complex16PTcomputational(3) LAPACK complex16PTcomputational(3)

# NAME¶

complex16PTcomputational - complex16

# SYNOPSIS¶

## Functions¶

subroutine zptcon (N, D, E, ANORM, RCOND, RWORK, INFO)
ZPTCON subroutine zpteqr (COMPZ, N, D, E, Z, LDZ, WORK, INFO)
ZPTEQR subroutine zptrfs (UPLO, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
ZPTRFS subroutine zpttrf (N, D, E, INFO)
ZPTTRF subroutine zpttrs (UPLO, N, NRHS, D, E, B, LDB, INFO)
ZPTTRS subroutine zptts2 (IUPLO, N, NRHS, D, E, B, LDB)
ZPTTS2 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 complex16 computational functions for PT matrices

# Function Documentation¶

## subroutine zptcon (integer N, double precision, dimension( * ) D, complex*16, dimension( * ) E, double precision ANORM, double precision RCOND, double precision, dimension( * ) RWORK, integer INFO)¶

ZPTCON

Purpose:

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

1-norm) of a complex Hermitian positive definite tridiagonal matrix

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

ZPTTRF.

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 ZPTTRF.```

E

```
E is COMPLEX*16 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 ZPTTRF.```

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.```

RWORK

```
RWORK 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.

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 zpteqr (character COMPZ, integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, complex*16, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer INFO)¶

ZPTEQR

Purpose:

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

symmetric positive definite tridiagonal matrix by first factoring the

matrix using DPTTRF and then calling ZBDSQR 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 positive definite Hermitian matrix

can also be found if ZHETRD, ZHPTRD, or ZHBTRD 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 Hermitian

matrix also.  Array Z contains the unitary 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 COMPLEX*16 array, dimension (LDZ, N)

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

reduction to tridiagonal form.

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

original Hermitian 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.

## subroutine zptrfs (character UPLO, integer N, integer NRHS, double precision, dimension( * ) D, complex*16, dimension( * ) E, double precision, dimension( * ) DF, complex*16, dimension( * ) EF, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldx, * ) X, integer LDX, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO)¶

ZPTRFS

Purpose:

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

equations when the coefficient matrix is Hermitian positive definite

and tridiagonal, and provides error bounds and backward error

estimates for the solution.```

Parameters

UPLO

```
UPLO is CHARACTER*1

Specifies whether the superdiagonal or the subdiagonal of the

tridiagonal matrix A is stored and the form of the

factorization:

= 'U':  E is the superdiagonal of A, and A = U**H*D*U;

= 'L':  E is the subdiagonal of A, and A = L*D*L**H.

(The two forms are equivalent if A is real.)```

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 real diagonal elements of the tridiagonal matrix A.```

E

```
E is COMPLEX*16 array, dimension (N-1)

The (n-1) off-diagonal elements of the tridiagonal matrix A

(see UPLO).```

DF

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

The n diagonal elements of the diagonal matrix D from

the factorization computed by ZPTTRF.```

EF

```
EF is COMPLEX*16 array, dimension (N-1)

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

factor U or L from the factorization computed by ZPTTRF

(see UPLO).```

B

```
B is COMPLEX*16 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 COMPLEX*16 array, dimension (LDX,NRHS)

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

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 COMPLEX*16 array, dimension (N)```

RWORK

```
RWORK 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```

Internal Parameters:

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

Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

## subroutine zpttrf (integer N, double precision, dimension( * ) D, complex*16, dimension( * ) E, integer INFO)¶

ZPTTRF

Purpose:

```
ZPTTRF computes the L*D*L**H factorization of a complex Hermitian

positive definite tridiagonal matrix A.  The factorization may also

be regarded as having the form A = U**H *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**H factorization of A.```

E

```
E is COMPLEX*16 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**H factorization of A.

E can also be regarded as the superdiagonal of the unit

bidiagonal factor U from the U**H *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.

## subroutine zpttrs (character UPLO, integer N, integer NRHS, double precision, dimension( * ) D, complex*16, dimension( * ) E, complex*16, dimension( ldb, * ) B, integer LDB, integer INFO)¶

ZPTTRS

Purpose:

```
ZPTTRS solves a tridiagonal system of the form

A * X = B

using the factorization A = U**H *D* U or A = L*D*L**H computed by ZPTTRF.

D is a diagonal matrix specified in the vector D, U (or L) is a unit

bidiagonal matrix whose superdiagonal (subdiagonal) is specified in

the vector E, and X and B are N by NRHS matrices.```

Parameters

UPLO

```
UPLO is CHARACTER*1

Specifies the form of the factorization and whether the

vector E is the superdiagonal of the upper bidiagonal factor

U or the subdiagonal of the lower bidiagonal factor L.

= 'U':  A = U**H *D*U, E is the superdiagonal of U

= 'L':  A = L*D*L**H, E is the subdiagonal of L```

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

factorization A = U**H *D*U or A = L*D*L**H.```

E

```
E is COMPLEX*16 array, dimension (N-1)

If UPLO = 'U', the (n-1) superdiagonal elements of the unit

bidiagonal factor U from the factorization A = U**H*D*U.

If UPLO = 'L', the (n-1) subdiagonal elements of the unit

bidiagonal factor L from the factorization A = L*D*L**H.```

B

```
B is COMPLEX*16 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.

## subroutine zptts2 (integer IUPLO, integer N, integer NRHS, double precision, dimension( * ) D, complex*16, dimension( * ) E, complex*16, dimension( ldb, * ) B, integer LDB)¶

ZPTTS2 solves a tridiagonal system of the form AX=B using the L D LH factorization computed by spttrf.

Purpose:

```
ZPTTS2 solves a tridiagonal system of the form

A * X = B

using the factorization A = U**H *D*U or A = L*D*L**H computed by ZPTTRF.

D is a diagonal matrix specified in the vector D, U (or L) is a unit

bidiagonal matrix whose superdiagonal (subdiagonal) is specified in

the vector E, and X and B are N by NRHS matrices.```

Parameters

IUPLO

```
IUPLO is INTEGER

Specifies the form of the factorization and whether the

vector E is the superdiagonal of the upper bidiagonal factor

U or the subdiagonal of the lower bidiagonal factor L.

= 1:  A = U**H *D*U, E is the superdiagonal of U

= 0:  A = L*D*L**H, E is the subdiagonal of L```

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

factorization A = U**H *D*U or A = L*D*L**H.```

E

```
E is COMPLEX*16 array, dimension (N-1)

If IUPLO = 1, the (n-1) superdiagonal elements of the unit

bidiagonal factor U from the factorization A = U**H*D*U.

If IUPLO = 0, the (n-1) subdiagonal elements of the unit

bidiagonal factor L from the factorization A = L*D*L**H.```

B

```
B is COMPLEX*16 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