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2.2 Gaussian Elimination with Backsubstitution41which (peeling of the C−1 ’s one at a time) implies a solutionx = C 1 · C2 · C3 · · · b(2.1.8)CITED REFERENCES AND FURTHER READING:Wilkinson, J.H. 1965, The Algebraic Eigenvalue Problem (New York: Oxford University Press). [1]Carnahan, B., Luther, H.A., and Wilkes, J.O. 1969, Applied Numerical Methods (New York:Wiley), Example 5.2, p. 282.Bevington, P.R. 1969, Data Reduction and Error Analysis for the Physical Sciences (New York:McGraw-Hill), Program B-2, p.
298.Westlake, J.R. 1968, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations(New York: Wiley).Ralston, A., and Rabinowitz, P. 1978, A First Course in Numerical Analysis, 2nd ed. (New York:McGraw-Hill), §9.3–1.2.2 Gaussian Elimination with BacksubstitutionThe usefulness of Gaussian elimination with backsubstitution is primarilypedagogical. It stands between full elimination schemes such as Gauss-Jordan, andtriangular decomposition schemes such as will be discussed in the next section.Gaussian elimination reduces a matrix not all the way to the identity matrix, butonly halfway, to a matrix whose components on the diagonal and above (say) remainnontrivial.
Let us now see what advantages accrue.Suppose that in doing Gauss-Jordan elimination, as described in §2.1, we ateach stage subtract away rows only below the then-current pivot element. When a22is the pivot element, for example, we divide the second row by its value (as before),but now use the pivot row to zero only a32 and a42 , not a12 (see equation 2.1.1).Suppose, also, that we do only partial pivoting, never interchanging columns, so thatthe order of the unknowns never needs to be modified.Then, when we have done this for all the pivots, we will be left with a reducedequation that looks like this (in the case of a single right-hand side vector): 0 0x1b1a11 a012 a013 a014 0 a022 a023 a024 x2 b02 (2.2.1)· = 0 00 a033 a034x3b3000 a044x4b04Here the primes signify that the a’s and b’s do not have their original numericalvalues, but have been modified by all the row operations in the elimination to thispoint.
The procedure up to this point is termed Gaussian elimination.Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited.
To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).Notice the essential difference between equation (2.1.8) and equation (2.1.6). In thelatter case, the C’s must be applied to b in the reverse order from that in which they becomeknown. That is, they must all be stored along the way.
This requirement greatly reducesthe usefulness of column operations, generally restricting them to simple permutations, forexample in support of full pivoting.42Chapter 2.Solution of Linear Algebraic EquationsBacksubstitutionBut how do we solve for the x’s? The last x (x4 in this example) is alreadyisolated, namelyx4 = b04 /a044(2.2.2)x3 =1 0[b − x4 a034 ]a033 3and then proceed with the x before that one.
The typical step isNX1 0a0ij xj x i = 0 bi −aii(2.2.3)(2.2.4)j=i+1The procedure defined by equation (2.2.4) is called backsubstitution. The combination of Gaussian elimination and backsubstitution yields a solution to the setof equations.The advantage of Gaussian elimination and backsubstitution over Gauss-Jordanelimination is simply that the former is faster in raw operations count: Theinnermost loops of Gauss-Jordan elimination, each containing one subtraction andone multiplication, are executed N 3 and N 2 M times (where there are N equationsand M unknowns).
The corresponding loops in Gaussian elimination are executedonly 13 N 3 times (only half the matrix is reduced, and the increasing numbers ofpredictable zeros reduce the count to one-third), and 12 N 2 M times, respectively.Each backsubstitution of a right-hand side is 12 N 2 executions of a similar loop (onemultiplication plus one subtraction). For M N (only a few right-hand sides)Gaussian elimination thus has about a factor three advantage over Gauss-Jordan.(We could reduce this advantage to a factor 1.5 by not computing the inverse matrixas part of the Gauss-Jordan scheme.)For computing the inverse matrix (which we can view as the case of M = Nright-hand sides, namely the N unit vectors which are the columns of the identitymatrix), Gaussian elimination and backsubstitution at first glance require 13 N 3 (matrixreduction) + 12 N 3 (right-hand side manipulations) + 12 N 3 (N backsubstitutions)= 43 N 3 loop executions, which is more than the N 3 for Gauss-Jordan.
However, theunit vectors are quite special in containing all zeros except for one element. If thisis taken into account, the right-side manipulations can be reduced to only 16 N 3 loopexecutions, and, for matrix inversion, the two methods have identical efficiencies.Both Gaussian elimination and Gauss-Jordan elimination share the disadvantagethat all right-hand sides must be known in advance. The LU decomposition methodin the next section does not share that deficiency, and also has an equally smalloperations count, both for solution with any number of right-hand sides, and formatrix inversion. For this reason we will not implement the method of Gaussianelimination as a routine.CITED REFERENCES AND FURTHER READING:Ralston, A., and Rabinowitz, P.
1978, A First Course in Numerical Analysis, 2nd ed. (New York:McGraw-Hill), §9.3–1.Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use.
Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).With the last x known we can move to the penultimate x,432.3 LU Decomposition and Its ApplicationsIsaacson, E., and Keller, H.B.
1966, Analysis of Numerical Methods (New York: Wiley), §2.1.Johnson, L.W., and Riess, R.D. 1982, Numerical Analysis, 2nd ed. (Reading, MA: AddisonWesley), §2.2.1.Westlake, J.R. 1968, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations(New York: Wiley).Suppose we are able to write the matrix A as a product of two matrices,L·U=A(2.3.1)where L is lower triangular (has elements only on the diagonal and below) and Uis upper triangular (has elements only on the diagonal and above). For the case ofa 4 × 4 matrix A, for example, equation (2.3.1) would look like this: α11 α21α31α410α22α32α4200α33α430β110 0· 00α440β12β2200β13β23β330β14β24 β34β44a11a= 21a31a41a12a22a32a42a13a23a33a43a14a24 a34a44(2.3.2)We can use a decomposition such as (2.3.1) to solve the linear setA · x = (L · U) · x = L · (U · x) = b(2.3.3)by first solving for the vector y such thatL·y=b(2.3.4)U·x=y(2.3.5)and then solvingWhat is the advantage of breaking up one linear set into two successive ones?The advantage is that the solution of a triangular set of equations is quite trivial, aswe have already seen in §2.2 (equation 2.2.4).
Thus, equation (2.3.4) can be solvedby forward substitution as follows,y1 =b1α11i−1X1 yi =αij yj bi −αiij=1(2.3.6)i = 2, 3, . . . , Nwhile (2.3.5) can then be solved by backsubstitution exactly as in equations (2.2.2)–(2.2.4),yNxN =βNNN(2.3.7)X1 xi =βij xj yi −i = N − 1, N − 2, . . . , 1βiij=i+1Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited.
To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).2.3 LU Decomposition and Its Applications.
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