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

652 HP MLIB LAPACK User’s Guide
Associated documentation
Associated documentation
The following documents provide supplemental material for this chapter:
• Anderson et al., LAPACK Users’ Guide. For the latest edition of the LAPACK
Users’ Guide, refer to the Netlib repository at the following URL:
http://www.netlib.org/lapack/lug/index.html
• Golub, G.H. and C.F. Van Loan. Matrix Computations, 2nd Edition.
Baltimore, MD: The Johns Hopkins University Press. 1989.
What you need to know to use these subprograms
Norms of vectors and matrices
To use the norm-computing subprograms, you need to understand the basics of
vector and matrix norms. For completeness, the following is a brief discussion of
vector and matrix norms. Most standard texts on linear algebra cover the
prerequisite material in greater detail.
Definition: A vector norm on R
n
, the vector space of n-dimensional real vectors,
is a function f:R
n
→ R that has the following properties:
Such a function is denoted with a double-bar notation: f(x)=||x||. Subscripts on
the double bar are used to distinguish between various norms. The most
important vector norms are the 1-, 2-, and ∞-norms, defined in Table 11-1.
The vector space of m-by-n matrices is isomorphic to the vector space of
mn-dimensional vectors. Therefore, the definition of a matrix norm follows from
the definition of a vector norm.
f(x) ≥ 0
x ε R
n
f(x) = 0 if and only if x =0
f(x+y) ≤ f(x)+f(y)
x, y ε R
n
f(αx) = |α|f(x)
αεR, x ε R
n