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.
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).
LDA Leading dimension of A exactly as specified
in the dimension statement of the calling program.
(Input)
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).
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
∞-norm.
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 L1 P1
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
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.
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.
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)
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