--- name: "laein" section: "3" platform: "linux" package: "lapack-doc" version: "3.12.1-2" release: "Arch Linux 2026-11-01" platforms: ["linux"] url: "https://new.carta.tech/laein" --- # laein(3) laein: eigvec by Hessenberg inverse iteration ## NAME laein - 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.