Product of a Symmetric Matrix in Banded Storage and a Vector

Calling Sequence

CALL DSBMV	(UPLO, N, NDIAG, DALPHA, ZA, LDA, ZX, INCX, DBETA, ZY, INCY)
CALL SSBMV	(UPLO, N, NDIAG, SALPHA, CA, LDA, CX, INCX, SBETA, CY, INCY)
void dsbmv	(char uplo, int n, int ndiag, double dalpha, double *da, int lda, double *dx, int incx, double dbeta, double *dy, int incy)
void ssbmv	(char uplo, int n, int ndiag, float salpha, float *sa, int lda, float *sx, int incx, float sbeta, float *sy, int incy)

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.
N	Size of a matrix with N rows and N columns. N 0.
NDIAG	Bandwidth of a symmetric matrix in banded storage with NDIAG superdiagonals and NDIAG subdiagonals. N-1 NDIAG 0 but if N = 0 then NDIAG = 0.
xALPHA	Scalar that scales the input value of the matrix A.
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.
xX	X and INCX describe a vector of length N. X contains an input vector.
INCX	Scalar that contains the storage spacing between successive elements of the vector. INCX 0. If INCX = 1, then elements of the vector are contiguous in memory. INCX 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 X is a one-dimensional array and INCX = -1 then the array will be accessed in reverse order.
	If X is a two-dimensional array and INCX = LDA then the vector will be a row of the array.
	If X is a two-dimensional array and INCX = LDA+1 then the vector will be a diagonal of the array.
xBETA	Scalar that scales the input value of the vector Y.
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 = 1, 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

INTEGER LDA, N, NDIAG

PARAMETER (N = 5)

PARAMETER (NDIAG = 1)

PARAMETER (LDA = NDIAG + 1)

C

INTEGER I, J

DOUBLE PRECISION A(LDA,N), ALPHA, BETA, X(N), Y(N)

C

EXTERNAL DSBMV

C

C Initialize the array A to store in symmetric banded form

C the matrix A shown below. Initialize the arrays X and Y

C to store the vectors X and Y shown below.

C

C 1 2 1 1

C 2 3 4 2 1

C A = 4 5 6 x = 3 y = 1

C 6 7 8 4 1

C 8 9 5 1

C

DATA A / 8D8, 1.0D0, 2.0D0, 3.0D0, 4.0D0, 5.0D0, 6.0D0,

$ 7.0D0, 8.0D0, 9.0D0 /

DATA X / 1.0D0, 2.0D0, 3.0D0, 4.0D0, 5.0D0 /

DATA Y / 1.0D0, 1.0D0, 1.0D0, 1.0D0, 1.0D0 /

C

PRINT 1000

PRINT 1010, A(2,1), A(1,2), 0.0D0, 0.0D0, 0.0D0

PRINT 1010, A(1,2), A(2,2), A(1,3), 0.0D0, 0.0D0

PRINT 1010, 0.0D0, A(1,3), A(2,3), A(1,4), 0.0D0

PRINT 1010, 0.0D0, 0.0D0, A(1,4), A(2,4), A(1,5)

PRINT 1010, 0.0D0, 0.0D0, 0.0D0, A(1,5), A(2,5)

PRINT 1020

DO 20, I = 1, LDA

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

20 CONTINUE

PRINT 1030

PRINT 1040, (X(I), Y(I), I = 1, N)

ALPHA = 2.0D0

BETA = 3.0D0

CALL DSBMV ('UPPER TRIANGULAR A', N, NDIAG, ALPHA, A, LDA,

$ X, 1, BETA, Y, 1)

PRINT 1050

PRINT 1060, (Y(I), I = 1, N)

C

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

1010 FORMAT (1X, 5(2X, F6.1))

1020 FORMAT (/1X, 'A in banded form: (* in unused fields)')

1030 FORMAT (/7X, 'x', 8X, 'y')

1040 FORMAT (3X, F6.1, 3X, F6.1)

1050 FORMAT (/1X, 'Ax + y:')

1060 FORMAT (1X, 3X, F6.1)

C

END

Sample Output

A in full form:
1.0 2.0 0.0 0.0 0.0
2.0 3.0 4.0 0.0 0.0
0.0 4.0 5.0 6.0 0.0
0.0 0.0 6.0 7.0 8.0
0.0 0.0 0.0 8.0 9.0
A in banded form: (* in unused fields)
****** 2.0 4.0 6.0 8.0
1.0 3.0 5.0 7.0 9.0
x y
1.0 1.0
2.0 1.0
3.0 1.0
4.0 1.0
5.0 1.0
Ax + y:
13.0
43.0
97.0
175.0
157.0