Provided by: liblapack-doc_3.8.0-2_all

#### NAME

```       complex16PTcomputational

```

#### 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.

```

#### DetailedDescription

```       This is the group of complex16 computational functions for PT matrices

```

#### FunctionDocumentation

```   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.

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 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.

Date:
December 2016

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.

Date:
December 2016

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.

Date:
December 2016

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.

Date:
June 2016

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

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