factorization of A,  as computed by SPTTRF.```

ANORM

```
ANORM is REAL

The 1-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 the

1-norm of inv(A) computed in this routine.```

WORK

```
WORK is REAL 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

Further Details:

```
The method used is described in Nicholas J. Higham, "Efficient

Algorithms for Computing the Condition Number of a Tridiagonal

Matrix", SIAM J. Sci. Stat. Comput., Vol. 7, No. 1, January 1986.```

## subroutine spteqr (character COMPZ, integer N, real, dimension( * ) D, real, dimension( * ) E, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer INFO)¶

SPTEQR

Purpose:

```
SPTEQR computes all eigenvalues and, optionally, eigenvectors of a

symmetric positive definite tridiagonal matrix by first factoring the

matrix using SPTTRF, and then calling SBDSQR to compute the singular

values of the bidiagonal factor.

This routine computes the eigenvalues of the positive definite

tridiagonal matrix to high relative accuracy.  This means that if the

eigenvalues range over many orders of magnitude in size, then the

small eigenvalues and corresponding eigenvectors will be computed

more accurately than, for example, with the standard QR method.

The eigenvectors of a full or band symmetric positive definite matrix

can also be found if SSYTRD, SSPTRD, or SSBTRD has been used to

reduce this matrix to tridiagonal form. (The reduction to tridiagonal

form, however, may preclude the possibility of obtaining high

relative accuracy in the small eigenvalues of the original matrix, if

these eigenvalues range over many orders of magnitude.)```

Parameters

COMPZ

```
COMPZ is CHARACTER*1

= 'N':  Compute eigenvalues only.

= 'V':  Compute eigenvectors of original symmetric

matrix also.  Array Z contains the orthogonal

matrix used to reduce the original matrix to

tridiagonal form.

= 'I':  Compute eigenvectors of tridiagonal matrix also.```

N

```
N is INTEGER

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

D

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

On entry, the n diagonal elements of the tridiagonal

matrix.

On normal exit, D contains the eigenvalues, in descending

order.```

E

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

On entry, the (n-1) subdiagonal elements of the tridiagonal

matrix.

On exit, E has been destroyed.```

Z

```
Z is REAL array, dimension (LDZ, N)

On entry, if COMPZ = 'V', the orthogonal matrix used in the

reduction to tridiagonal form.

On exit, if COMPZ = 'V', the orthonormal eigenvectors of the

original symmetric matrix;

if COMPZ = 'I', the orthonormal eigenvectors of the

tridiagonal matrix.

If INFO > 0 on exit, Z contains the eigenvectors associated

with only the stored eigenvalues.

If  COMPZ = 'N', then Z is not referenced.```

LDZ

```
LDZ is INTEGER

The leading dimension of the array Z.  LDZ >= 1, and if

COMPZ = 'V' or 'I', LDZ >= max(1,N).```

WORK

```
WORK is REAL array, dimension (4*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  the Cholesky factorization of the matrix could

not be performed because the i-th principal minor

was not positive definite.

> N   the SVD algorithm failed to converge;

if INFO = N+i, i off-diagonal elements of the

bidiagonal factor did not converge to zero.```

Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date

December 2016

## subroutine sptrfs (integer N, integer NRHS, real, dimension( * ) D, real, dimension( * ) E, real, dimension( * ) DF, real, dimension( * ) EF, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer INFO)¶

SPTRFS

Purpose:

```
SPTRFS improves the computed solution to a system of linear

equations when the coefficient matrix is symmetric positive definite

and tridiagonal, and provides error bounds and backward error

estimates for the solution.```

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.```

D

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

The n diagonal elements of the tridiagonal matrix A.```

E

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

The (n-1) subdiagonal elements of the tridiagonal matrix A.```

DF

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

The n diagonal elements of the diagonal matrix D from the

factorization computed by SPTTRF.```

EF

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

The (n-1) subdiagonal elements of the unit bidiagonal factor

L from the factorization computed by SPTTRF.```

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 SPTTRS.

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 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).```

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 (2*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 spttrs (integer N, integer NRHS, real, dimension( * ) D, real, dimension( * ) E, real, dimension( ldb, * ) B, integer LDB, integer INFO)¶

SPTTRS

Purpose:

```
SPTTRS solves a tridiagonal system of the form

A * X = B

using the L*D*L**T factorization of A computed by SPTTRF.  D is a

diagonal matrix specified in the vector D, L is a unit bidiagonal

matrix whose subdiagonal is specified in the vector E, and X and B

are N by NRHS matrices.```

Parameters

N

```
N is INTEGER

The order of the tridiagonal 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.```

D

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

The n diagonal elements of the diagonal matrix D from the

L*D*L**T factorization of A.```

E

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

The (n-1) subdiagonal elements of the unit bidiagonal factor

L from the L*D*L**T factorization of A.  E can also be regarded

as the superdiagonal of the unit bidiagonal factor U from the

factorization A = U**T*D*U.```

B

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

On entry, the right hand side vectors B for the system of

linear equations.

On exit, 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 = -k, the k-th argument had an illegal value```

Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date

December 2016

## subroutine sptts2 (integer N, integer NRHS, real, dimension( * ) D, real, dimension( * ) E, real, dimension( ldb, * ) B, integer LDB)¶

SPTTS2 solves a tridiagonal system of the form AX=B using the L D LH factorization computed by spttrf.

Purpose:

```
SPTTS2 solves a tridiagonal system of the form

A * X = B

using the L*D*L**T factorization of A computed by SPTTRF.  D is a

diagonal matrix specified in the vector D, L is a unit bidiagonal

matrix whose subdiagonal is specified in the vector E, and X and B

are N by NRHS matrices.```

Parameters

N

```
N is INTEGER

The order of the tridiagonal 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.```

D

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

The n diagonal elements of the diagonal matrix D from the

L*D*L**T factorization of A.```

E

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

The (n-1) subdiagonal elements of the unit bidiagonal factor

L from the L*D*L**T factorization of A.  E can also be regarded

as the superdiagonal of the unit bidiagonal factor U from the

factorization A = U**T*D*U.```

B

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

On entry, the right hand side vectors B for the system of

linear equations.

On exit, 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