F 5:40 exams-library utoronto-camyaccess library utoronto.ca Question 3 [15 mark

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F 5:40 exams-library utoronto-camyaccess library utoronto.ca Question 3 [15 marks] Let A e R"x" be a non-singular matrix a. Show how Gaussian elimination with partial pivoting can be used to solve the k linear systems Ari ti, Aa2 in m3/3 o(km2) flops. You must show precisely how the matrix components of the PA LU factorization are used in each stage of your algorithm, and justify the final operation count (Note: You do not need to give details of the Gaussian elimination algorithm-you may assume the factorization is available.) 89%

Explanation / Answer

a)While solving a set of simultaneous linear equations, using Gaussian elimination includes

1)Factorization of the matrix into lower and upper triangular matrx which involves (n^3)/3 operational steps.

2)Back substitution involving k(n^2)/2 operational steps.

3)Forward substitution involving k(n^2)/2 operational steps

so this operational steps sums up to ((n^3)/3)+k(n^2).

b)When k=n, and in finding the inverse of A, using Gaussian elimunation involves

1)Forward elimination with computational steps n(n^3)/3

2)Back substitution with computational steps n(n^2)/2.

so total number of computational steps are ((n^4)/3)+((n^3)/2)

c)Ax=b,this type of linear equations can be solved using gaussian elimination algorithm.

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