c11-0 (Numerical Recipes in C), страница 3
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The efficient way to form C isfirst to solve the equationY · LT = A(11.0.22)for the lower triangle of the matrix Y. Then solveL·C=Y(11.0.23)for the lower triangle of the symmetric matrix C.Another generalization of the standard eigenvalue problem is to problemsnonlinear in the eigenvalue λ, for example,(Aλ2 + Bλ + C) · x = 0(11.0.24)This can be turned into a linear problem by introducing an additional unknowneigenvector y and solving the 2N × 2N eigensystem,0−1−A1· C −A−1 xx·=λ·Byy(11.0.25)This technique generalizes to higher-order polynomials in λ. A polynomial of degreeM produces a linear M N × M N eigensystem (see [7]).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).Many eigenpackages also deal with the so-called generalized eigenproblem, [6]46311.1 Jacobi Transformations of a Symmetric MatrixCITED REFERENCES AND FURTHER READING:Stoer, J., and Bulirsch, R. 1980, Introduction to Numerical Analysis (New York: Springer-Verlag),Chapter 6. [1]Wilkinson, J.H., and Reinsch, C.
1971, Linear Algebra, vol. II of Handbook for Automatic Computation (New York: Springer-Verlag). [2]IMSL Math/Library Users Manual (IMSL Inc., 2500 CityWest Boulevard, Houston TX 77042). [4]NAG Fortran Library (Numerical Algorithms Group, 256 Banbury Road, Oxford OX27DE, U.K.),Chapter F02. [5]Golub, G.H., and Van Loan, C.F. 1989, Matrix Computations, 2nd ed.
(Baltimore: Johns HopkinsUniversity Press), §7.7. [6]Wilkinson, J.H. 1965, The Algebraic Eigenvalue Problem (New York: Oxford University Press). [7]Acton, F.S. 1970, Numerical Methods That Work; 1990, corrected edition (Washington: Mathematical Association of America), Chapter 13.Horn, R.A., and Johnson, C.R. 1985, Matrix Analysis (Cambridge: Cambridge University Press).11.1 Jacobi Transformations of a SymmetricMatrixThe Jacobi method consists of a sequence of orthogonal similarity transformations of the form of equation (11.0.14). Each transformation (a Jacobi rotation) isjust a plane rotation designed to annihilate one of the off-diagonal matrix elements.Successive transformations undo previously set zeros, but the off-diagonal elementsnevertheless get smaller and smaller, until the matrix is diagonal to machine precision.
Accumulating the product of the transformations as you go gives the matrixof eigenvectors, equation (11.0.15), while the elements of the final diagonal matrixare the eigenvalues.The Jacobi method is absolutely foolproof for all real symmetric matrices. Formatrices of order greater than about 10, say, the algorithm is slower, by a significantconstant factor, than the QR method we shall give in §11.3.
However, the Jacobialgorithm is much simpler than the more efficient methods. We thus recommend itfor matrices of moderate order, where expense is not a major consideration.The basic Jacobi rotation Ppq is a matrix of the formPpq=1···c...··· s.1 ..−s··· c···(11.1.1)1Here all the diagonal elements are unity except for the two elements c in rows (andcolumns) p and q. All off-diagonal elements are zero except the two elements s and−s. The numbers c and s are the cosine and sine of a rotation angle φ, so c2 + s2 = 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).Smith, B.T., et al. 1976, Matrix Eigensystem Routines — EISPACK Guide, 2nd ed., vol. 6 ofLecture Notes in Computer Science (New York: Springer-Verlag). [3].