# pteqr - Man Page

pteqr: eig, positive definite tridiagonal

## Synopsis

### Functions

subroutine cpteqr (compz, n, d, e, z, ldz, work, info)
CPTEQR
subroutine dpteqr (compz, n, d, e, z, ldz, work, info)
DPTEQR
subroutine spteqr (compz, n, d, e, z, ldz, work, info)
SPTEQR
subroutine zpteqr (compz, n, d, e, z, ldz, work, info)
ZPTEQR

## Function Documentation

### 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 leading principal
minor of order i was not positive.
> 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.

Definition at line 144 of file cpteqr.f.

### 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 leading principal
minor of order i was not positive.
> 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.

Definition at line 144 of file dpteqr.f.

### 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 leading principal
minor of order i was not positive.
> 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.

Definition at line 144 of file spteqr.f.

### 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 leading principal
minor of order i was not positive.
> 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