Computational Routines for the Generalized Nonsymmetric Eigenproblem

LA_GGHRD
Real and complex versions.

SUBROUTINE LA_GGHRD( COMPQ, COMPZ, & 


N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, & 


LDZ, INFO ) 


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


COMPQ, COMPZ 


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


LDB, LDQ, LDZ, N 


INTEGER, INTENT(OUT) :: INFO 

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


B(LDB,*), Q(LDQ,*), Z(LDZ,*) 


where 

type ::= REAL $\mid$ COMPLEX 

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

LA_GGHRD reduces a pair of real / complex matrices $( A, B )$ to generalized upper Hessenberg form using orthogonal / unitary transformations.
References: See  [1] and [9,20].
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LA_GGBAL
Real and complex versions.

SUBROUTINE LA_GGBAL( JOB, N, A, LDA, B, & 


LDB, ILO, IHI, LSCALE, RSCALE, WORK, & 


INFO ) 


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


INTEGER, INTENT(IN) :: LDA, LDB, N 


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


REAL(wp), INTENT(OUT) :: LSCALE(*), & 


RSCALE(*), WORK(*) 

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


where 

type ::= REAL $\mid$ COMPLEX 

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

LA_GGBAL balances a pair of general real / complex matrices $( A, B )$.
References: See  [1] and [9,20,42].
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LA_GGBAK
Real and complex versions.

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


IHI, LSCALE, RSCALE, 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) :: LSCALE(*), & 


RSCALE(*) 

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


where 

type ::= REAL $\mid$ COMPLEX 

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

LA_GGBAK forms the right or left eigenvectors of a real / complex generalized eigenvalue problem $A x = \lambda B x$, by backward transformation on the computed eigenvectors of the balanced pair of matrices output by LA_GGBAL.
References: See  [1] and [9,20,42].
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LA_HGEQZ
Real version.

SUBROUTINE LA_HGEQZ( JOB, COMPQ, & 


COMPZ, N, ILO, IHI, A, LDA, B, LDB, & 


ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, & 


WORK, LWORK, INFO ) 


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


COMPZ, JOB 


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


LDQ, LDZ, LWORK, N 


INTEGER, INTENT(OUT) :: INFO 


REAL(wp), INTENT(INOUT) :: A(LDA,*), & 


B(LDB,*), Q(LDQ,*), Z(LDZ,*) 


REAL(wp), INTENT(OUT) :: ALPHAR(*), & 


ALPHAI(*), BETA(*), WORK(LWORK) 


where 

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

Complex version.

 SUBROUTINE LA_HGEQZ( JOB, COMPQ, & 


COMPZ, N, ILO, IHI, A, LDA, B, LDB, & 


ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, & 


LWORK, RWORK, INFO ) 


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


COMPZ, JOB 


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


LDQ, LDZ, LWORK, N 


INTEGER, INTENT(OUT) :: INFO 


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


COMPLEX(wp), INTENT(INOUT) :: A(LDA,*), & 


B(LDB,*), Q(LDQ,*), Z(LDZ,*) 


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


BETA(*), WORK(LWORK) 


where 

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

LA_HGEQZ implements a single-shift version of the $QZ$ method for finding the generalized eigenvalues $w_i=\alpha_i/\beta_i$ of the equation $det( A - w_i B ) = 0$
References: See  [1] and [9,20,32].
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LA_TGEVC
Real version.

SUBROUTINE LA_TGEVC( SIDE, HOWMNY, & 


SELECT, N, A, LDA, B, LDB, VL, LDVL, & 


VR, LDVR, MM, M, WORK, INFO ) 


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


HOWMNY, SIDE 


INTEGER, INTENT(IN) :: LDA, LDB, LDVL, 		 


LDVR, MM, N 


INTEGER, INTENT(OUT) :: INFO, M 


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


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


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_TGEVC( SIDE, HOWMNY, & 


SELECT, N, A, LDA, B, LDB, VL, LDVL, & 


VR, LDVR, MM, M, WORK, RWORK, INFO ) 


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


HOWMNY, SIDE 


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


LDVR, MM, N 


INTEGER, INTENT(OUT) :: INFO, M 


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


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


COMPLEX(wp), INTENT(IN) :: A(LDA,*), & 


B(LDB,*) 


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


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


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


where 

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

LA_TGEVC computes some or all of the right and/or left generalized eigenvectors of a pair of real / complex upper triangular matrices $( A, B )$.
References: See  [1] and [9,20].
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LA_TGEXC
Real version.

SUBROUTINE LA_TGEXC( WANTQ, WANTZ, & 


N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, IFST, & 


ILST, WORK, LWORK, INFO ) 


LOGICAL, INTENT(IN) :: WANTQ, WANTZ 


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


LDZ, LWORK, N 


INTEGER, INTENT(INOUT) :: IFST, ILST 


INTEGER, INTENT(OUT) :: INFO 


REAL(wp), INTENT(INOUT) :: A(LDA,*), & 


B(LDB,*), Q(LDQ,*), Z(LDZ,*) 


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


where 

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

Complex version.

 SUBROUTINE LA_TGEXC( WANTQ, WANTZ, & 


N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, IFST, & 


ILST, INFO ) 


LOGICAL, INTENT(IN) :: WANTQ, WANTZ 


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


LDZ, N 


INTEGER, INTENT(INOUT) :: IFST, ILST 


INTEGER, INTENT(OUT) :: INFO 


COMPLEX(wp), INTENT(INOUT) :: A(LDA,*), & 


B(LDB,*), Q(LDQ,*), Z(LDZ,*) 


where 

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

LA_TGEXC reorders the generalized Schur decomposition of a complex matrix pair $( A, B )$, using an orthogonal / unitary equivalence transformation $(A, B) := Q (A, B) Z^H$, so that the diagonal block of $( A, B )$ with row index IFST is moved to row ILST.
References: See [1] and [9,20,28].
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LA_TGSYL
Real and complex versions.

SUBROUTINE LA_TGSYL( TRANS, IJOB, M, N, & 


A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, & 


F, LDF, SCALE, DIF, WORK, LWORK, & 


IWORK, INFO ) 


CHARACTER, INTENT(IN) :: TRANS 


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


LDC, LDD, LDE, LDF, LWORK, M, N 


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


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

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


D(LDD,*), E(LDF,*) 

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


F(LDF,*) 

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


where 

type ::= REAL $\mid$ COMPLEX 

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

LA_TGSYL solves the generalized Sylvester equation.
References: See  [1] and [9,20,26,24,27].
-----------------------------------

LA_TGSNA
Real and complex versions.

SUBROUTINE LA_TGSNA( JOB, HOWMNY, & 


SELECT, N, A, LDA, B, LDB, VL, LDVL, & 


VR, LDVR, S, DIF, MM, M, WORK, & 


LWORK, IWORK, INFO ) 


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


HOWMNY, JOB 


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


LDVR, LWORK, MM, N 


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


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


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

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


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

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


where 

type ::= REAL $\mid$ COMPLEX 

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

LA_TGSNA estimates reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a matrix pair $( A, B )$.
References: See  [1] and [9,20,25,28,26].
-----------------------------------

LA_TGSEN
Real version.

SUBROUTINE LA_TGSEN( IJOB, WANTQ, & 


WANTZ, SELECT, N, A, LDA, B, LDB, & 


ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, & 


M, PL, PR, DIF, WORK, LWORK, IWORK, & 


LIWORK, INFO ) 


LOGICAL, INTENT(IN) :: WANTQ, WANTZ 


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


LDQ, LDZ, LIWORK, LWORK, N 


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


IWORK(LIWORK) 


REAL(wp), INTENT(OUT) :: PL, PR 


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


REAL(wp), INTENT(INOUT) :: A(LDA,*), & 


B(LDB,*), Q(LDQ,*), Z(LDZ,*) 


REAL(wp), INTENT(OUT) :: ALPHAI(*), & 


ALPHAR(*), BETA(*), DIF(2), & 


WORK(LWORK) 


where 

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

Complex version.

 SUBROUTINE LA_TGSEN( IJOB, WANTQ, & 


WANTZ, SELECT, N, A, LDA, B, LDB, & 


ALPHA, BETA, Q, LDQ, Z, LDZ, & 


M, PL, PR, DIF, WORK, LWORK, IWORK, & 


LIWORK, INFO ) 


LOGICAL, INTENT(IN) :: WANTQ, WANTZ 


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


LDQ, LDZ, LIWORK, LWORK, N 


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


IWORK(LIWORK) 


REAL(wp), INTENT(OUT) :: PL, PR 


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


REAL(wp), INTENT(OUT) :: DIF(2) 


COMPLEX(wp), INTENT(INOUT) :: A(LDA,*), & 


B(LDB,*), Q(LDQ,*), Z(LDZ,*) 


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


BETA(*), WORK(LWORK) 


where 

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

LA_TGSEN reorders the generalized Schur decomposition of a real / complex matrix pair $( A, B )$ (in terms of an orthogonal / unitary equivalence transformation $Q^H (A, B) Z)$, so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the pair $( A, B )$.
References: See  [1] and [9,20,28,25,26].
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