Computational Routines for the Singular Value Decomposition

LA_GEBRD
Real and complex versions.

SUBROUTINE LA_GEBRD( M, N, A, LDA, D, E, & 


TAUQ, TAUP, WORK, LWORK, INFO ) 


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


INTEGER, INTENT(OUT) :: INFO 


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

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

type(wp), INTENT(OUT) :: TAUP(*), TAUQ(*), & 


WORK(LWORK) 


where 

type ::= REAL $\mid$ COMPLEX 

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

LA_GEBRD reduces a general real / complex $m \times n$ matrix $A$ to upper or lower bidiagonal form $B$ by an orthogonal / unitary transformation: $Q^H A P = B$.
References: See  [1] and [9,20].
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LA_GBBRD
Real version.

SUBROUTINE LA_GBBRD( VECT, M, N, NCC, & 


KL, KU, AB, LDAB, D, E, Q, LDQ, PT, & 


LDPT, C, LDC, WORK, INFO ) 


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


INTEGER, INTENT(IN) :: KL, KU, LDAB, & 


LDC, LDPT, LDQ, M, N, NCC 


INTEGER, INTENT(OUT) :: INFO 


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


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


C(LDC,*) 


REAL(wp), INTENT(OUT) :: PT(LDPT,*), & 


Q(LDQ,*), WORK(*) 


where 

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

Complex version.

 SUBROUTINE LA_GBBRD( VECT, M, N, NCC, & 


KL, KU, AB, LDAB, D, E, Q, LDQ, PT, & 


LDPT, C, LDC, WORK, RWORK, INFO ) 


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


INTEGER, INTENT(IN) :: KL, KU, LDAB, & 


LDC, LDPT, LDQ, M, N, NCC 


INTEGER, INTENT(OUT) :: INFO 


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


RWORK(*) 


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


AB(LDAB,*), C(LDC,*) 


COMPLEX(wp), INTENT(OUT) :: PT(LDPT,*), & 


Q(LDQ,*), WORK(*) 


where 

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

LA_GBBRD reduces a real / complex general $m \times n$ band matrix $A$ to real upper bidiagonal form $B$ by an orthogonal / unitary transformation: $Q^H A P = B$.
References: See  [1] and [9,20].
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LA_ORGBR / LA_UNGBR
Real and complex versions.

SUBROUTINE LA_ORGBR / LA_UNGBR( & 


VECT, M, N, K, A, LDA, TAU, WORK, & 


LWORK, INFO ) 


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


INTEGER, INTENT(IN) :: K, LDA, LWORK, M, & 


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_ORGBR / LA_UNGBR generates one of the real / complex orthogonal / unitary matrices $Q$ or $P^H$ determined by LA_GEBRD when reducing a complex matrix $A$ to bidiagonal form: $A = Q B P^H$. $Q$ and $P^H$ are defined as products of elementary reflectors $H_i$ or $G_i$ respectively.
References: See  [1] and [9,20].
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LA_ORMBR / LA_UNMBR
Real and complex versions.

SUBROUTINE LA_ORMBR / LA_UNMBR( & 


VECT, SIDE, TRANS, M, N, K, A, LDA, & 


TAU, C, LDC, WORK, LWORK, INFO ) 


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


TRANS, VECT 


INTEGER, INTENT(IN) :: K, 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_ORMBR / LA_UNMBR overwrites the general real / complex $m \times n$ matrix $C$ with the products $Q C$, $C Q$, $Q^H C$, $C Q^H$, $P C$, $C P$, $Q^H C$ or $C Q^H$, respectively, where $Q$ and $P^H$ are the orthogonal / unitary matrices determined by LA_GEBRD when reducing a real / complex matrix $A$ to bidiagonal form: $A = Q B P^H$.
References: See  [1] and [9,20].
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LA_BDSQR
Real and complex versions.

SUBROUTINE LA_BDSQR( UPLO, N, NCVT, & 


NRU, NCC, D, E, VT, LDVT, U, LDU, C, & 


LDC, RWORK, INFO ) 


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


INTEGER, INTENT(IN) :: LDC, LDU, LDVT, & 


N, NCC, NCVT, NRU 


INTEGER, INTENT(OUT) :: INFO 


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


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

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


U(LDU,*), VT(LDVT,*) 


where 

type ::= REAL $\mid$ COMPLEX 

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

LA_BDSQR computes the singular value decomposition (SVD) of a real $n\times n$ (upper or lower) bidiagonal matrix $B$: $B = Q S P^H$ where $S$ is a diagonal matrix with non-negative diagonal elements (the singular values of $B$), and $Q$ and $P$ are orthogonal matrices.
References: See  [1] and [9,20,8,34].
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LA_BDSDC
Real version.

SUBROUTINE LA_BDSDC( UPLO, COMPQ, N, & 


D, E, U, LDU, VT, LDVT, Q, IQ, WORK, & 


IWORK, INFO )  


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


COMPQ, UPLO 


INTEGER, INTENT(IN) :: LDU, LDVT, N  


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


IWORK( * ) 


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


REAL(wp), INTENT(OUT) :: Q(*), U(LDU,*), & 


VT(LDVT,*), WORK(*) 


where 

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

LA_BDSDC computes the singular value decomposition (SVD) of a real $n\times n$ (upper or lower) bidiagonal matrix $B$: $B = U S VT$, using a divide and conquer method, where $S$ is a diagonal matrix with non-negative diagonal elements (the singular values of $B$), and $U$ and $VT$ are orthogonal matrices of left and right singular vectors, respectively. LA_BDSDC can be used to compute all singular values, and optionally, singular vectors or singular vectors in compact form.
References: See  [1] and [9,20].
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