Using the and approach, computing requires (n-i) multiplies Ly=f Ux=y xi plus 1 divide. Hence, only multiplies are required to solve the triangular systems. operations are required to solve systems , by Gaussian Lemma: m j=1, m …, elimination. To compute requires operations. In general, operations are Note: A-1 n3 n2 required to compute Thus, to solve systems requires operations. A-1f(j). m mn2 Hence, operations are necessary to solve systems. n3+mn2 m Thus, it is more efficient to use Gaussian elimination instead of always computing the inverse! We can always compute by solving and then the are A-1 Axi=ei, xi’s i=1,2,…,n the columns of A-1.
Using the Ly=f and Ux=y approach, computing xi requires (n-i) multiplies plus 1 divide. Hence, only multiplies are required to solve the triangular systems. Lemma: operations are required to solve m systems , j=1, …, m by Gaussian elimination. Note: To compute A-1 requires n3 operations. In general, n2 operations are required to compute A-1f(j). Thus, to solve m systems requires mn2 operations. Hence, n3+mn2 operations are necessary to solve m systems. Thus, it is always more efficient to use Gaussian elimination instead of computing the inverse! We can always compute A-1 by solving Axi=ei, i=1,2,…,n and then the xi’s are the columns of A-1.
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