General Linear Systems

LA_GETRF
Real and complex versions.

 SUBROUTINE LA_GETRF( M, N, A, LDA, & 


IPIV, INFO ) 


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


INTEGER, INTENT(OUT) :: INFO 


INTEGER, INTENT( OUT ) :: IPIV( * ) 

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


where 

type ::= REAL $\mid$ COMPLEX 

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

LA_GETRF computes an $LU$ factorization of a general $m \times n$ real / complex matrix $A$ using partial pivoting with row interchanges.
References: See  [1] and [9,20].
-----------------------------------

LA_GETRS
Real and complex versions.

SUBROUTINE LA_GETRS( TRANS, N, NRHS, & 


A, LDA, IPIV, B, LDB, INFO ) 


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


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


INTEGER, INTENT(OUT) :: INFO 


INTEGER, INTENT(IN) :: IPIV(*) 

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

type(wp), INTENT(INOUT) :: rhs 


where 

type ::= REAL $\mid$ COMPLEX 

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

rhs  ::= B(LDB,*) $\mid$ B(*)

LA_GETRS solves a system of linear equations $AX = B$ or $A^T X = B$ with a general $n\times n$ real / complex matrix $A$ using the $LU$ factorization computed by LA_GETRF.
References: See  [1] and [9,20].
-----------------------------------

LA_GECON
Real version.

SUBROUTINE LA_GECON( NORM, N, A, & 


LDA, ANORM, RCOND, WORK, & 


IWORK, INFO ) 


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


NORM 


INTEGER, INTENT(IN) :: LDA, N 


INTEGER, INTENT(OUT) :: INFO 


REAL(wp), INTENT(IN) :: ANORM 


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


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


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


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


where 

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

Complex version.

 SUBROUTINE LA_GECON( NORM, N, A, & 


LDA, ANORM, RCOND, WORK, & 


RWORK, INFO ) 


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


NORM 


INTEGER, INTENT(IN) :: LDA, N 


INTEGER, INTENT(OUT) :: INFO 


REAL(wp), INTENT(IN) :: ANORM 


REAL(wp), INTENT(OUT) :: RCOND, & 


RWORK(*) 


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


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


where 

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

LA_GECON estimates the reciprocal of the condition number of a general real / complex matrix $A$, using the $LU$ factorization computed by LA_GETRF.
References: See  [1] and [9,20,21].
-----------------------------------

LA_GERFS
Real version.

 SUBROUTINE LA_GERFS( TRANS, N, NRHS, & 


A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, & 


FERR, BERR, WORK, IWORK, INFO ) 


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


TRANS 


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


LDX, NRHS, N 


INTEGER, INTENT(OUT) :: INFO 


INTEGER, INTENT(IN) :: IPIV(*) 


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


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


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


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


AF(LDAF,*), rhs 


REAL(wp), INTENT(INOUT) :: sol 


where 

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

rhs  ::= B(LDB,*) $\mid$ B(*) 

sol  ::= X(LDX,*) $\mid$ X(*) 

err  ::= FERR(*), BERR(*) $\mid$ FERR, BERR

Complex version.

 SUBROUTINE LA_GERFS( TRANS, N, NRHS, & 


A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, & 


FERR, BERR, WORK, RWORK, INFO ) 


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


TRANS 


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


LDX, NRHS, N 


INTEGER, INTENT(OUT) :: INFO 


INTEGER, INTENT(IN) :: IPIV(*) 


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


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


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


AF(LDAF,*), rhs 


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


where 

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

rhs  ::= B(LDB,*) $\mid$ B(*) 

sol  ::= X(LDX,*) $\mid$ X(*) 

err  ::= FERR(*), BERR(*) $\mid$ FERR, BERR

LA_GERFS improves the computed solution to a system of linear equations and provides error bounds and backward error estimates for the solution.
References: See  [1] and [9,21,20].
-----------------------------------

LA_GETRI
Real and complex versions.

 SUBROUTINE LA_GETRI( N, A, LDA, IPIV, & 


WORK, LWORK, INFO ) 


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


INTEGER, INTENT(OUT) :: INFO 


INTEGER, INTENT(IN) :: IPIV(*) 

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

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


where 

type ::= REAL $\mid$ COMPLEX 

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

LA_GETRI computes the inverse of a matrix using the $LU$ factorization computed by LA_GETRF.
References: See  [1] and [9,20].
-----------------------------------

LA_GEEQU
Real and complex versions.

 SUBROUTINE LA_GEEQU( M, N, A, LDA, R, C, & 


ROWCND, COLCND, AMAX, INFO ) 


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


INTEGER, INTENT(OUT) :: INFO 


REAL(wp), INTENT(OUT) :: AMAX, COLCND, & 


ROWCND 

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


REAL(wp), INTENT(OUT) :: C( * ), R( * ) 


where 

type ::= REAL $\mid$ COMPLEX 

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

LA_GEEQU computes row and column scalings intended to equilibrate an $m \times n$ general real / complex matrix $A$ and reduce its condition number.
References: See  [1] and [9,21,20].
-----------------------------------

LA_GBTRF
Real and complex versions.

SUBROUTINE LA_GBTRF( M, N, KL, KU, AB, & 


LDAB, IPIV, INFO ) 


INTEGER, INTENT(IN) :: KL, KU, LDAB, M, N 


INTEGER, INTENT(OUT) :: INFO 


INTEGER, INTENT(INOUT) :: IPIV(*) 

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


where 

type ::= REAL $\mid$ COMPLEX 

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

LA_GBTRF computes an $LU$ factorization of a real / complex $m \times n$ band matrix $A$ using partial pivoting with row interchanges.
References: See  [1] and [9,20].
-----------------------------------

LA_GBTRS
Real and complex versions.

SUBROUTINE LA_GBTRS( TRANS, N, KL, KU, & 


NRHS, AB, LDAB, IPIV, B, LDB, INFO ) 


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


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


N, NRHS 


INTEGER, INTENT(OUT) :: INFO 


INTEGER, INTENT(IN) :: IPIV(*) 

type(wp), INTENT(IN) :: AB( LDAB,*) 

type(wp), INTENT(INOUT) ::  rhs 


where 

type ::= REAL $\mid$ COMPLEX 

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

rhs  ::= B(LDB,*) $\mid$ B(*)

LA_GBTRS solves a system of linear equations $AX = B$ or $A^T X = B$ with a general real / complex band matrix $A$ using the $LU$ factorization computed by LA_GBTRF.
References: See  [1] and [9,20].
-----------------------------------

LA_GBCON
Real version.

 SUBROUTINE LA_GBCON( NORM, N, KL, & 


KU, AB, LDAB, IPIV, ANORM, & 


RCOND, WORK, IWORK, INFO ) 


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


NORM 


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


INTEGER, INTENT(OUT) :: INFO 


REAL(wp), INTENT(IN) :: ANORM 


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


INTEGER, INTENT(IN) :: IPIV( * ) 


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


REAL(wp), INTENT(IN) :: AB( LDAB, * ) 


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


where 

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

Complex version.

SUBROUTINE LA_GBCON( NORM, N, KL, & 


KU, AB, LDAB, IPIV, ANORM, & 


RCOND, WORK, RWORK, INFO ) 


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


NORM 


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


INTEGER, INTENT(OUT) :: INFO 


REAL(wp), INTENT(IN) :: ANORM 


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


INTEGER, INTENT(IN) :: IPIV( * ) 


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


COMPLEX(wp), INTENT(IN) :: AB( LDAB, * ) 


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


where 

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

LA_GBCON estimates the reciprocal of the condition number of a real / complex band matrix $A$, using the $LU$ factorization computed by LA_GBTRF.
References: See  [1] and [9,21,20].
-----------------------------------

LA_GBRFS
Real version.

SUBROUTINE LA_GBCON( TRANS, N, KL, KU, & 


NRHS, AB, LDAB, AFB, LDAFB, IPIV, B, & 


LDB, X, LDX, FERR, BERR, WORK, & 


IWORK, INFO ) 


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


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


LDAFB, LDB, LDX, N, NRHS 


INTEGER, INTENT(OUT) :: INFO 


INTEGER, INTENT(IN) :: IPIV(*) 


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


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


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


AFB( LDAFB,*), rhs 


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


REAL(wp), INTENT(INOUT) :: sol 


where 

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

rhs  ::= B(LDB,*) $\mid$ B(*) 

sol  ::= X(LDX,*) $\mid$ X(*) 

err  ::= FERR(*), BERR(*) $\mid$ FERR, BERR

Complex version.

SUBROUTINE LA_GBRFS( TRANS, N, KL, KU, & 


NRHS, AB, LDAB, AFB, LDAFB, IPIV, B, & 


LDB, X, LDX, FERR, BERR, WORK, & 


RWORK, INFO ) 


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


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


LDAFB, LDB, LDX, N, NRHS 


INTEGER, INTENT(OUT) :: INFO 


INTEGER, INTENT(IN) :: IPIV(*) 


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


COMPLEX(wp), INTENT(IN) :: AB( LDAB,*), & 


AFB( LDAFB,*), rhs 


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


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


where 

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

rhs  ::= B(LDB,*) $\mid$ B(*) 

sol  ::= X(LDX,*) $\mid$ X(*) 

err  ::= FERR(*), BERR(*) $\mid$ FERR, BERR

LA_GBRFS improves the computed solution to a system of linear equations when the coefficient matrix is banded, and provides error bounds and backward error estimates for the solution.
References: See  [1] and [9,21,20].
-----------------------------------

LA_GBEQU
Real and complex versions.

SUBROUTINE LA_GBEQU( M, N, KL, KU, AB, & 


LDAB, R, C, ROWCND, COLCND, AMAX, & 


INFO ) 


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


M, N 


INTEGER, INTENT(OUT) :: INFO 


REAL(wp), INTENT(OUT) :: AMAX, & 


COLCND, ROWCND 


REAL(wp), INTENT(OUT) :: C(*), R(*) 

type(wp), INTENT(IN) :: AB( LDAB,*) 


where 

type ::= REAL $\mid$ COMPLEX 

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

LA_GBEQU computes row and column scalings intended to equilibrate an $m \times n$ real / complex band matrix $A$ and reduce its condition number.
References: See  [1] and [9,21,20].
-----------------------------------

LA_GTTRF
Real and complex versions.

SUBROUTINE LA_GTTRF( N, DL, D, DU,  


DU2, IPIV, INFO ) 


INTEGER, INTENT(IN) :: N 


INTEGER, INTENT(OUT) :: INFO 


INTEGER, INTENT(OUT) :: IPIV(*) 

type(wp), INTENT(INOUT) :: D(*), DL(*), DU(*) 

type(wp), INTENT(OUT) :: DU2(*) 


where 

type ::= REAL $\mid$ COMPLEX 

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

LA_SGTTRF computes an $LU$ factorization of a real / complex tridiagonal matrix $A$ using elimination with partial pivoting and row interchanges.
References: See  [1] and [9,20].
-----------------------------------

LA_GTTRS
Real and complex versions.

SUBROUTINE LA_GTTRS( TRANS, N, NRHS, & 


DL, D, DU, DU2, IPIV, B, LDB, INFO ) 


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


TRANS 


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


INTEGER, INTENT(OUT) :: INFO 


INTEGER, INTENT(IN) :: IPIV(*) 

type(wp), INTENT(IN) :: D(*), DL(*), DU(*), & 


DU2(*) 

type(wp), INTENT(INOUT) :: rhs 


where 

type ::= REAL $\mid$ COMPLEX 

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

rhs  ::= B(LDB,*) $\mid$ B(*)

LA_GTTRS solves one of the systems of equations $AX = B$ or $A^T X = B$, with a tridiagonal real / complex matrix $A$ using the $LU$ factorization computed by LA_GTTRF.
References: See  [1] and [9,20].
-----------------------------------

LA_GTCON
Real version.

SUBROUTINE LA_GTCON( NORM, N, DL, & 


D, DU, DU2, IPIV, ANORM, RCOND, & 


WORK, IWORK, INFO ) 


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


NORM 


INTEGER, INTENT(IN) :: N 


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


REAL(wp), INTENT(IN) :: ANORM 


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


INTEGER, INTENT(IN) :: IPIV(*) 


REAL(wp), INTENT(IN) :: D(*), DL(*), & 


DU(*), DU2(*) 


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


where 

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

Complex version.

SUBROUTINE LA_GTCON( NORM, N, DL, & 


D, DU, DU2, IPIV, ANORM, RCOND, & 


WORK, INFO ) 


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


NORM 


INTEGER, INTENT(IN) :: N 


INTEGER, INTENT(OUT) :: INFO 


REAL(wp), INTENT(IN) :: ANORM 


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


INTEGER, INTENT(IN) :: IPIV(*) 


COMPLEX(wp), INTENT(IN) :: D(*), DL(*), & 


DU(*), DU2(*) 


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


where 

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

LA_GTCON estimates the reciprocal of the condition number of a real / complex tridiagonal matrix $A$ using the $LU$ factorization as computed by LA_GTTRF.
References: See  [1] and [9,21,20].
-----------------------------------

LA_GTRFS
Real version.

SUBROUTINE LA_GTRFS( TRANS, N, NRHS, & 


DL, D, DU, DLF, DF, DUF, DU2, IPIV, B, & 


LDB, X, LDX, FERR, BERR, WORK, & 


IWORK, INFO ) 


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


INTEGER, INTENT(IN) :: LDB, LDX, N, & 


NRHS, IPIV(*) 


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


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


REAL(wp), INTENT(IN) :: rhs, D(*), DF(*), DL(*), & 


DLF(*), DU(*), DU2(*), DUF(*) 


REAL(wp), INTENT(INOUT) :: sol 


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


where 

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

rhs  ::= B(LDB,*) $\mid$ B(*) 

sol  ::= X(LDX,*) $\mid$ X(*) 

err  ::= FERR(*), BERR(*) $\mid$ FERR, BERR

Complex version.

SUBROUTINE LA_GTRFS( TRANS, N, NRHS, & 


DL, D, DU, DLF, DF, DUF, DU2, IPIV, B, & 


LDB, X, LDX, FERR, BERR, WORK, & 


RWORK, INFO ) 


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


TRANS 


INTEGER, INTENT(IN) :: LDB, LDX, N, & 


NRHS, IPIV(*) 


INTEGER, INTENT(OUT) :: INFO 


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


COMPLEX(wp), INTENT(IN) :: rhs, D(*), DF(*), DL(*), & 


DLF(*), DU(*), DU2(*), DUF(*) 


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


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


where 

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

rhs  ::= B(LDB,*) $\mid$ B(*) 

sol  ::= X(LDX,*) $\mid$ X(*) 

err  ::= FERR(*), BERR(*) $\mid$ FERR, BERR

LA_GTRFS improves the computed solution to a system of linear equations when the coefficient matrix is tridiagonal, and provides error bounds and backward error estimates for the solution.
References: See  [1] and [9,21,20].
-----------------------------------