# laein - Man Page

laein: eigvec by Hessenberg inverse iteration

## Synopsis

### Functions

subroutine **claein** (rightv, noinit, n, h, ldh, w, v, b, ldb, rwork, eps3, smlnum, info)**CLAEIN** computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration.

subroutine **dlaein** (rightv, noinit, n, h, ldh, wr, wi, vr, vi, b, ldb, work, eps3, smlnum, bignum, info)**DLAEIN** computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration.

subroutine **slaein** (rightv, noinit, n, h, ldh, wr, wi, vr, vi, b, ldb, work, eps3, smlnum, bignum, info)**SLAEIN** computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration.

subroutine **zlaein** (rightv, noinit, n, h, ldh, w, v, b, ldb, rwork, eps3, smlnum, info)**ZLAEIN** computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration.

## Detailed Description

## Function Documentation

### subroutine claein (logical rightv, logical noinit, integer n, complex, dimension( ldh, * ) h, integer ldh, complex w, complex, dimension( * ) v, complex, dimension( ldb, * ) b, integer ldb, real, dimension( * ) rwork, real eps3, real smlnum, integer info)

**CLAEIN** computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration.

**Purpose:**

CLAEIN uses inverse iteration to find a right or left eigenvector corresponding to the eigenvalue W of a complex upper Hessenberg matrix H.

**Parameters***RIGHTV*RIGHTV is LOGICAL = .TRUE. : compute right eigenvector; = .FALSE.: compute left eigenvector.

*NOINIT*NOINIT is LOGICAL = .TRUE. : no initial vector supplied in V = .FALSE.: initial vector supplied in V.

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

*H*H is COMPLEX array, dimension (LDH,N) The upper Hessenberg matrix H.

*LDH*LDH is INTEGER The leading dimension of the array H. LDH >= max(1,N).

*W*W is COMPLEX The eigenvalue of H whose corresponding right or left eigenvector is to be computed.

*V*V is COMPLEX array, dimension (N) On entry, if NOINIT = .FALSE., V must contain a starting vector for inverse iteration; otherwise V need not be set. On exit, V contains the computed eigenvector, normalized so that the component of largest magnitude has magnitude 1; here the magnitude of a complex number (x,y) is taken to be |x| + |y|.

*B*B is COMPLEX array, dimension (LDB,N)

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*RWORK*RWORK is REAL array, dimension (N)

*EPS3*EPS3 is REAL A small machine-dependent value which is used to perturb close eigenvalues, and to replace zero pivots.

*SMLNUM*SMLNUM is REAL A machine-dependent value close to the underflow threshold.

*INFO*INFO is INTEGER = 0: successful exit = 1: inverse iteration did not converge; V is set to the last iterate.

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **147** of file **claein.f**.

### subroutine dlaein (logical rightv, logical noinit, integer n, double precision, dimension( ldh, * ) h, integer ldh, double precision wr, double precision wi, double precision, dimension( * ) vr, double precision, dimension( * ) vi, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) work, double precision eps3, double precision smlnum, double precision bignum, integer info)

**DLAEIN** computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration.

**Purpose:**

DLAEIN uses inverse iteration to find a right or left eigenvector corresponding to the eigenvalue (WR,WI) of a real upper Hessenberg matrix H.

**Parameters***RIGHTV*RIGHTV is LOGICAL = .TRUE. : compute right eigenvector; = .FALSE.: compute left eigenvector.

*NOINIT*NOINIT is LOGICAL = .TRUE. : no initial vector supplied in (VR,VI). = .FALSE.: initial vector supplied in (VR,VI).

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

*H*H is DOUBLE PRECISION array, dimension (LDH,N) The upper Hessenberg matrix H.

*LDH*LDH is INTEGER The leading dimension of the array H. LDH >= max(1,N).

*WR*WR is DOUBLE PRECISION

*WI*WI is DOUBLE PRECISION The real and imaginary parts of the eigenvalue of H whose corresponding right or left eigenvector is to be computed.

*VR*VR is DOUBLE PRECISION array, dimension (N)

*VI*VI is DOUBLE PRECISION array, dimension (N) On entry, if NOINIT = .FALSE. and WI = 0.0, VR must contain a real starting vector for inverse iteration using the real eigenvalue WR; if NOINIT = .FALSE. and WI.ne.0.0, VR and VI must contain the real and imaginary parts of a complex starting vector for inverse iteration using the complex eigenvalue (WR,WI); otherwise VR and VI need not be set. On exit, if WI = 0.0 (real eigenvalue), VR contains the computed real eigenvector; if WI.ne.0.0 (complex eigenvalue), VR and VI contain the real and imaginary parts of the computed complex eigenvector. The eigenvector is normalized so that the component of largest magnitude has magnitude 1; here the magnitude of a complex number (x,y) is taken to be |x| + |y|. VI is not referenced if WI = 0.0.

*B*B is DOUBLE PRECISION array, dimension (LDB,N)

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= N+1.

*WORK*WORK is DOUBLE PRECISION array, dimension (N)

*EPS3*EPS3 is DOUBLE PRECISION A small machine-dependent value which is used to perturb close eigenvalues, and to replace zero pivots.

*SMLNUM*SMLNUM is DOUBLE PRECISION A machine-dependent value close to the underflow threshold.

*BIGNUM*BIGNUM is DOUBLE PRECISION A machine-dependent value close to the overflow threshold.

*INFO*INFO is INTEGER = 0: successful exit = 1: inverse iteration did not converge; VR is set to the last iterate, and so is VI if WI.ne.0.0.

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **170** of file **dlaein.f**.

### subroutine slaein (logical rightv, logical noinit, integer n, real, dimension( ldh, * ) h, integer ldh, real wr, real wi, real, dimension( * ) vr, real, dimension( * ) vi, real, dimension( ldb, * ) b, integer ldb, real, dimension( * ) work, real eps3, real smlnum, real bignum, integer info)

**SLAEIN** computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration.

**Purpose:**

SLAEIN uses inverse iteration to find a right or left eigenvector corresponding to the eigenvalue (WR,WI) of a real upper Hessenberg matrix H.

**Parameters***RIGHTV*RIGHTV is LOGICAL = .TRUE. : compute right eigenvector; = .FALSE.: compute left eigenvector.

*NOINIT*NOINIT is LOGICAL = .TRUE. : no initial vector supplied in (VR,VI). = .FALSE.: initial vector supplied in (VR,VI).

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

*H*H is REAL array, dimension (LDH,N) The upper Hessenberg matrix H.

*LDH*LDH is INTEGER The leading dimension of the array H. LDH >= max(1,N).

*WR*WR is REAL

*WI*WI is REAL The real and imaginary parts of the eigenvalue of H whose corresponding right or left eigenvector is to be computed.

*VR*VR is REAL array, dimension (N)

*VI*VI is REAL array, dimension (N) On entry, if NOINIT = .FALSE. and WI = 0.0, VR must contain a real starting vector for inverse iteration using the real eigenvalue WR; if NOINIT = .FALSE. and WI.ne.0.0, VR and VI must contain the real and imaginary parts of a complex starting vector for inverse iteration using the complex eigenvalue (WR,WI); otherwise VR and VI need not be set. On exit, if WI = 0.0 (real eigenvalue), VR contains the computed real eigenvector; if WI.ne.0.0 (complex eigenvalue), VR and VI contain the real and imaginary parts of the computed complex eigenvector. The eigenvector is normalized so that the component of largest magnitude has magnitude 1; here the magnitude of a complex number (x,y) is taken to be |x| + |y|. VI is not referenced if WI = 0.0.

*B*B is REAL array, dimension (LDB,N)

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= N+1.

*WORK*WORK is REAL array, dimension (N)

*EPS3*EPS3 is REAL A small machine-dependent value which is used to perturb close eigenvalues, and to replace zero pivots.

*SMLNUM*SMLNUM is REAL A machine-dependent value close to the underflow threshold.

*BIGNUM*BIGNUM is REAL A machine-dependent value close to the overflow threshold.

*INFO*INFO is INTEGER = 0: successful exit = 1: inverse iteration did not converge; VR is set to the last iterate, and so is VI if WI.ne.0.0.

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **170** of file **slaein.f**.

### subroutine zlaein (logical rightv, logical noinit, integer n, complex*16, dimension( ldh, * ) h, integer ldh, complex*16 w, complex*16, dimension( * ) v, complex*16, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) rwork, double precision eps3, double precision smlnum, integer info)

**ZLAEIN** computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration.

**Purpose:**

ZLAEIN uses inverse iteration to find a right or left eigenvector corresponding to the eigenvalue W of a complex upper Hessenberg matrix H.

**Parameters***RIGHTV*RIGHTV is LOGICAL = .TRUE. : compute right eigenvector; = .FALSE.: compute left eigenvector.

*NOINIT*NOINIT is LOGICAL = .TRUE. : no initial vector supplied in V = .FALSE.: initial vector supplied in V.

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

*H*H is COMPLEX*16 array, dimension (LDH,N) The upper Hessenberg matrix H.

*LDH*LDH is INTEGER The leading dimension of the array H. LDH >= max(1,N).

*W*W is COMPLEX*16 The eigenvalue of H whose corresponding right or left eigenvector is to be computed.

*V*V is COMPLEX*16 array, dimension (N) On entry, if NOINIT = .FALSE., V must contain a starting vector for inverse iteration; otherwise V need not be set. On exit, V contains the computed eigenvector, normalized so that the component of largest magnitude has magnitude 1; here the magnitude of a complex number (x,y) is taken to be |x| + |y|.

*B*B is COMPLEX*16 array, dimension (LDB,N)

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*RWORK*RWORK is DOUBLE PRECISION array, dimension (N)

*EPS3*EPS3 is DOUBLE PRECISION A small machine-dependent value which is used to perturb close eigenvalues, and to replace zero pivots.

*SMLNUM*SMLNUM is DOUBLE PRECISION A machine-dependent value close to the underflow threshold.

*INFO*INFO is INTEGER = 0: successful exit = 1: inverse iteration did not converge; V is set to the last iterate.

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **147** of file **zlaein.f**.

## Author

Generated automatically by Doxygen for LAPACK from the source code.