LFTCG

Computes the LU factorization of a complex general matrix.

Required Arguments

A — Complex N by N matrix to be factored. (Input)

FACT — Complex N × N matrix containing the LU factorization of the matrix A. (Output)

If A is not needed, A and FACT can share the same storage locations.

If A is not needed, A and FACT can share the same storage locations.

IPVT — Vector of length N containing the pivoting information for the LU factorization. (Output)

Optional Arguments

N — Order of the matrix. (Input)

Default: N = size (A,2).

Default: N = size (A,2).

LDA — Leading dimension of A exactly as specified in the dimension statement of the calling program. (Input)

Default: LDA = size (A,1).

Default: LDA = size (A,1).

LDFACT — Leading dimension of FACT exactly as specified in the dimension statement of the calling program. (Input)

Default: LDFACT = size (FACT,1).

Default: LDFACT = size (FACT,1).

FORTRAN 90 Interface

Generic: CALL LFTCG (A, FACT, IPVT [, …])

Specific: The specific interface names are S_LFTCG and D_LFTCG.

FORTRAN 77 Interface

Single: CALL LFTCG (N, A, LDA, FACT, LDFACT, IPVT)

Double: The double precision name is DLFTCG.

ScaLAPACK Interface

Generic: CALL LFTCG (A0, FACT0, IPVT0 [, …])

Specific: The specific interface names are S_LFTCG and D_LFTCG.

See the ScaLAPACK Usage Notes below for a description of the arguments for distributed computing.

Description

Routine LFTCG performs an LU factorization of a complex general coefficient matrix. The LU factorization is done using scaled partial pivoting. Scaled partial pivoting differs from partial pivoting in that the pivoting strategy is the same as if each row were scaled to have the same .

LFTCG fails if U, the upper triangular part of the factorization, has a zero diagonal element. This can occur only if A either is singular or is very close to a singular matrix.

The LU factors are returned in a form that is compatible with routines LFICG, LFSCG and LFDCG. To solve systems of equations with multiple right-hand-side vectors, use LFTCG followed by either LFICG or LFSCG called once for each right-hand side. The routine LFDCG can be called to compute the determinant of the coefficient matrix after LFCCG has performed the factorization.

Let F be the matrix FACT and let p be the vector IPVT. The triangular matrix U is stored in the upper triangle of F. The strict lower triangle of F contains the information needed to reconstruct L using

L = LN-1PN-1 … L1P1

where Pk is the identity matrix with rows k and Pk interchanged and Lk is the identity with Fik for i = k + 1, …, N inserted below the diagonal. The strict lower half of F can also be thought of as containing the negative of the multipliers.

The underlying code is based on either LINPACK , LAPACK, or ScaLAPACK code depending upon which supporting libraries are used during linking. For a detailed explanation see “Using ScaLAPACK, LAPACK, LINPACK, and EISPACK” in the Introduction section of this manual.

Comments

1. Workspace may be explicitly provided, if desired, by use of L2TCG/DL2TCG. The reference is:

CALL L2TCG (N, A, LDA, FACT, LDFACT, IPVT, WK)

The additional argument is:

WK — Complex work vector of length N.

2. Informational error

Type | Code | Description |
---|---|---|

4 | 2 | The input matrix is singular. |

ScaLAPACK Usage Notes

The arguments which differ from the standard version of this routine are:

A0 — MXLDA by MXCOL complex local matrix containing the local portions of the distributed matrix A. A contains the matrix to be factored. (Input)

FACT0 — MXLDA by MXCOL complex local matrix containing the local portions of the distributed matrix FACT. FACT contains the LU factorization of the matrix A. (Output)

If A is not needed, A and FACT can share the same storage locations.

If A is not needed, A and FACT can share the same storage locations.

IPVT0 — Local vector of length MXLDA containing the local portions of the distributed vector IPVT. IPVT contains the pivoting information for the LU factorization. (Output)

All other arguments are global and are the same as described for the standard version of the routine. In the argument descriptions above, MXLDA and MXCOL can be obtained through a call to SCALAPACK_GETDIM (see Utilities) after a call to SCALAPACK_SETUP (see Utilities) has been made. See the ScaLAPACK Example below.

Examples

Example

A linear system with multiple right-hand sides is solved. LFTCG is called to factor the coefficient matrix. LFSCG is called to compute the two solutions for the two right-hand sides. In this case the coefficient matrix is assumed to be well-conditioned and correctly scaled. Otherwise, it would be better to call LFCCG to perform the factorization, and LFICG to compute the solutions.

USE LFTCG_INT

USE LFSCG_INT

USE WRCRN_INT

! Declare variables

PARAMETER (LDA=3, LDFACT=3, N=3)

INTEGER IPVT(N)

COMPLEX A(LDA,LDA), B(N,2), X(N,2), FACT(LDFACT,LDFACT)

! Set values for A

! A = ( 1.0+1.0i 2.0+3.0i 3.0-3.0i)

! ( 2.0+1.0i 5.0+3.0i 7.0-5.0i)

! (-2.0+1.0i -4.0+4.0i 5.0+3.0i)

!

DATA A/(1.0,1.0), (2.0,1.0), (-2.0,1.0), (2.0,3.0), (5.0,3.0),&

(-4.0,4.0), (3.0,-3.0), (7.0,-5.0), (5.0,3.0)/

!

! Set the right-hand sides, B

! B = ( 3.0+ 5.0i 9.0+ 0.0i)

! ( 22.0+10.0i 13.0+ 9.0i)

! (-10.0+ 4.0i 6.0+10.0i)

!

DATA B/(3.0,5.0), (22.0,10.0), (-10.0,4.0), (9.0,0.0),&

(13.0,9.0), (6.0,10.0)/

!

! Factor A

CALL LFTCG (A, FACT, IPVT)

! Solve for the two right-hand sides

DO 10 J=1, 2

CALL LFSCG (FACT, IPVT, B(:,J), X(:,J))

10 CONTINUE

! Print results

CALL WRCRN (’X’, X)

END

Output

X

1 2

1 ( 1.000,-1.000) ( 0.000, 2.000)

2 ( 2.000, 4.000) (-2.000,-1.000)

3 ( 3.000, 0.000) ( 1.000, 3.000)

ScaLAPACK Example

The same linear system with multiple right-hand sides is solved as a distributed example. LFTCG is called to factor the matrix. LFSCG is called to compute the two solutions for the two right-hand sides. SCALAPACK_MAP and SCALAPACK_UNMAP are IMSL utility routines (see Chapter 11, “Utilities”) used to map and unmap arrays to and from the processor grid. They are used here for brevity. DESCINIT is a ScaLAPACK tools routine which initializes the descriptors for the local arrays.

USE MPI_SETUP_INT

USE LFTCG_INT

USE LFSCG_INT

USE WRCRN_INT

USE SCALAPACK_SUPPORT

IMPLICIT NONE

INCLUDE ‘mpif.h’

! Declare variables

INTEGER J, LDA, N, DESCA(9), DESCL(9)

INTEGER INFO, MXCOL, MXLDA

INTEGER, ALLOCATABLE :: IPVT0(:)

COMPLEX, ALLOCATABLE :: A(:,:), B(:,:), X(:,:), X0(:)

COMPLEX, ALLOCATABLE :: A0(:,:), FACT0(:,:), B0(:)

PARAMETER (LDA=3, N=3)

! Set up for MPI

MP_NPROCS = MP_SETUP()

IF(MP_RANK .EQ. 0) THEN

ALLOCATE (A(LDA,N), B(N,2), X(N,2))

! Set values for A and B

A(1,:) = (/ ( 1.0, 1.0), ( 2.0, 3.0), ( 3.0,-3.0)/)

A(2,:) = (/ ( 2.0, 1.0), ( 5.0, 3.0), ( 7.0,-5.0)/)

A(3,:) = (/ (-2.0, 1.0), (-4.0, 4.0), ( 5.0, 3.0)/)

!

B(1,:) = (/ ( 3.0, 5.0), ( 9.0, 0.0)/)

B(2,:) = (/ ( 22.0, 10.0), (13.0, 9.0)/)

B(3,:) = (/ (-10.0, 4.0), ( 6.0, 10.0)/)

ENDIF

! Set up a 1D processor grid and define

! its context ID, MP_ICTXT

CALL SCALAPACK_SETUP(N, N, .TRUE., .TRUE.)

! Get the array descriptor entities MXLDA,

! and MXCOL

CALL SCALAPACK_GETDIM(N, N, MP_MB, MP_NB, MXLDA, MXCOL)

! Set up the array descriptors

CALL DESCINIT(DESCA, N, N, MP_MB, MP_NB, 0, 0, MP_ICTXT, MXLDA, INFO)

CALL DESCINIT(DESCL, N, 1, MP_MB, 1, 0, 0, MP_ICTXT, MXLDA, INFO)

! Allocate space for the local arrays

ALLOCATE(A0(MXLDA,MXCOL), X0(MXLDA),FACT0(MXLDA,MXCOL), &

B0(MXLDA), IPVT0(MXLDA))

! Map input array to the processor grid

CALL SCALAPACK_MAP(A, DESCA, A0)

! Factor A

CALL LFTCG (A0, FACT0, IPVT0)

! Solve for the two right-hand sides

DO 10 J=1, 2

CALL SCALAPACK_MAP(B(:,J), DESCL, B0)

CALL LFSCG (FACT0, IPVT0, B0, X0)

CALL SCALAPACK_UNMAP(X0, DESCL, X(:,J))

10 CONTINUE

! Print results.

! Only Rank=0 has the solution, X.

IF(MP_RANK.EQ.0) CALL WRCRN (’X’, X)

IF (MP_RANK .EQ. 0) DEALLOCATE(A, B, X)

DEALLOCATE(A0, B0, FACT0, IPVT0, X0)

! Exit ScaLAPACK usage

CALL SCALAPACK_EXIT(MP_ICTXT)

! Shut down MPI

MP_NPROCS = MP_SETUP(‘FINAL’)

END

Output

X

1 2

1 ( 1.000,-1.000) ( 0.000, 2.000)

2 ( 2.000, 4.000) (-2.000,-1.000)

3 ( 3.000, 0.000) ( 1.000, 3.000)