complexPTcomputational(3) LAPACK complexPTcomputational(3)

# NAME¶

complexPTcomputational

# SYNOPSIS¶

## Functions¶

subroutine cptcon (N, D, E, ANORM, RCOND, RWORK, INFO)
CPTCON subroutine cpteqr (COMPZ, N, D, E, Z, LDZ, WORK, INFO)
CPTEQR subroutine cptrfs (UPLO, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
CPTRFS subroutine cpttrf (N, D, E, INFO)
CPTTRF subroutine cpttrs (UPLO, N, NRHS, D, E, B, LDB, INFO)
CPTTRS subroutine cptts2 (IUPLO, N, NRHS, D, E, B, LDB)
CPTTS2 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 complex computational functions for PT matrices

# Function Documentation¶

## subroutine cptcon (integer N, real, dimension( * ) D, complex, dimension( * ) E, real ANORM, real RCOND, real, dimension( * ) RWORK, integer INFO)¶

CPTCON

Purpose:

``` CPTCON 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
CPTTRF.
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 CPTTRF.
```

E

```          E is COMPLEX 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 CPTTRF.
```

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.
```

RWORK

```          RWORK 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

Univ. of Colorado Denver

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 cpteqr (character COMPZ, integer N, real, dimension( * ) D, real, dimension( * ) E, complex, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer INFO)¶

CPTEQR

Purpose:

``` CPTEQR computes all eigenvalues and, optionally, eigenvectors of a
symmetric positive definite tridiagonal matrix by first factoring the
matrix using SPTTRF and then calling CBDSQR 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 CHETRD, CHPTRD, or CHBTRD 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 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 COMPLEX 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 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

Univ. of Colorado Denver

NAG Ltd.

Date

December 2016

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

CPTRFS

Purpose:

``` CPTRFS 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 REAL array, dimension (N)
The n real diagonal elements of the tridiagonal matrix A.
```

E

```          E is COMPLEX array, dimension (N-1)
The (n-1) off-diagonal elements of the tridiagonal matrix A
(see UPLO).
```

DF

```          DF is REAL array, dimension (N)
The n diagonal elements of the diagonal matrix D from
the factorization computed by CPTTRF.
```

EF

```          EF is COMPLEX array, dimension (N-1)
The (n-1) off-diagonal elements of the unit bidiagonal
factor U or L from the factorization computed by CPTTRF
(see UPLO).
```

B

```          B is COMPLEX 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 array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by CPTTRS.
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 COMPLEX array, dimension (N)
```

RWORK

```          RWORK is REAL 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

Univ. of Colorado Denver

NAG Ltd.

Date

December 2016

## subroutine cpttrf (integer N, real, dimension( * ) D, complex, dimension( * ) E, integer INFO)¶

CPTTRF

Purpose:

``` CPTTRF 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 REAL 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 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

Univ. of Colorado Denver

NAG Ltd.

Date

December 2016

## subroutine cpttrs (character UPLO, integer N, integer NRHS, real, dimension( * ) D, complex, dimension( * ) E, complex, dimension( ldb, * ) B, integer LDB, integer INFO)¶

CPTTRS

Purpose:

``` CPTTRS 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 CPTTRF.
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 REAL 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 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 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

Univ. of Colorado Denver

NAG Ltd.

Date

June 2016

## subroutine cptts2 (integer IUPLO, integer N, integer NRHS, real, dimension( * ) D, complex, dimension( * ) E, complex, dimension( ldb, * ) B, integer LDB)¶

CPTTS2 solves a tridiagonal system of the form AX=B using the L D LH factorization computed by spttrf.

Purpose:

``` CPTTS2 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 CPTTRF.
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 REAL 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 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 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

Univ. of Colorado Denver

NAG Ltd.

Date

June 2016

# Author¶

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