Solution to a Linear System in a Triangular Matrix

Calling Sequence

CALL DTRTRS	(UPLO, TRANSA, DIAG, N, NRHS, DA, LDA, DB, LDB, INFO)
CALL STRTRS	(UPLO, TRANSA, DIAG, N, NRHS, SA, LDA, SB, LDB, INFO)
CALL ZTRTRS	(UPLO, TRANSA, DIAG, N, NRHS, ZA, LDA, ZB, LDB, INFO)
CALL CTRTRS	(UPLO, TRANSA, DIAG, N, NRHS, CA, LDA, CB, LDB, INFO)
void dtrtrs	(char uplo, char trans, char diag, int n, int nrhs, double *da, int lda, double *db, int ldb, int *info)
void strtrs	(char uplo, char trans, char diag, int n, int nrhs, float *sa, int lda, float *sb, int ldb, int *info)
void ztrtrs	(char uplo, char trans, char diag, int n, int nrhs, doublecomplex *za, int lda, doublecomplex *zb, int ldb, int *info)
void ctrtrs	(char uplo, char trans, char diag, int n, int nrhs, complex *ca, int lda, complex *cb, int ldb, int *info)

Arguments

UPLO	Indicates whether xA contains the upper or lower triangle of the matrix. The legal values for UPLO are listed below. Any values not listed below are illegal.
	U' or 'u'	xA contains the upper triangle.
	'L' or 'l'	xA contains the lower triangle.
TRANSA	Indicates the form of the system of equations. The legal values for TRANSA are listed below. Any values not listed below are illegal.
	N' or 'n'	No transpose, solve AX = B.
	'T' or 't'	Transpose, solve ATX = B.
	'C' or 'c'	Conjugate transpose, solve AHX = B.
	Note that AT and AH are the same for real matrices.
DIAG	Indicates whether or not A is unit triangular. The legal values for DIAG are listed below. Any values not listed below are illegal.
	'N' or 'n'	A is not unit triangular.
	'U' or 'u'	A is unit triangular.
N	Order of the matrix A. N 0.
NRHS	Number of right-hand sides, equal to the number of columns of the matrix B. NRHS 0.
xA	Upper or lower triangular matrix A.
	If DIAG = 'U' or 'u', the diagonal elements of A are assumed to be 1 and are not used.
LDA	Leading dimension of the array A as specified in a dimension or type statement. LDA max(1, N).
xB	On entry, the N×NRHS right-hand side matrix B. On exit, the N×NRHS solution matrix X.
LDB	Leading dimension of the array B as specified in a dimension or type statement. LDB max(1, N).
INFO	On exit:
	INFO = 0	Subroutine completed normally.
	INFO < 0	The ith argument, where i = |INFO|, had an illegal value.
	INFO > 0	The ith diagonal element, where i = INFO, of A is zero. The matrix is therefore singular and the solution could not be computed.

Sample Program

PROGRAM TEST

IMPLICIT NONE

C

DOUBLE PRECISION ZERO

INTEGER LDA, LDB, N, NRHS

PARAMETER (N = 4)

PARAMETER (LDA = N)

PARAMETER (LDB = N)

PARAMETER (NRHS = 1)

PARAMETER (ZERO = 0.0D0)

C

DOUBLE PRECISION A(LDA,N), B(LDB,NRHS)

INTEGER ICOL, INFO, IROW

C

EXTERNAL DTRTRS

INTRINSIC ABS

C

C Initialize the array A to store the coefficient matrix A

C shown below. Initialize the array B to store the right

C hand side vector b shown below.

C

C 1 4

C A = 1 1 b = 3

C 1 1 1 2

C 1 1 1 1 1

C

DATA A / 1.0D0, 1.0D0, 1.0D0, 1.0D0,

$ 8.8D8, 1.0D0, 1.0D0, 1.0D0,

$ 8.8D8, 8.8D8, 1.0D0, 1.0D0,

$ 8.8D8, 8.8D8, 8.8D8, 1.0D0 /

DATA B / 4.0D0, 3.0D0, 2.0D0, 1.0D0 /

C

C Print the initial value of the arrays.

C

PRINT 1000

DO 100, IROW = 1, N

PRINT 1010, (A(IROW,ICOL), ICOL = 1, IROW),

$ (ZERO, ICOL = IROW + 1, N)

100 CONTINUE

PRINT 1020

PRINT 1010, ((A(IROW,ICOL), ICOL = 1, N), IROW = 1, LDA)

PRINT 1030

PRINT 1060, B

C

C Solve Ax=b and print the result.

C

CALL DTRTRS ('LOWER TRIANGULAR A', 'TRANSPOSE A',

$ 'NO UNIT DIAGONAL A', N, NRHS, A, LDA, B, LDB,

$ INFO)

IF (INFO .NE. 0) THEN

PRINT 1040, INFO

STOP 1

END IF

PRINT 1050

PRINT 1060, B

C

1000 FORMAT (1X, 'A in full form:')

1010 FORMAT (4(3X, F8.4))

1020 FORMAT (/1X, 'A in triangular form: ',

$ ' (* in unused elements)')

1030 FORMAT (/1X, 'b:')

1040 FORMAT (/1X, 'Error solving Ax=b, INFO = ', I5)

1050 FORMAT (/1X, 'x:')

1060 FORMAT (1X, F8.4)

C

END

Sample Output

A in full form:
1.0000 0.0000 0.0000 0.0000
1.0000 1.0000 0.0000 0.0000
1.0000 1.0000 1.0000 0.0000
1.0000 1.0000 1.0000 1.0000
A in triangular form: (* in unused elements)
1.0000 ******** ******** ********
1.0000 1.0000 ******** ********
1.0000 1.0000 1.0000 ********
1.0000 1.0000 1.0000 1.0000
b:
4.0000
3.0000
2.0000
1.0000
x:
1.0000
1.0000
1.0000
1.0000