Introduction to Computational Linear Algebra

Basic Linear Algebra Subprograms - BLAS 11

Using (1.3) and (1.5), one obtains:

|z − zc| ≤ n(|A||x| + |y|)O(

M

) (1.7)

The proof of this inequality is left to Exercise 1.13.

• For vector external product, if A

c

is the computed result of A = xy

T

,

then using also (1.2), one easily obtains:

(A

c

)(i, j) = A(i, j)(1 + γ

ij

), |γ

ij

| = O(

M

).

1.6 Matrix-Matrix Operations: Level-3 BLAS

Note that matrix additions and multiplication by a scalar, typically of the

form: aA+B, a ∈ R , A, B ∈ R

n×n

are level-2 BLAS, since they require O(n

2

)

arithmetic operations.

On the other hand, matrix multiplications are level-3 BLAS. Let A ∈ R

m×n

,

B ∈ R

n×p

. The product of the two matrices ...