HP MLIB User's Guide Vol. 1 7th Ed.

Chapter 3 Basic Matrix Operations 301
Solve triangular band system STBSV/DTBSV/CTBSV/ZTBSV
Name STBSV/DTBSV/CTBSV/ZTBSV
Solve triangular band system
Purpose Given an n-by-n upper- or lower-triangular band matrix A and an n-vector x,
these subprograms overwrite x with the solution y to the system of linear
equations Ay = x. This operation is the forward elimination or back
substitution step of Gaussian elimination for band matrices. Optionally, A can
be replaced by A
T
, the transpose of A, or by A*, the conjugate transpose of A.
A lower-triangular band matrix is a matrix whose strict upper triangle is zero
and whose nonzero lower-triangular elements all are on, or fairly near, the
principal diagonal. Specifically, a
ij
≠ 0 only if 0 ≤ i−j ≤ kd for some integer kd.
In contrast, an upper-triangular band matrix is a matrix whose strict lower
triangle is zero and whose nonzero upper-triangular elements all are on, or
fairly near, the principal diagonal, but with a
ij
≠ 0 only if 0 ≤ j−i ≤ kd.
Specifically, these subprograms compute
where A
−T
is the inverse of the transpose of A, and is the inverse of the
conjugate transpose of A.
These subprograms are more primitive than the LAPACK band equation
solvers. As such, they are intended to supplement the equation solvers but not
replace them, serving instead as building blocks in constructing optimized
linear algebra software. In fact, many of the LAPACK subprograms have been
recoded to call these routines.
Refer to “F_STBSV/F_DTBSV/F_CTBSV/F_ZTBSV” on page 400 for details
about the equivalent BLAS Standard subprograms.
Matrix
Storage
Triangular band matrices are stored in a compressed form that takes
advantage of knowing the positions of the only elements that can be nonzero.
The following examples illustrate the storage of triangular band matrices.
xA
1–
x← xA
T–
x← and xA
*–
x←,,
A
*–