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Scientific Computing Linear Systems – Gaussian Elimination

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Linear Systems

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Solve Ax=b, where A is an n n matrix and b is an n 1 column vector We can also talk about non-square systems where A is m n, b is m 1, and x is n 1 – Overdetermined if m>n: “ more equations than unknowns ” – Underdetermined if n>m: “ more unknowns than equations ”

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Singular Systems A is singular if some row is linear combination of other rows Singular systems can be underdetermined: or inconsistent:

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Using Matrix Inverses To solve Ax = b it would be nice to use the inverse to A, that is, A -1 However, it is usually not a good idea to compute x=A -1 b – Inefficient – Prone to roundoff error

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Gaussian Elimination Fundamental operations: 1.Replace one equation with linear combination of other equations 2.Interchange two equations 3.Multiply one equation by a scalar These are called elementary row operations. Do these operations again and again to reduce the system to a “ trivial ” system

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Triangular Form Two special forms of matrices are especially nice for solving Ax=b: In both cases, successive substitution leads to a solution

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Triangular Form A is lower triangular

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Triangular Form Solve by forward substitution:

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Triangular Form Solve by forward substitution:

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Triangular Form Solve by forward substitution: Etc

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Triangular Form If A is upper triangular, solve by back- substitution:

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Triangular Form Solve by back-substitution: Etc

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Gaussian Elimination Algorithm Do elementary row operations on the augmented system [A|b] to reduce the system to upper triangular form. Then, use back-substitution to find the answer.

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Gaussian Elimination Example: Augmented Matrix form:

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Gaussian Elimination Row Ops: What do we do to zero out first column under first pivot? Zero out below second pivot:

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Gaussian Elimination Back-substitute

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Gaussian Elimination

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Matlab Implementation Task: Implement Gaussian Elimination (without pivoting) in a Matlab M-file. Notes Input = Coefficient matrix A, rhs b Output = solution vector x

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Matlab Implementation Class Exercise: We will go through this code line by line, using the example in Pav section 3.3.2 to see how the code works.

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Matlab Implementation Matlab: >> A=[2 1 1 3; 4 4 0 7; 6 5 4 17; 2 -1 0 7] A = 2 1 1 3 4 4 0 7 6 5 4 17 2 -1 0 7 >> b = [7 11 31 15]' b = 7 11 31 15 >> gauss_no_pivot(A,b) ans = 1.5417 -1.4167 0.8333 1.5000

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Matlab Implementation Class Exercise: How would we change the Matlab function gauss_no_pivot so we could see the result of each step of the row reduction?

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Gaussian Elimination - Pivoting Consider this system: We immediately run into a problem: we cannot zero out below pivot, or back- substitute! More subtle problem:

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Gaussian Elimination - Pivoting Conclusion: small diagonal elements are bad! Remedy: swap the row with the small diagonal element with a row below, this is called pivoting

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Gaussian Elimination - Pivoting Our Example: Swap rows 1 and 2: Now continue:

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Gaussian Elimination - Pivoting Two strategies for pivoting: – Partial Pivoting – Scaled Partial Pivoting

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Partial Pivoting

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Matlab – Partial Pivoting Partial Pivoting: At step k, we are working on kth row, pivot = A kk. Search for largest A ik in kth column below (and including) A kk. Let p = index of row containing largest entry. If p ≠ k, swap rows p and k. Continue with Gaussian Elimination.

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Matlab – Partial Pivoting Finding largest entry in a column: >> A A = 2 1 1 3 4 4 0 7 6 5 4 17 2 -1 0 7 >> [r,m] = max(A(2:4,2)) r = 5 m = 2 Why isn’t m = 3?

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Matlab – Partial Pivoting Swapping rows m and k: BB = 2 1 1 3 4 4 0 7 6 5 4 17 2 -1 0 7 >> BB([1 3],:) = BB([3 1], :) BB = 6 5 4 17 4 4 0 7 2 1 1 3 2 -1 0 7

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Matlab – Partial Pivoting Code change? All that is needed is in main loop (going from rows k ->n-1) we add % Find max value (M) and index (m) of max entry below AAkk [M,m] = max(abs(AA(k:n,k))); m = m + (k-1); % row offset % swap rows, if needed if m ~= k, AA([k m],:) = AA([m k],:); end

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Matlab – Partial Pivoting Class Exercise Review example in Moler Section 2.6

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Scaled Partial Pivoting Pav, Section 3.2

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Practice Class Exercise: We will work through the example in Pav, Section 3.2.2

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Practice Class Exercise: Do Pav Ex 3.7 for partial pivoting (“naïve Gaussian Elimination”). Do Pav Ex 3.7 for scaled partial pivoting.

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