Computational Routines for the Nonsymmetric eigenproblem

LA_GEHRD
Real and complex versions.

SUBROUTINE LA_GEHRD( N, ILO, IHI, A, & 


LDA, TAU, WORK, LWORK, INFO ) 


INTEGER, INTENT(IN) :: IHI, ILO, LDA, & 


LWORK, N 


INTEGER, INTENT(OUT) :: INFO 

type(wp), INTENT(INOUT) :: A(LDA,*) 

type(wp), INTENT(OUT) :: TAU(*), 		 


WORK(LWORK) 


where 

type ::= REAL $\mid$ COMPLEX 

wp   ::= KIND(1.0) $\mid$ KIND(1.0D0)

LA_GEHRD reduces a real / complex general matrix $A$ to upper Hessenberg form $H$ by an orthogonal / unitary similarity transformation: $Q^H A Q = H$ .
References: See  [1] and [9,20].
-----------------------------------

LA_GEBAL
Real and complex versions.

SUBROUTINE LA_GEBAL( JOB, N, A, LDA, & 


ILO, IHI, SCALE, INFO ) 


CHARACTER(LEN=1), INTENT(IN) :: JOB 


INTEGER, INTENT(IN) :: LDA, N 


INTEGER, INTENT(OUT) :: IHI, ILO, INFO 


REAL(wp), INTENT(OUT) :: SCALE(*) 

type(wp), INTENT(INOUT) :: A(LDA,*) 


where 

type ::= REAL $\mid$ COMPLEX 

wp   ::= KIND(1.0) $\mid$ KIND(1.0D0)

LA_GEBAL balances a general real / complex matrix $A$.
References: See  [1] and [9,20,37].
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LA_GEBAK
Real and complex versions.

SUBROUTINE LA_GEBAK( JOB, SIDE, N, ILO, & 


IHI, SCALE, M, V, LDV, INFO ) 


CHARACTER(LEN=1), INTENT(IN) :: JOB, & 


SIDE 


INTEGER, INTENT(IN) :: IHI, ILO, LDV, M, N 


INTEGER, INTENT(OUT) :: INFO 


REAL(wp), INTENT(IN) :: SCALE(*) 

type(wp), INTENT(INOUT) :: V(LDV,*) 


where 

type ::= REAL $\mid$ COMPLEX 

wp   ::= KIND(1.0) $\mid$ KIND(1.0D0)

LA_GEBAK forms the right or left eigenvectors of a real / complex general matrix by backward transformation on the computed eigenvectors of the balanced matrix output by LA_GEBAL.
References: See  [1] and [9,20].
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LA_ORGHR / LA_UNGHR
Real and complex versions.

SUBROUTINE LA_ORGHR / LA_UNGHR( N, & 


ILO, IHI, A, LDA, TAU, WORK, LWORK, & 


INFO ) 


INTEGER, INTENT(IN) :: IHI, ILO, LDA, & 


LWORK, N 


INTEGER, INTENT(OUT) :: INFO 

type(wp), INTENT(IN) :: TAU(*) 

type(wp), INTENT(INOUT) :: A(LDA,*) 

type(wp), INTENT(OUT) :: WORK(LWORK) 


where 

type ::= REAL $\mid$ COMPLEX 

wp   ::= KIND(1.0) $\mid$ KIND(1.0D0)

LA_ORGHR / LA_UNGHR generate a real orthogonal / complex unitary matrix $Q$ which is defined as the product of elementary reflectors of order $n$, as returned by LA_GEHRD.
References: See  [1] and [9,20].
-----------------------------------

LA_ORMHR / LA_UNMHR
Real and complex versions.

SUBROUTINE LA_ORMHR / LA_UNMHR( SIDE, & 


TRANS, M, N, ILO, IHI, A, LDA, TAU, C, & 


LDC, WORK, LWORK, INFO ) 


CHARACTER(LEN=1),  INTENT(IN) :: SIDE, & 


TRANS 


INTEGER, INTENT(IN) :: IHI, ILO, LDA, LDC, & 


LWORK, M, N 


INTEGER, INTENT(OUT) :: INFO 

type(wp), INTENT(IN) :: A(LDA,*), TAU(*) 

type(wp), INTENT(INOUT) :: C(LDA,*) 

type(wp), INTENT(OUT) :: WORK(LWORK) 


where 

type ::= REAL $\mid$ COMPLEX 

wp   ::= KIND(1.0) $\mid$ KIND(1.0D0)

LA_ORMHR / LA_UNMHR overwrites the general real / complex $m \times n$ $C$ with a real orthogonal / complex unitary matrix $Q$ of order $nq$ where $Q$ is defined as the product of elementary reflectors, as returned by LA_GEHRD.
References: See  [1] and [9,20].
-----------------------------------

LA_HSEQR
Real version.

SUBROUTINE LA_HSEQR( JOB, COMPZ, N, & 


ILO, IHI, H, LDH, WR, WI, Z, LDZ, WORK, & 


LWORK, INFO ) 


CHARACTER(LEN=1), INTENT(IN) :: COMPZ, & 


JOB 


INTEGER, INTENT(IN) :: IHI, ILO, LDH, LDZ, 		 


LWORK, N 


INTEGER, INTENT(OUT) :: INFO 


REAL(wp), INTENT(OUT) :: WR(*), WI(*) 


REAL(wp), INTENT(INOUT) :: H(LDH,*), Z(LDZ,*) 


REAL(wp), INTENT(OUT) :: WORK(LWORK) 


where 

wp   ::= KIND(1.0) $\mid$ KIND(1.0D0)

Complex version.

 SUBROUTINE LA_HSEQR( JOB, COMPZ, N, & 


ILO, IHI, H, LDH, W, Z, LDZ, WORK, & 


LWORK, INFO ) 


CHARACTER(LEN=1), INTENT(IN) :: COMPZ, & 


JOB 


INTEGER, INTENT(IN) :: IHI, ILO, LDH, LDZ, & 


LWORK, N 


INTEGER, INTENT(OUT) :: INFO 


COMPLEX(wp), INTENT(INOUT) :: H(LDH,*), & 


Z(LDZ,*) 


COMPLEX(wp), INTENT(OUT) :: W(*), & 


WORK(LWORK) 


where 

wp   ::= KIND(1.0) $\mid$ KIND(1.0D0)

LA_HSEQR computes the eigenvalues of a real / complex upper Hessenberg matrix $H$, and, optionally, the matrices $T$ and $Z$ from the Schur decomposition $H = Z T Z^H$, where $T$ is an upper triangular matrix (the Schur form), and $Z$ is the unitary matrix of Schur vectors.

References: See  [1] and [9,20].
-----------------------------------

LA_HSEIN
Real version.

SUBROUTINE LA_HSEIN( VR, LDVR, MM, M, & 


WORK, IFAILL, IFAILR, INFO ) 


CHARACTER(LEN=1), INTENT(IN) :: & 


EIGSRC, INITV, SIDE 


INTEGER, INTENT(IN) :: LDH, LDVL, LDVR, & 


MM, N 


INTEGER, INTENT(OUT) :: INFO, M, & 


IFAILL(*), IFAILR(*) 


LOGICAL, INTENT(IN) :: SELECT(*) 


REAL(wp), INTENT(INOUT) :: WR(*), WI(*) 


REAL(wp), INTENT(IN) :: H(LDH,*) 


REAL(wp), INTENT(INOUT) :: VL(LDVL,*), 		 


VR(LDVR,*) 


REAL(wp), INTENT(OUT) :: WORK(*) 


where 

wp   ::= KIND(1.0) $\mid$ KIND(1.0D0)

Complex version.

 SUBROUTINE LA_HSEIN( VR, LDVR, MM, M, & 


WORK, RWORK, IFAILL, IFAILR, INFO ) 


CHARACTER(LEN=1), INTENT(IN) :: & 


EIGSRC, INITV, SIDE 


INTEGER, INTENT(IN) :: LDH, LDVL, LDVR, & 


MM, N 


INTEGER, INTENT(OUT) :: INFO, M, & 


IFAILL(*), IFAILR(*) 


LOGICAL, INTENT(IN) :: SELECT(*) 


REAL(wp), INTENT(OUT) :: RWORK( * ) 


COMPLEX(wp), INTENT(IN) :: H(LDH,*) 


COMPLEX(wp), INTENT(INOUT) :: VL(LDVL,*), & 


VR(LDVR,*), W(*) 


COMPLEX(wp), INTENT(OUT) :: WORK(*) 


where 

wp   ::= KIND(1.0) $\mid$ KIND(1.0D0)

LA_HSEIN uses inverse iteration to find specified right and/or left eigenvectors of a real / complex upper Hessenberg matrix $H$.
References: See  [1] and [9,20].
-----------------------------------

LA_TREVC
Real version.

SUBROUTINE LA_TREVC( SIDE, HOWMNY, & 


SELECT, N, T, LDT, VL, LDVL, VR, LDVR, & 


MM, M, WORK, INFO ) 


CHARACTER(LEN=1), INTENT(IN) :: & 


HOWMNY, SIDE 


INTEGER, INTENT(IN) :: LDT, LDVL, LDVR, & 


MM, N 


INTEGER, INTENT(OUT) :: INFO, M 


LOGICAL, INTENT(INOUT) :: SELECT(*) 


REAL(wp), INTENT(IN) :: T(LDT,*) 


REAL(wp), INTENT(INOUT) :: VL(LDVL,*), & 


VR(LDVR,*) 


REAL(wp), INTENT(OUT) :: WORK(*) 


where 

wp   ::= KIND(1.0) $\mid$ KIND(1.0D0)

Complex version.

 SUBROUTINE LA_TREVC( SIDE, HOWMNY, & 


SELECT, N, T, LDT, VL, LDVL, VR, LDVR, & 


MM, M, WORK, RWORK, INFO ) 


CHARACTER(LEN=1), INTENT(IN) :: & 


HOWMNY, SIDE 


INTEGER, INTENT(IN) :: LDT, LDVL, LDVR, & 


MM, N 


INTEGER, INTENT(OUT) :: INFO, M 


LOGICAL, INTENT(INOUT) :: SELECT(*) 


REAL(wp), INTENT(OUT) :: RWORK(*) 


COMPLEX(wp), INTENT(INOUT) :: T(LDT,*), & 


VL(LDVL,*), VR(LDVR,*) 


COMPLEX(wp), INTENT(OUT) :: WORK(*) 


where 

wp   ::= KIND(1.0) $\mid$ KIND(1.0D0)

LA_TREVC computes some or all of the right and/or left eigenvectors of a real / complex upper quasi-triangular / triangular matrix $T$.
References: See  [1] and [9,20].
-----------------------------------

LA_TREXC
Real version.

SUBROUTINE LA_TREXC( COMPQ, N, T, LDT, & 


Q, LDQ, IFST, ILST, WORK, INFO ) 


CHARACTER(LEN=1), INTENT(IN) :: COMPQ 


INTEGER, INTENT(IN) :: IFST, ILST, LDQ, & 


LDT, N 


INTEGER, INTENT(OUT) :: INFO 


REAL(wp), INTENT(INOUT) :: Q(LDQ,*), & 


T(LDT,*), WORK(*) 


where 

wp   ::= KIND(1.0) $\mid$ KIND(1.0D0)

Complex version.

 SUBROUTINE LA_TREXC( COMPQ, N, T, LDT, & 


Q, LDQ, IFST, ILST, INFO ) 


CHARACTER(LEN=1), INTENT(IN) :: COMPQ 


INTEGER, INTENT(IN) :: IFST, ILST, LDQ, & 


LDT, N 


INTEGER, INTENT(OUT) :: INFO 


COMPLEX(wp), INTENT(INOUT) :: Q(LDQ,*), & 


T(LDT,*)


where 

wp   ::= KIND(1.0) $\mid$ KIND(1.0D0)

LA_TREXC reorders the Schur factorization of a real / complex matrix $A = Q T Q^H$, so that the diagonal block of $T$ with row index IFST is moved to row ILST.
References: See  [1] and [9,20].
-----------------------------------

LA_TRSYL
Real and complex versions.

SUBROUTINE LA_TRSYL( TRANA, TRANB, & 


ISGN, M, N, A, LDA, B, LDB, C, LDC, & 


SCALE, INFO ) 


CHARACTER(LEN=1), INTENT(IN) :: & 


TRANA, TRANB 


INTEGER, INTENT(IN) :: ISGN, LDA, LDB, & 


LDC, M, N 


INTEGER, INTENT(OUT) :: INFO 


REAL(wp), INTENT(OUT) :: SCALE 

type(wp), INTENT(IN) :: A(LDA,*), B(LDB,*) 

type(wp), INTENT(INOUT) :: C(LDC,*) 


where 

type ::= REAL $\mid$ COMPLEX 

wp   ::= KIND(1.0) $\mid$ KIND(1.0D0)

LA_TRSYL solves the real / complex Sylvester matrix equation.
References: See  [1] and [9,20].
-----------------------------------

LA_TRSNA
Real version.

SUBROUTINE LA_TRSNA( JOB, HOWMNY, & 


SELECT, N, T, LDT, VL, LDVL, VR, LDVR, & 


S, SEP, MM, M, WORK, LDWORK, IWORK, & 


INFO ) 


CHARACTER(LEN=1), INTENT(IN) :: & 


HOWMNY, JOB 


INTEGER, INTENT(IN) :: LDT, LDVL, LDVR, & 


LDWORK, MM, N 


INTEGER, INTENT(OUT) :: INFO, M, IWORK(*) 


LOGICAL, INTENT(IN) :: SELECT(*) 


REAL(wp), INTENT(OUT) :: S(*), SEP(*) 


REAL(wp), INTENT(IN) :: T(LDT,*),  & 


VL(LDVL,*), VR(LDVR,*) 


REAL(wp), INTENT(OUT) :: WORK(LDWORK,*) 


where 

wp   ::= KIND(1.0) $\mid$ KIND(1.0D0)

Complex version.

 SUBROUTINE LA_TRSNA( JOB, HOWMNY, & 


SELECT, N, T, LDT, VL, LDVL, VR, LDVR, & 


S, SEP, MM, M, WORK, LDWORK, RWORK, & 


INFO ) 


CHARACTER(LEN=1), INTENT(IN) :: & 


HOWMNY, JOB 


INTEGER, INTENT(IN) :: LDT, LDVL, LDVR, & 


LDWORK, MM, N 


INTEGER, INTENT(OUT) :: INFO, M 


LOGICAL, INTENT(IN) :: SELECT(*) 


REAL(wp), INTENT(OUT) :: RWORK(*), S(*), & 


SEP(*) 


COMPLEX(wp), INTENT(IN) :: T(LDT,*), & 


VL(LDVL,*), VR(LDVR,*) 


COMPLEX(wp), INTENT(OUT) :: & 


WORK(LDWORK,*) 


where 

wp   ::= KIND(1.0) $\mid$ KIND(1.0D0)

LA_TRSNA estimates reciprocal condition numbers for specified eigenvalues and/or right eigenvectors of a real / complex upper quasi-triangular / triangular matrix $T$ (or of any matrix $Q T Q^H$ with $Q$ orthogonal / unitary).
References: See  [1] and [9,20].
-----------------------------------

LA_TRSEN
Real version.

SUBROUTINE LA_TRSEN( JOB, COMPQ, & 


SELECT, N, T, LDT, Q, LDQ, WR, WI, M, S, & 


SEP, WORK, LWORK, IWORK, LIWORK, & 


INFO ) 


CHARACTER(LEN=1), INTENT(IN) :: COMPQ, & 


JOB 


INTEGER, INTENT(IN) :: LDQ, LDT, LWORK, & 


N, LIWORK 


INTEGER, INTENT(OUT) :: INFO, M, & 


IWORK(LIWORK) 


REAL(wp), INTENT(OUT) :: S, SEP 


LOGICAL, INTENT(IN) :: SELECT(*) 


REAL(wp), INTENT(INOUT) :: Q(LDQ,*), T(LDT,*) 


REAL(wp), INTENT(IN) :: WR(*), WI(*) 


REAL(wp), INTENT(OUT) :: WORK(LWORK) 


where 

wp   ::= KIND(1.0) $\mid$ KIND(1.0D0)

Complex version.

 SUBROUTINE LA_TRSEN( JOB, COMPQ, & 


SELECT, N, T, LDT, Q, LDQ, W, M, S, & 


SEP, WORK, LWORK, INFO ) 


CHARACTER(LEN=1), INTENT(IN) :: COMPQ, & 


JOB 


INTEGER, INTENT(IN) :: LDQ, LDT, LWORK, N 


INTEGER, INTENT(OUT) :: INFO, M 


REAL(wp), INTENT(OUT) :: S, SEP 


LOGICAL, INTENT(IN) :: SELECT(*) 


COMPLEX(wp), INTENT(INOUT) :: Q(LDQ,*), & 


T(LDT,*) 


COMPLEX(wp), INTENT(IN) :: W(*), & 


WORK(LWORK) 


where 

wp   ::= KIND(1.0) $\mid$ KIND(1.0D0)

LA_TRSEN reorders the Schur factorization of a real / complex matrix $A = Q T Q^H$, so that a selected cluster of eigenvalues appears in the leading positions on the diagonal of the upper triangular matrix $T$, and the leading columns of $Q$ form an orthonormal basis of the corresponding right invariant subspace.
References: See  [1] and [9,20].
-----------------------------------