Page 747 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

additions required is excessive. In short, there is currently no practical method for the direct solution of general linear systems that
significantly improves on the operation counts for Gaussian elimination and the text method of Gauss–Jordan elimination.

Operation counts are not the only criterion by which to judge a method for the computer solution of a linear system. As the speed
of computers has increased, the time it takes to move entries of the matrix from memory to the processing unit has become

increasingly important. For very large matrices, the time for memory accesses greatly exceeds the time required to do the actual

computations! Despite this, the conclusion above still stands: Except for extremely large matrices, Gaussian elimination or a

variant thereof is nearly always the method of choice for solving . It is almost never necessary to compute , and we

should avoid doing so whenever possible. Solving by Cramer's rule would be senseless for numerical purposes, despite its

theoretical value.

EXAMPLE 1 Avoiding the Inverse . The result is a vector y. Rather than computing as given, it

Suppose we needed to compute the product
would be more efficient to write this as

that is, as

and to compute the result as follows: First, compute the vector ; second, solve for z using Gaussian elimination;

third, compute the vector .

For extremely large matrices, such as the ones that occur in numerical weather prediction, approximate methods for solving
are often employed. In such cases, the matrix is typically sparse; that is, it has very few nonzero entries. These techniques are
beyond the scope of this text.

Exercise Set 9.8

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Find the number of additions and multiplications required to compute if A is an matrix and B is an matrix.
1.

Use the result in Exercise 1 to find the number of additions and multiplications required to compute by direct multiplication

2. if A is an matrix.

Assuming A to be an matrix, use the formulas in Table 1 to determine the number of operations required for the
3. procedures in Table 3.

Table 3