LU Factorization of a Tridiagonal Matrix

Calling Sequence

CALL DGTTRF	(N, DLOW, DDIAG, DUP1, DUP2, IPIVOT, INFO)
CALL SGTTRF	(N, SLOW, SDIAG, SUP1, SUP2, IPIVOT, INFO)
CALL ZGTTRF	(N, ZLOW, ZDIAG, ZUP1, ZUP2, IPIVOT, INFO)
CALL CGTTRF	(N, CLOW, CDIAG, CUP1, CUP2, IPIVOT, INFO)
void dgttrf	(int n, double *dlow, double *ddiag, double *dup1, double *dup2, int *ipivot, int *info)
void sgttrf	(int n, float *slow, float *sdiag, float *sup1, float *sup2, int *ipivot, int *info)
void zgttrf	(int n, doublecomplex *zlow, doublecomplex *zdiag, doublecomplex *zup1, doublecomplex *zup2, int *ipivot, int *info)
void cgttrf	(int n, complex *clow, complex *cdiag, complex *cup1, complex *cup2, int *ipivot, int *info)

Arguments

N	Order of the matrix A. N 0.
xLOW	On entry, the N-1 subdiagonal elements of A. On exit, the N-1 multipliers that define the matrix L from the LU factorization of A.
xDIAG	On entry, the N diagonal elements of A. On exit, the N diagonal elements of the upper triangular matrix U from the LU factorization of A.
xUP1	On entry, the N-1 superdiagonal elements of A. On exit, the N-1 elements of the first superdiagonal of U.
xUP2	On exit, the N-2 elements of the second superdiagonal of U.
IPIVOT	On exit, the pivot indices. During factorization, row i was exchanged with row IPIVOT(i) for 1 i N.
INFO	On exit:
	INFO = 0	Subroutine completed normally.
	INFO < 0	The ith argument, where i = |INFO|, had an illegal value.
	INFO > 0	U(i,i), where i = INFO, is exactly zero and U is therefore singular. The factorization has been completed, but division by zero will occur if this factorization is used to solve a linear system.

Sample Program

PROGRAM TEST

IMPLICIT NONE

C

INTEGER LDB, LDIWRK, LDWORK, N, NRHS

PARAMETER (N = 5)

PARAMETER (LDB = N)

PARAMETER (LDIWRK = N)

PARAMETER (LDWORK = 2 * N)

PARAMETER (NRHS = 1)

C

DOUBLE PRECISION ANORM, B(LDB,NRHS), DIAG(N), DLOWER(N-1)

DOUBLE PRECISION DUP1(N-1), DUP2(N-2), RCOND, WORK(LDWORK)

INTEGER ICOL, INFO, IPIVOT(N), IROW, IWORK(LDIWRK)

C

EXTERNAL DGTCON, DGTTRF, DGTTRS

INTRINSIC ABS, MAX

C

C Initialize the arrays DLOWER, DIAG, and DUP1 to store

C the first subdiagonal, the diagonal, and the first

C superdiagonal of the coefficient matrix A shown below.

C Initialize the array B to store the right hand side

C matrix b shown below.

C

C -3 1 6

C 1 -2 1 12

C A = 1 -2 1 b = 18

C 1 -2 1 12

C 1 -1 6

C

DATA DLOWER / 1.0D0, 1.0D0, 1.0D0, 1.0D0 /

DATA DIAG / -3.0D0, -2.0D0, -2.0D0, -2.0D0, -1.0D0 /

DATA DUP1 / 1.0D0, 1.0D0, 1.0D0, 1.0D0 /

DATA B / 6.0D0, 1.2D1, 1.8D1, 1.2D1, 6.0D0 /

C

C Print the initial value of A.

C

PRINT 1000

DO 100, IROW = 1, N

PRINT 1010, (0.0D0, ICOL = 1, IROW - 2),

$ (DLOWER(ICOL + 1), ICOL = ABS(IROW - 2), IROW - 2),

$ DIAG(IROW),

$ (DUP1(IROW), ICOL = 1, MIN(1, N - IROW)),

$ (0.0D0, ICOL = IROW + 2, N)

100 CONTINUE

PRINT 1020

PRINT 1030, B

C

C LU factor A.

C

CALL DGTTRF (N, DLOWER, DIAG, DUP1, DUP2, IPIVOT, INFO)

IF (INFO .NE. 0) THEN

PRINT 1040, INFO

STOP 1

END IF

ANORM = 4.0D0

C

C Estimate and print the condition number of A.

C

CALL DGTCON ('ONE-NORM', N, DLOWER, DIAG, DUP1, DUP2,

$ IPIVOT, ANORM, RCOND, WORK, IWORK, INFO)

IF (INFO .NE. 0) THEN

PRINT 1050, ABS (INFO)

STOP 2

END IF

PRINT 1060, 1.0D0 / RCOND

C

C Solve the system and print the solution.

C

CALL DGTTRS ('NO TRANSPOSE A', N, NRHS, DLOWER, DIAG, DUP1,

$ DUP2, IPIVOT, B, LDB, INFO)

IF (INFO .NE. 0) THEN

PRINT 1070, INFO

STOP 3

END IF

PRINT 1080

PRINT 1030, B

C

1000 FORMAT (1X, 'A:')

1010 FORMAT (5(3X, F5.2))

1020 FORMAT (/1X, 'b:')

1030 FORMAT (1X, F8.2)

1040 FORMAT (1X, 'Error factoring A, INFO = ', I5)

1050 FORMAT (1X, 'Illegal argument to DGTCON, argument #', I2)

1060 FORMAT (/1X, 'Estimated condition number of A: ', F6.2)

1070 FORMAT (1X, 'Error solving Ax=b, INFO = ', I5)

1080 FORMAT (/1X, 'x:')

C

END

Sample Output

A:
-3.00 1.00 0.00 0.00 0.00
1.00 -2.00 1.00 0.00 0.00
0.00 1.00 -2.00 1.00 0.00
0.00 0.00 1.00 -2.00 1.00
0.00 0.00 0.00 1.00 -1.00
b:
6.00
12.00
18.00
12.00
6.00
Estimated condition number of A: 50.00
x:
-27.00
-75.00
-111.00
-129.00
-135.00