Computational Routines for the Symmetric Eigenproblem

LA_SYTRD / LA_HETRD
Real and complex versions.

SUBROUTINE LA_SYTRD / LA_HETRD( UPLO, & 


N, A, LDA, D, E, TAU, WORK, LWORK, & 


INFO ) 


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


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


INTEGER, INTENT(OUT) :: INFO 

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


REAL(wp), INTENT(OUT) :: D(*), E(*) 

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


WORK(LWORK) 


where 

type ::= REAL $\mid$ COMPLEX 

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

LA_SYTRD / LA_HETRD reduces a real symmetric / complex Hermitian matrix $A$ to real symmetric tridiagonal form $T$ by an orthogonal / unitary similarity transformation: $Q^H A Q = T$.
References: See  [1] and [9,20].
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LA_SPTRD / LA_HPTRD
Real and complex versions.

SUBROUTINE LA_SPTRD / LA_HPTRD( UPLO, & 


N, AP, D, E, TAU, INFO ) 


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


INTEGER, INTENT(IN) :: N 


INTEGER, INTENT(OUT) :: INFO 


REAL(wp), INTENT(OUT) :: D(*), E(*) 

type(wp), INTENT(INOUT) :: AP(*) 

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


where 

type ::= REAL $\mid$ COMPLEX 

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

LA_SPTRD / LA_HPTRD reduces a real symmetric / complex Hermitian matrix $A$ stored in packed storage to real symmetric tridiagonal form $T$ by an orthogonal / unitary similarity transformation: $Q^H A Q = T$.
References: See  [1] and [9,20].
-----------------------------------

LA_SBTRD / LA_HBTRD
Real and complex versions.

SUBROUTINE LA_SBTRD / LA_HBTRD( VECT, & 


UPLO, N, KD, AB, LDAB, D, E, Q, LDQ, & 


WORK, INFO ) 


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


VECT 


INTEGER, INTENT(IN) :: KD, LDAB, LDQ, N 


INTEGER, INTENT(OUT) :: INFO 


REAL(wp), INTENT(OUT) :: D(*), E(*) 

type(wp), INTENT(INOUT) :: AB(LDAB,*), & 


Q(LDQ,*) 

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


where 

type ::= REAL $\mid$ COMPLEX 

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

LA_SBTRD / LA_HBTRD reduces a real symmetric / complex Hermitian band matrix $A$ to real symmetric tridiagonal form $T$ by an orthogonal / unitary similarity transformation: $Q^H A Q = T$.
References: See  [1] and [9,20].
-----------------------------------

LA_ORGTR / LA_UNGTR
Real and complex versions.

SUBROUTINE LA_ORGTR / LA_UNGTR( UPLO, & 


N, A, LDA, TAU, WORK, LWORK, INFO ) 


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


INTEGER, INTENT(IN) :: 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_ORGTR / LA_UNGTR generates a real orthogonal / complex unitary matrix $Q$ which is defined as the product of $n - 1$ elementary reflectors of order $n$, as returned by LA_SYTRD / LA_HETRD.
References: See  [1] and [9,20].
-----------------------------------

LA_ORMTR / LA_UNMTR
Real and complex versions.

SUBROUTINE LA_ORMTR / LA_UNMTR( SIDE, & 


UPLO, TRANS, M, N, A, LDA, TAU, C, LDC, & 


WORK, LWORK, INFO ) 


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


TRANS, UPLO 


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


LWORK, M, N 


INTEGER, INTENT(OUT) :: INFO 

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

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

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


where 

type ::= REAL $\mid$ COMPLEX 

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

LA_ORMTR / LA_UNMTR overwrite the general real / complex $m \times n$ matrix $C$ with a real orthogonal / complex unitary matrix $Q$ of order $nq$.
References: See  [1] and [9,20].
-----------------------------------

LA_OPGTR / LA_UPGTR
Real and complex versions.

SUBROUTINE LA_OPGTR / LA_UPGTR( UPLO, & 


N, AP, TAU, Q, LDQ, WORK, INFO ) 


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


INTEGER, INTENT(IN) :: LDQ, N 


INTEGER, INTENT(OUT) :: INFO 

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

type(wp), INTENT(OUT) :: Q(LDQ,*), WORK(*) 


where 

type ::= REAL $\mid$ COMPLEX 

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

LA_OPGTR / LA_UPGTR generate a real orthogonal / complex unitary matrix $Q$ which is defined as the product of $n - 1$ elementary reflectors $H_i$ of order $n$, as returned by LA_SPTRD / LA_HPTRD using packed storage.
References: See  [1] and [9,20].
-----------------------------------

LA_OPMTR / LA_UPMTR
Real and complex versions.

SUBROUTINE LA_OPMTR / LA_UPMTR( SIDE, & 


UPLO, TRANS, M, N, AP, TAU, C, LDC, & 


WORK, INFO ) 


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


TRANS, UPLO 


INTEGER, INTENT(IN) :: LDC, M, N 


INTEGER, INTENT(OUT) :: INFO 

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

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

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


where 

type ::= REAL $\mid$ COMPLEX 

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

LA_OPMTR / LA_UPMTR overwrite the general real / complex $m \times n$ matrix $C$ with a real orthogonal / complex unitary matrix $Q$ of order $nq$, defined as the product of $nq-1$ elementary reflectors, as returned by LA_SPTRD / LA_HPTRD.
References: See  [1] and [9,20].
-----------------------------------

LA_STEQR
Real and complex versions.

SUBROUTINE LA_STEQR( COMPZ, N, D, E, Z, & 


LDZ, WORK, INFO ) 


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


INTEGER, INTENT(IN) :: LDZ, N 


INTEGER, INTENT(OUT) :: INFO 


REAL(wp), INTENT(INOUT) :: D(*), E(*) 


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

type(wp), INTENT(INOUT) :: Z(LDZ,*) 


where 

type ::= REAL $\mid$ COMPLEX 

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

LA_STEQR computes all eigenvalues and, optionally, eigenvectors of a symmetric tridiagonal matrix using the implicit $QL$ or $QR$ method.
References: See  [1] and [9,20].
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LA_STERF
Real version.

SUBROUTINE LA_STERF( N, D, E, INFO ) 


INTEGER, INTENT(IN) :: N 


INTEGER, INTENT(OUT) :: INFO 


REAL(wp), INTENT(INOUT) :: D(*), E(*) 


where 

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

LA_STERF computes all eigenvalues of a symmetric tridiagonal matrix using the Pal-Walker-Kahan variant of the $QL$ or $QR$ algorithm.
References: See  [1] and [9,20].
-----------------------------------

LA_STEDC
Real version.

SUBROUTINE LA_STEDC( COMPZ, N, D, E, Z, & 


LDZ, WORK, LWORK, IWORK, LIWORK, & 


INFO ) 


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


INTEGER, INTENT(IN) :: LDZ, LIWORK, & 


LWORK, N 


INTEGER, INTENT(OUT) :: INFO,  & 


IWORK(LIWORK) 


REAL(wp), INTENT(INOUT) :: D(*), E(*) 


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


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


where 

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

Complex version.

 SUBROUTINE LA_STEDC( COMPZ, N, D, E, Z, & 


LDZ, WORK, LWORK, RWORK, LRWORK, & 


IWORK, LIWORK, INFO ) 


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


INTEGER, INTENT(IN) :: LDZ, LIWORK, & 


LRWORK, LWORK, N 


INTEGER, INTENT(OUT) :: INFO,  & 


IWORK(LIWORK) 


REAL(wp), INTENT(INOUT) :: D(*), E(*) 


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


COMPLEX(wp), INTENT(INOUT) :: Z(LDZ,*) 


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


WORK(LWORK) 


where 

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

LA_STEDC computes all eigenvalues and, optionally, eigenvectors of a symmetric tridiagonal matrix using the divide and conquer method.
References: See  [1] and [9,20].
-----------------------------------

LA_STEGR
Real and complex versions.

SUBROUTINE LA_STEGR( JOBZ, RANGE, N, & 


D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, & 


LDZ, ISUPPZ, WORK, LWORK, & 


IWORK, LIWORK, INFO ) 


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


RANGE 


INTEGER, INTENT(IN) :: IL, IU, LDZ, & 


LIWORK, LWORK, N 


INTEGER, INTENT(OUT) :: INFO, M 


INTEGER, INTENT(OUT) :: ISUPPZ( * ), & 


IWORK(LIWORK) 


REAL(wp), INTENT(IN) :: ABSTOL, VL, VU 


REAL(wp), INTENT(INOUT) :: D( * ), E( * ) 


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


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

type(wp), INTENT(OUT) :: Z( LDZ, * ) 


where 

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

LA_STEGR computes selected eigenvalues and, optionally, eigenvectors of a real symmetric / complex Hermitian tridiagonal matrix $T$.
References: See  [1] and [9,20,11].
-----------------------------------

LA_STEBZ
Real version.

SUBROUTINE LA_STEBZ( RANGE, ORDER, & 


N, VL, VU, IL, IU, ABSTOL, D, E, M, & 


NSPLIT, W, IBLOCK, ISPLIT, WORK, & 


IWORK, INFO ) 


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


ORDER, RANGE 


INTEGER, INTENT(IN) :: IL, IU, M, N 


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


IBLOCK(*), ISPLIT(*), IWORK(*) 


REAL(wp), INTENT(IN) :: ABSTOL, VL, VU, & 


D(*), E(*) 


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


where 

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

LA_STEBZ computes the eigenvalues of a symmetric tridiagonal matrix $T$. The user may ask for all eigenvalues in the half-open interval ($VL, VU$], or the $IL^{th}$ through $IU^{th}$ eigenvalues.
References: See  [1] and [9,20,29].
-----------------------------------

LA_STEIN
Real and complex versions.

SUBROUTINE LA_STEIN( N, D, E, M, W, & 


IBLOCK, ISPLIT, Z, LDZ, WORK, IWORK, & 


IFAIL, INFO ) 


INTEGER, INTENT(IN) :: LDZ, M, N, & 


IBLOCK(*), ISPLIT(*) 


INTEGER, INTENT(OUT) :: INFO, IFAIL(*), & 


IWORK(*) 


REAL(wp), INTENT(IN) :: D(*), E(*), W(*) 


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

type(wp), INTENT(OUT) :: Z( LDZ, * ) 


where 

type ::= REAL $\mid$ COMPLEX 

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

LA_STEIN computes the eigenvectors of a real symmetric tridiagonal matrix $T$ corresponding to specified eigenvalues, using inverse iteration.
References: See  [1] and [9,20].
-----------------------------------

LA_PTEQR
Real and complex versions.

SUBROUTINE LA_PTEQR( COMPZ, N, D, E, Z, & 


LDZ, WORK, INFO ) 


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


INTEGER, INTENT(IN) :: INFO, LDZ, N 


REAL(wp), INTENT(INOUT) :: D(*), E(*) 


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

type(wp), INTENT(INOUT) :: Z(LDZ,*) 


where 

type ::= REAL $\mid$ COMPLEX 

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

LA_PTEQR computes all eigenvalues and, optionally, eigenvectors of a symmetric positive definite tridiagonal matrix by first factoring the matrix using LA_PTTRF and then calling LA_BDSQR to compute the singular values of the bidiagonal factor.
References: See  [1] and [9,20].
-----------------------------------