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Chapter 13 Gaussian Elimination (III) Bunch-Parlett diagonal pivoting Reference: James R. Bunch and Linda Kaufman, Some Stable Methods for Calculating Inertia and Solving Symmetric Linear Systems, Mathematics of Computation, volume 31, number 137, January 1977, page Speaker: Lung-Sheng Chien

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OutLine Preliminary - 1x1, 2x2 pivoting in Gaussian Elimination - real symmetric indefinite matrix Symmetric permutation LDL’ decomposition (diagonal pivoting) Example of complete diagonal pivoting Algorithm of complete diagonal pivoting

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It is easy to show from Gaussian Elimination Pivoting in Traditional Gaussian Elimination consider where and then one step LU-factorization leads to We use principal submatrix as pivoting, calledpivoting since Question: can we use or higherpivoting, i.e.

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then one-step GE leads to If principal sub-matrixis non-singular, Example 1: pivoting 2x2 pivoting in block Gaussian Elimination where and

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Example 2: pivoting where and 3x3 pivoting in block Gaussian Elimination Question: what is criterion to choose 1x1, 2x2 or 3x3 pivoting? stabilityperformance Question: what is the cost to estimate the criterion ?

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Real symmetry indefinite matrix A is symmetric impliesand From linear algebra, if A is real symmetric, then A has real spectrum decomposition ( 譜分解 ) and or write in matrix formand without loss of generality, we can rearrange eigen-values to be increasing Definition: suppose matrix A is real symmetric, then we say “A is indefinite” if there exists an eigen-value of A less than zero. If A is nonsingular, then Thereafter we focus on LU-decomposition of symmetry indefinite matrix A

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LU-factorization for real symmetry indefinite matrix A factorization where and Question: why not LU-decomposition? A is real symmetric, we only store lower triangle part of A, say

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row-major based Pointer array to store lower triangular part of A col-major based Question: can you find another representation for lower part of matrix A ? Question: what is the cost to fetch one element of A ? That is, operation count for Question: if one uses row-major to store lower part of matrix A, then how to fetch a column of A efficiently?

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OutLine Preliminary Symmetric permutation - change diagonal element to (1,1) position - change off-diagonal element to (2,1) position - implementation of symmetric permutation LDL’ decomposition (diagonal pivoting) Example of complete diagonal pivoting Algorithm of complete diagonal pivoting

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Define permutation matrix How to do permutation on real symmetry indefinite matrix A interchange row 2, 3 interchange column 2, 3 Let interchange row 2, interchange column 2, Symmetric permutation:

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Change diagonal element to (1,1) position [1] suppose we want to changeto position (1,1), then consider permutation and do symmetric permutation on A, say interchange col 1, 3 interchange row 1, 3 Question: we only store lower triangle part of matrix A, above permutation does not work, how to modify it? Observation 1:

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Observation 2: we only need to update lower triangle part of A (diagonal term is excluded ) diagonal term does not changed we have changed does not changed Change diagonal element to (1,1) position [2]

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interchange col 1, interchange row 1, Keep lower triangle part Fill-in diagonal term Fill-in A(3,1) Change diagonal element to (1,1) position [3] Question: any compact form to represent above interchange?

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Change diagonal element to (1,1) position [4] case 1: case 2: 2 6

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interchanges rows or columns Change diagonal element to (1,1) position [5] Step 1: interchange position (1,1) and (k,k), i.e. For general real symmetric matrix A with dimension n, suppose we want to change keep position (k,1) and (1,k), i.e. does not changed

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Change diagonal element to (1,1) position [6] Step 2: interchange column 1 and k below row k+1, i.e Therefore

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Change diagonal element to (1,1) position [7] Step 3: interchange column 1 (from row-2 to row-(k-1)) and row k (from col-2 to col-(k-1)) Therefore

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Implementation of symmetric permutation: swapping overhead [1] Suppose we use col-major based pointer array to store real symmetric matrix A Step 2: interchanging column 1 and k is O.K since memory block is contiguous. Contiguous memory block

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Implementation of symmetric permutation: swapping overhead [2] Step 3: interchanging column 1 and row k, i.e. is NOT efficient sinceis NOT contiguous. NOT contiguous, swapping is very slow. Observation: the situation is the same even we choose row-major based pointer array as storage strategy. Solution: in order to avoid high swapping overhead, we should avoid permutation.

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Change off-diagonal element to (2,1) position [1] suppose we want to changeto position (2,1), then consider first permutation which change first index from 4 to interchange col 2, 4 interchange row 2, 4 and second permutationwhich change second index from 3 to interchange col 1, interchange row 1,

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interchange row and col 2, interchange row and col 1, Change off-diagonal element to (2,1) position [2] Observation: two permutation matrices can be computed easily since 2 4 and 1 3 are independent.

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Change off-diagonal element to (2,1) position [3] HOW to transformto position (2,1), under only lower triangle part of A is stored? interchange row/col 2, 4 Observation 1:anddoes not changed Step 1: interchange position (4,4) and (2,2),

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Change off-diagonal element to (2,1) position [4] Observation 2: we only need to update lower triangle part of A (diagonal term is excluded ) Step 2: interchange row 2 and 4 below col 2, i.e interchange row 2, interchange col 2, NOTE: Interchanging column 2, 4 does not change position (2,1) and (4,1), it suffices to interchange position (2,1) and (4,1) first.

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Change off-diagonal element to (2,1) position [5] Step 3: interchange column 2 (from row-3 to row-3) and row 4 (from col-3 to col-3)

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interchanges rows or columns Change off-diagonal element to (2,1) position [6] Step 1: interchange position (2,2) and (k,k), i.e. For general real symmetric matrix A with dimension n, suppose we want to change keep position (k,2) and (2,k), i.e. does not changed

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Change off-diagonal element to (2,1) position [7] Step 2: interchange column 2 and k below row k+1, i.e Therefore

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Change off-diagonal element to (2,1) position [8] Step 3: interchange row 2 (column 1) and row k (column1), i.e Therefore

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Change off-diagonal element to (2,1) position [9] Step 4: interchange column 2 (from row-3 to row-(k-1)) and row k (from col-3 to col-(k-1)) Therefore

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OutLine Preliminary Symmetric permutation LDL’ decomposition (diagonal pivoting) - growth rate of 1x1, 2x2 pivoting - criterion for pivot strategy - comparison to complete pivoting in GE Example of complete diagonal pivoting Algorithm of complete diagonal pivoting

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LDL’ decomposition: 1x1, 2x2 pivoting diagonal pivoting method with complete pivoting: Bunch-Parlett, “Direct methods fro solving symmetric indefinite systems of linear equations,” SIAM J. Numer. Anal., v. 8, 1971, pp diagonal pivoting method with partial pivoting: Bunch-Kaufman, “Some Stable Methods for Calculating Inertia and Solving Symmetric Linear Systems,” Mathematics of Computation, volume 31, number 137, January 1977, page

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maximal element of matrix A maximal diagonal element of matrix A Case 1: s = 1 Growth rate in LDL’ decomposition [1] is of order Define and constant with and where

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Growth rate in LDL’ decomposition [2] Therefore Observation: in PA=LU, we choose such that implies Therefore Observation: large element growth will not occur for apivot if is large relative to

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Growth rate in LDL’ decomposition [3] Case 2: s = 2 where with and with and implies and

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Growth rate in LDL’ decomposition [4] for Therefore

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Growth rate in LDL’ decomposition [5] Therefore

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Fix a number( later on, we will determine optimal value of ) Case 1: maximal element of matrix A maximal diagonal element of matrix A supposewe interchange 1st and k-th rows and columns by introducing permutation matrixand do symmetric permutation Then doonwithpivot, written as Criterion for pivot strategy [1]

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Criterion for pivot strategy [2] Recall forpivot,, we have growth rate and Now for and Case 2: suppose, we interchange q-th and 1st rows and columns, and then r-th and 2nd rows and columns by introducing permutation matrix and do symmetric permutation Question: we must change q 1 first, then change r 2, why?

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Criterion for pivot strategy [3] but (boundary case) transforms

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Criterion for pivot strategy [4] Then doonwithpivot, written as pivot Recall forpivot,, we have growth rate and

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Criterion for pivot strategy [5] Now for To sum up Case 1: and Case 2: and Question: How to choose value of

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Criterion for pivot strategy [6] worst case analysis : choosesuch that growth rate ofpivot +pivotgrowth rate ofpivot or equivalentlygrowth rate ofpivotgrowth rate ofpivot Define wheresatisfies Exercise: verify and

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Diagonal pivoting versus complete pivoting of Gaussian Elimination [1] We must search for the largest element in each reduced matrix, this is a complete pivoting strategy analogous to Gaussian Elimination with complete pivoting maximal element of matrix A maximal diagonal element of matrix A 1 (1) find a such that (2) swap rowand row thenwith (2) swap columnand column complete pivoting of GE (Gaussian Elimination)

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growth rate 2 diagonal pivoting complete pivoting in GE Diagonal pivoting versus complete pivoting of Gaussian Elimination [2] Remark: Bunch [1] proves that the element growth in the diagonal pivoting method in comparison with with complete pivoting is bounded by for Gaussian Elimination with complete pivoting, where [1] J.R. Bunch, “Analysis of the diagonal pivoting method”, SIAM J. Numer. Anal., v. 8, 1971, pp

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OutLine Preliminary Symmetric permutation LDL’ decomposition (diagonal pivoting) Example of complete diagonal pivoting Algorithm of complete diagonal pivoting

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swap row/column we choosepivot andby permutation matrix Example (complete pivoting) [1]

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Example (complete pivoting) [2] we choosepivot swap row/columnby permutation matrix Recursively do the same procedure for such that

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Example (complete pivoting) [3] Do symmetry permutation forand we dopivot on

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Example (complete pivoting) [4] Or write in original matrix form

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Example (complete pivoting) [5] In practice, we have two key issues we only store lower part of matrix col-major based

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Example (complete pivoting) [6] we store decomposition 2 andinto original Question: How can you judge correct decomposition from

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Example (complete pivoting) [7] Case 1: four 1x1 pivot Case 2: two 2x2 pivot

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Example (complete pivoting) [8] Case 3: 1x1 pivot + 2x2 pivot + 1x1 pivot Case 4: 2x2 pivot + 1x1 pivot + 1x1 pivot Solution: we need an array to record pivot sequence pivot x2 pivot 1x1 pivot

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OutLine Preliminary Symmetric permutation LDL’ decomposition (diagonal pivoting) Example of complete diagonal pivoting Algorithm of complete diagonal pivoting

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we have compute update original matrix combines all lower triangle matrix and store in Algorithm ( PAP’ = LDL’ ) [1] Given symmetric indefinite matrix, construct initial lower triangle matrix while use permutation vector to record permutation matrix let, and, where

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compute and 1 Algorithm ( PAP’ = LDL’ ) [2] Case 1: define permutationto do symmetric permutation To compute, we only update lower triangle of 3 then 2

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Algorithm ( PAP’ = LDL’ ) [3] do 1x1 pivot : To compute We only update lower triangle matrix 4 then 5 6 and

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Algorithm ( PAP’ = LDL’ ) [4] Case 2: define permutation to interchange q-th and k-th rows and columns,and then r-th and (k+1)-th rows and columns 7 and then To compute, we only update lower triangle of (1) do interchange row/col then

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Algorithm ( PAP’ = LDL’ ) [5] (2) do interchange row/col then where 3

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Algorithm ( PAP’ = LDL’ ) [6] To compute 10 then (1) do interchange row 11 (2) do interchange row

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Algorithm ( PAP’ = LDL’ ) [7] do 2x2 pivot : 12 then 13 and means is 2x2 pivot endwhile

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Question: recursion implementation Initialization - check algorithm holds for k=1 Recursion formula - check algorithm holds for k or k+1, if k-1 is true Termination condition - check algorithm holds for k=n-1 Breakdown of algorithm - check which condition PAP’=LDL’ fails No extension: algorithm works only for square, symmetric indefinite matrix. normal abnormal Extension of algorithm

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MATLAB implementation [1] Given a (full) symmetric indefinite matrix, compute factorization return four quantities Example : return : Remark: try to neglect upper triangle part ofin MATLAB implementation

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MATLAB implementation [2] When factorization is complete,, you need to provide linear solver define, then by forward substitution define, then by diagonal block inversion Example:, then define, thenby backward substitution However you cannot transposeexplicitly in MATLAB , you must scanonce to determine

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