Matrix-vector multiply

SGBMV/DGBMV/CGBMV/ZGBMV

The following example illustrates the storage of general band matrices.

Consider the following matrix A of size m = 9 by n = 8, with lower and upper bandwidths kl = 2 and ku = 3, respectively:

11

12

13

14

0

0

0

0

21

22

23

24

25

0

0

0

31

32

33

34

35

36

0

0

0

42

43

44

45

46

47

0

0

0

53

54

55

56

57

58

0

0

0

64

65

66

67

68

0

0

0

0

75

76

77

78

0

0

0

0

0

86

87

88

0

0

0

0

0

0

97

98

A is given in an array ab with at least kl+ku+1 = 6rows and n = 8 columns as follows:

*

*

*

14

25

36

47

58

*

*

13

24

35

46

57

68

*

12

23

34

45

56

67

78

11

22

33

44

55

66

77

88

21

32

43

54

65

76

87

98

31

42

53

64

75

86

97

*

The asterisks in the ku-by-kutriangle at the upper left corner and in the

(kl+n−m)-by-(kl+n−m) triangle at the lower right corner represent elements of ab that are not referenced. Thus, if aij is an element within the band of A, then

it is stored in ab(ku+1+i−j,j). Therefore, the columns of A are stored in the columns of ab, and the diagonals of A are stored in the rows of ab, such that the principal diagonal is stored in row ku+1 of ab.

Usage

VECLIB:

 

 

CHARACTER*1

trans

 

INTEGER*4

m, n, kl, ku, ldab, incx, incy

 

REAL*4

alpha, beta, ab(ldab, n), x(lenx), y(leny)

CALL SGBMV(trans, m, n, kl, ku, alpha, ab, ldab, x, incx, beta, y, incy)

CHARACTER*1

trans

INTEGER*4

m, n, kl, ku, ldab, incx, incy

REAL*8

alpha, beta, ab(ldab, n), x(lenx), y(leny)

CALL DGBMV(trans, m, n, kl, ku, alpha, ab, ldab, x, incx, beta, y, incy)

Chapter 3 Basic Matrix Operations 213