Solution to a Triangular System in Banded Storage of Linear Equations

Calling Sequence

CALL DTBSV	(UPLO, TRANSA, DIAG, N, NDIAG, DA, LDA, DY, INCY)
CALL STBSV	(UPLO, TRANSA, DIAG, N, NDIAG, SA, LDA, SY, INCY)
CALL ZTBSV	(UPLO, TRANSA, DIAG, N, NDIAG, ZA, LDA, ZY, INCY)
CALL CTBSV	(UPLO, TRANSA, DIAG, N, NDIAG, CA, LDA, CY, INCY)
void dtbsv	(char uplo, char trans, char diag, int n, int ndiag, double *da, int lda, double *dx, int incx)
void stbsv	(char uplo, char trans, char diag, int n, int ndiag, float *sa, int lda, float *sx, int incx)
void ztbsv	(char uplo, char trans, char diag, int n, int ndiag, doublecomplex *za, int lda, doublecomplex *zx, int incx)
void ctbsv	(char uplo, char trans, char diag, int n, int ndiag, complex *ca, int lda, complex *cx, int incx)

Arguments

UPLO	Indicates whether the values in a matrix reside in the upper or lower triangle of the array in which the matrix is stored. The legal values for UPLO are listed below. Any value not listed below is illegal.
	'L' or 'l'	Only the lower triangle of the array will be referenced.
	'U' or 'u'	Only the upper triangle of the array will be referenced.
TRANSA	Indicates how to use the matrix stored in the array argument. The legal values for TRANSA are listed below. Any value not listed below is illegal.
	'N' or 'n'	Use the matrix as it is stored in the array.
	'T' or 't'	Use the transpose of the matrix.
	'C' or 'c'	Use the conjugate transpose of the matrix.
DIAG	Indicates whether or not a triangular matrix is unit triangular. If the matrix is unit triangular then none of the diagonal elements of the array will be accessed. The legal values for DIAG are listed below. Any value not listed below is illegal.
	'N' or 'n'	The matrix is not unit triangular.
	'U' or 'u'	The matrix is unit triangular.
N	Size of a matrix with N rows and N columns. N 0.
NDIAG	Bandwidth of a matrix with NDIAG superdiagonals or subdiagonals. N-1 NDIAG 0 but if N = 0 then NDIAG = 0.
xA	Two-dimensional array that contains the input matrix.
LDA	Leading dimension of the array A as specified in a dimension or type statement. LDA NDIAG + 1.
xY	Y and INCY describe a vector of length N.
	On entry, an input vector.
	On exit, a result vector.
INCY	Scalar that contains the storage spacing between successive elements of the vector Y. INCY 0.
	If INCY is one then elements of the vector are contiguous in memory. INCY may take on values besides 1 to allow the programmer to extract from a matrix a vector that is not stored in contiguous memory locations.
	If Y is a one-dimensional array and INCY = -1 then the array will be accessed in reverse order.
	If Y is a two-dimensional array and INCY = LDA then the vector will be a row of the array.
	If Y is a two-dimensional array and INCY = LDA+1 then the vector will be a diagonal of the array.

Sample Program

PROGRAM TEST

IMPLICIT NONE

C

DOUBLE PRECISION ZERO

INTEGER LDA, N, NDIAG

PARAMETER (N = 4)

PARAMETER (NDIAG = 1)

PARAMETER (LDA = NDIAG + 1)

PARAMETER (ZERO = 0.0D0)

C

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

INTEGER I, J

C

EXTERNAL DTBSV

C

C Initialize the array A to store in triangular banded form

C the triangular banded matrix A shown below. Initialize

C the array B to store the vector b shown below.

C

C 1 8

C A = 2 3 b = 43

C 4 5 86

C 6 7 137

C

DATA A / 1.0D0, 2.0D0, 3.0D0, 4.0D0, 5.0D0, 6.0D0, 7.0D0,

$ 8D8 /

DATA B / 8.0D0, 4.3D1, 8.6D1, 1.37D2 /

C

PRINT 1000

PRINT 1010, A(1,1)

PRINT 1010, A(2,1), A(1,2)

PRINT 1020, A(2,2), A(1,3)

PRINT 1030, A(2,3), A(1,4)

PRINT 1040

DO 10, I = 1, LDA

PRINT 1010, (A(I,J), J = 1, N)

10 CONTINUE

PRINT 1050

PRINT 1060, B

CALL DTBSV ('LOWER TRIANGULAR A', 'NOT TRANSPOSED A',

$ 'NOT UNIT DIAGONAL A', N, 1, A, LDA, B, 1)

PRINT 1070

PRINT 1060, B

C

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

1010 FORMAT (1X, 4(2X, F5.1))

1020 FORMAT (8X, 2(2X, F5.1))

1030 FORMAT (15X, 2(2X, F5.1))

1040 FORMAT (/1X, 'A in triangular banded form: ',

$ '(* in unused elements)')

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

1060 FORMAT (3X, F5.1)

1070 FORMAT (/1X, 'A**(-1) * b:')

C

END

Sample Output

A in full form:
1.0
2.0 3.0
4.0 5.0
6.0 7.0
A in triangular banded form: (* in unused elements)
1.0 3.0 5.0 7.0
2.0 4.0 6.0 *****
b:
8.0
43.0
86.0
137.0
A**(-1) * b:
8.0
9.0
10.0
11.0