doublePTcomputational(3) LAPACK doublePTcomputational(3)

# NAME¶

doublePTcomputational

# SYNOPSIS¶

## Functions¶

subroutine dptcon (N, D, E, ANORM, RCOND, WORK, INFO)
DPTCON subroutine dpteqr (COMPZ, N, D, E, Z, LDZ, WORK, INFO)
DPTEQR subroutine dptrfs (N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR, WORK, INFO)
DPTRFS subroutine dpttrf (N, D, E, INFO)
DPTTRF subroutine dpttrs (N, NRHS, D, E, B, LDB, INFO)
DPTTRS subroutine dptts2 (N, NRHS, D, E, B, LDB)
DPTTS2 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 double computational functions for PT matrices

# Function Documentation¶

## subroutine dptcon (integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision ANORM, double precision RCOND, double precision, dimension( * ) WORK, integer INFO)¶

DPTCON

Purpose:

```
DPTCON 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

DPTTRF.

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 DOUBLE PRECISION array, dimension (N)

The n diagonal elements of the diagonal matrix D from the

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

E

```
E is DOUBLE PRECISION 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 DPTTRF.```

ANORM

```
ANORM is DOUBLE PRECISION

The 1-norm of the original matrix A.```

RCOND

```
RCOND is DOUBLE PRECISION

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 DOUBLE PRECISION 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 dpteqr (character COMPZ, integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer INFO)¶

DPTEQR

Purpose:

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

symmetric positive definite tridiagonal matrix by first factoring the

matrix using DPTTRF, and then calling DBDSQR 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 DSYTRD, DSPTRD, or DSBTRD 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N-1)

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

matrix.

On exit, E has been destroyed.```

Z

```
Z is DOUBLE PRECISION 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 DOUBLE PRECISION 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 dptrfs (integer N, integer NRHS, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision, dimension( * ) DF, double precision, dimension( * ) EF, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldx, * ) X, integer LDX, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, double precision, dimension( * ) WORK, integer INFO)¶

DPTRFS

Purpose:

```
DPTRFS 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 DOUBLE PRECISION array, dimension (N)

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

E

```
E is DOUBLE PRECISION array, dimension (N-1)

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

DF

```
DF is DOUBLE PRECISION array, dimension (N)

The n diagonal elements of the diagonal matrix D from the

factorization computed by DPTTRF.```

EF

```
EF is DOUBLE PRECISION array, dimension (N-1)

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

L from the factorization computed by DPTTRF.```

B

```
B is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDX,NRHS)

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

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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 dpttrf (integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, integer INFO)¶

DPTTRF

Purpose:

```
DPTTRF computes the L*D*L**T factorization of a real symmetric

positive definite tridiagonal matrix A.  The factorization may also

be regarded as having the form A = U**T*D*U.```

Parameters

N

```
N is INTEGER

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

D

```
D is DOUBLE PRECISION array, dimension (N)

On entry, the n diagonal elements of the tridiagonal matrix

A.  On exit, the n diagonal elements of the diagonal matrix

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

E

```
E is DOUBLE PRECISION array, dimension (N-1)

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

matrix A.  On exit, 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 U**T*D*U factorization of A.```

INFO

```
INFO is INTEGER

= 0: successful exit

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

> 0: if INFO = k, the leading minor of order k is not

positive definite; if k < N, the factorization could not

be completed, while if k = N, the factorization was

completed, but D(N) <= 0.```

Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date

December 2016

## subroutine dpttrs (integer N, integer NRHS, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision, dimension( ldb, * ) B, integer LDB, integer INFO)¶

DPTTRS

Purpose:

```
DPTTRS solves a tridiagonal system of the form

A * X = B

using the L*D*L**T factorization of A computed by DPTTRF.  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 DOUBLE PRECISION array, dimension (N)

The n diagonal elements of the diagonal matrix D from the

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

E

```
E is DOUBLE PRECISION 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 DOUBLE PRECISION 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 dptts2 (integer N, integer NRHS, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision, dimension( ldb, * ) B, integer LDB)¶

DPTTS2 solves a tridiagonal system of the form AX=B using the L D LH factorization computed by spttrf.

Purpose:

```
DPTTS2 solves a tridiagonal system of the form

A * X = B

using the L*D*L**T factorization of A computed by DPTTRF.  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 DOUBLE PRECISION array, dimension (N)

The n diagonal elements of the diagonal matrix D from the

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

E

```
E is DOUBLE PRECISION 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 DOUBLE PRECISION 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

NAG Ltd.

Date

December 2016

# Author¶

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