HP MLIB User's Guide Vol. 2 7th Ed.

Chapter 9 LAPACK Auxiliary Subprograms 657
Compute norm of general band matrix SLANGB/DLANGB/CLANGB/ZLANGB
Name SLANGB/DLANGB/CLANGB/ZLANGB
Compute norm of general band matrix
Purpose These subprograms compute the norm of an m-by-n general band matrix A. A
band matrix is a matrix whose nonzero elements all lie near the principal
diagonal. Specifically, a
ij
=0 ifi−j > kl or j−i > ku for some integers kl and ku.
The smallest such kl and ku for a given matrix are called the lower and upper
bandwidths, respectively, and k = kl+ku+1 is the total bandwidth.
Matrix
Storage
Because it is not necessary to store or operate on the zeros outside the band of
A, you need only provide the elements within the band of A. Compared to
storing the entire matrix, this can save memory if kl+ku+1 < n.
The following example illustrates the storage of general band matrices.
Consider the following matrix A of order n = 9 and lower and upper
bandwidths kl = 2 and ku = 3, respectively:
A is given in an array
ab with at least kl+ku+1 = 6 rows and n = 9 columns as
follows:
The asterisks in the ku-by-ku triangle at the upper left corner and in the
kl-by-kl triangle at the lower right corner represent elements of ab that are not
referenced. Thus, if a
ij
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, the
diagonals of A are stored in the rows of ab, and the principal diagonal is stored
in row ku+1 of ab.
1112131400000
21222324250000
31 32 33 34 35 36 0 0 0
0424344454647 0 0
0 0 53 54 55 56 57 58 0
0 0 0 64 65 66 67 68 69
00007576777879
0000086878889
000000979899
* * * 14 25 36 47 58 69
* * 13 24 35 46 57 68 79
*1223344556677889
11 22 33 44 55 66 77 88 99
21 32 43 54 65 76 87 98 *
31 42 53 64 75 86 97 * *