# steqr - Man Page

steqr: eig, QR iteration

## Synopsis

### Functions

subroutine **csteqr** (compz, n, d, e, z, ldz, work, info)**CSTEQR**

subroutine **dsteqr** (compz, n, d, e, z, ldz, work, info)**DSTEQR**

subroutine **ssteqr** (compz, n, d, e, z, ldz, work, info)**SSTEQR**

subroutine **zsteqr** (compz, n, d, e, z, ldz, work, info)**ZSTEQR**

## Detailed Description

## Function Documentation

### subroutine csteqr (character compz, integer n, real, dimension( * ) d, real, dimension( * ) e, complex, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer info)

**CSTEQR**

**Purpose:**

CSTEQR computes all eigenvalues and, optionally, eigenvectors of a symmetric tridiagonal matrix using the implicit QL or QR method. The eigenvectors of a full or band complex Hermitian matrix can also be found if CHETRD or CHPTRD or CHBTRD has been used to reduce this matrix to tridiagonal form.

**Parameters***COMPZ*COMPZ is CHARACTER*1 = 'N': Compute eigenvalues only. = 'V': Compute eigenvalues and eigenvectors of the original Hermitian matrix. On entry, Z must contain the unitary matrix used to reduce the original matrix to tridiagonal form. = 'I': Compute eigenvalues and eigenvectors of the tridiagonal matrix. Z is initialized to the identity matrix.

*N*N is INTEGER The order of the matrix. N >= 0.

*D*D is REAL array, dimension (N) On entry, the diagonal elements of the tridiagonal matrix. On exit, if INFO = 0, the eigenvalues in ascending 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', then Z contains the unitary matrix used in the reduction to tridiagonal form. On exit, if INFO = 0, then if COMPZ = 'V', Z contains the orthonormal eigenvectors of the original Hermitian matrix, and if COMPZ = 'I', Z contains the orthonormal eigenvectors of the symmetric tridiagonal matrix. If COMPZ = 'N', then Z is not referenced.

*LDZ*LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if eigenvectors are desired, then LDZ >= max(1,N).

*WORK*WORK is REAL array, dimension (max(1,2*N-2)) If COMPZ = 'N', then WORK is not referenced.

*INFO*INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: the algorithm has failed to find all the eigenvalues in a total of 30*N iterations; if INFO = i, then i elements of E have not converged to zero; on exit, D and E contain the elements of a symmetric tridiagonal matrix which is unitarily similar to the original matrix.

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **131** of file **csteqr.f**.

### subroutine dsteqr (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)

**DSTEQR**

**Purpose:**

DSTEQR computes all eigenvalues and, optionally, eigenvectors of a symmetric tridiagonal matrix using the implicit QL or QR method. The eigenvectors of a full or band symmetric matrix can also be found if DSYTRD or DSPTRD or DSBTRD has been used to reduce this matrix to tridiagonal form.

**Parameters***COMPZ*COMPZ is CHARACTER*1 = 'N': Compute eigenvalues only. = 'V': Compute eigenvalues and eigenvectors of the original symmetric matrix. On entry, Z must contain the orthogonal matrix used to reduce the original matrix to tridiagonal form. = 'I': Compute eigenvalues and eigenvectors of the tridiagonal matrix. Z is initialized to the identity matrix.

*N*N is INTEGER The order of the matrix. N >= 0.

*D*D is DOUBLE PRECISION array, dimension (N) On entry, the diagonal elements of the tridiagonal matrix. On exit, if INFO = 0, the eigenvalues in ascending 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', then Z contains the orthogonal matrix used in the reduction to tridiagonal form. On exit, if INFO = 0, then if COMPZ = 'V', Z contains the orthonormal eigenvectors of the original symmetric matrix, and if COMPZ = 'I', Z contains the orthonormal eigenvectors of the symmetric tridiagonal matrix. If COMPZ = 'N', then Z is not referenced.

*LDZ*LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if eigenvectors are desired, then LDZ >= max(1,N).

*WORK*WORK is DOUBLE PRECISION array, dimension (max(1,2*N-2)) If COMPZ = 'N', then WORK is not referenced.

*INFO*INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: the algorithm has failed to find all the eigenvalues in a total of 30*N iterations; if INFO = i, then i elements of E have not converged to zero; on exit, D and E contain the elements of a symmetric tridiagonal matrix which is orthogonally similar to the original matrix.

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **130** of file **dsteqr.f**.

### subroutine ssteqr (character compz, integer n, real, dimension( * ) d, real, dimension( * ) e, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer info)

**SSTEQR**

**Purpose:**

SSTEQR computes all eigenvalues and, optionally, eigenvectors of a symmetric tridiagonal matrix using the implicit QL or QR method. The eigenvectors of a full or band symmetric matrix can also be found if SSYTRD or SSPTRD or SSBTRD has been used to reduce this matrix to tridiagonal form.

**Parameters***COMPZ*COMPZ is CHARACTER*1 = 'N': Compute eigenvalues only. = 'V': Compute eigenvalues and eigenvectors of the original symmetric matrix. On entry, Z must contain the orthogonal matrix used to reduce the original matrix to tridiagonal form. = 'I': Compute eigenvalues and eigenvectors of the tridiagonal matrix. Z is initialized to the identity matrix.

*N*N is INTEGER The order of the matrix. N >= 0.

*D*D is REAL array, dimension (N) On entry, the diagonal elements of the tridiagonal matrix. On exit, if INFO = 0, the eigenvalues in ascending 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', then Z contains the orthogonal matrix used in the reduction to tridiagonal form. On exit, if INFO = 0, then if COMPZ = 'V', Z contains the orthonormal eigenvectors of the original symmetric matrix, and if COMPZ = 'I', Z contains the orthonormal eigenvectors of the symmetric tridiagonal matrix. If COMPZ = 'N', then Z is not referenced.

*LDZ*LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if eigenvectors are desired, then LDZ >= max(1,N).

*WORK*WORK is REAL array, dimension (max(1,2*N-2)) If COMPZ = 'N', then WORK is not referenced.

*INFO*INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: the algorithm has failed to find all the eigenvalues in a total of 30*N iterations; if INFO = i, then i elements of E have not converged to zero; on exit, D and E contain the elements of a symmetric tridiagonal matrix which is orthogonally similar to the original matrix.

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **130** of file **ssteqr.f**.

### subroutine zsteqr (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)

**ZSTEQR**

**Purpose:**

ZSTEQR computes all eigenvalues and, optionally, eigenvectors of a symmetric tridiagonal matrix using the implicit QL or QR method. The eigenvectors of a full or band complex Hermitian matrix can also be found if ZHETRD or ZHPTRD or ZHBTRD has been used to reduce this matrix to tridiagonal form.

**Parameters***COMPZ*COMPZ is CHARACTER*1 = 'N': Compute eigenvalues only. = 'V': Compute eigenvalues and eigenvectors of the original Hermitian matrix. On entry, Z must contain the unitary matrix used to reduce the original matrix to tridiagonal form. = 'I': Compute eigenvalues and eigenvectors of the tridiagonal matrix. Z is initialized to the identity matrix.

*N*N is INTEGER The order of the matrix. N >= 0.

*D*D is DOUBLE PRECISION array, dimension (N) On entry, the diagonal elements of the tridiagonal matrix. On exit, if INFO = 0, the eigenvalues in ascending 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', then Z contains the unitary matrix used in the reduction to tridiagonal form. On exit, if INFO = 0, then if COMPZ = 'V', Z contains the orthonormal eigenvectors of the original Hermitian matrix, and if COMPZ = 'I', Z contains the orthonormal eigenvectors of the symmetric tridiagonal matrix. If COMPZ = 'N', then Z is not referenced.

*LDZ**WORK*WORK is DOUBLE PRECISION array, dimension (max(1,2*N-2)) If COMPZ = 'N', then WORK is not referenced.

*INFO*INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: the algorithm has failed to find all the eigenvalues in a total of 30*N iterations; if INFO = i, then i elements of E have not converged to zero; on exit, D and E contain the elements of a symmetric tridiagonal matrix which is unitarily similar to the original matrix.

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **131** of file **zsteqr.f**.

## Author

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