# Least Squares. 给定一组测量或观测值 假设 y 是 n 个函数的线性组合 设 m×n 矩阵 X 定义为 所以 β=X\y.

## Presentation on theme: "Least Squares. 给定一组测量或观测值 假设 y 是 n 个函数的线性组合 设 m×n 矩阵 X 定义为 所以 β=X\y."— Presentation transcript:

Least Squares

Log-linear: Gaussians:

p = 1e4*[ ]'; t = (1949:1:2005)'; x = (1940: 0.2 :2019)'; w = 2010; d = 3; s = (t-1979)/30; c = polyfit(s,p,d); s = (x-1979)/30; y = polyval(c,s); s = (w-1979)/30; z = polyval(c,s); plot(t,p,'o',x,y,'b-',w,z,'r*') s = (t-1979)/30; c = polyfit(s,p,3); X = [s.^3, s.^2, s, ones(length(s),1)]; X\p 中国人口预测

1 、求逆的过程困难 2 、

Matlab 中避免用法方程求解 QR 分解： Q 正交阵 (orthogonal) R 上三角阵 求解：

195055196 196066207 197082992 198098705 1990114333 2000 12674 3 s: -0.9667 -0.6333 -0.3000 0.0333 0.3667 0.7000 X: 0.9344 -0.9667 1.0000 0.4011 -0.6333 1.0000 0.0900 -0.3000 1.0000 0.0011 0.0333 1.0000 0.1344 0.3667 1.0000 0.4900 0.7000 1.0000 y: 55196 66207 82992 98705 114333 126743

u(1) = u(1) + sigma; rho = 1/( sigma *u(1)); X(i,k) = 0; X(k,k) = -sigma; j = k+1:n; v = rho*(u'* X(i,j)); X(i,j) = X(i,j) - u*v; if ~isempty(y) tau = rho*(u'* y(i)); y(i) = y(i) - tau*u; end beta = X(1:n,1:n)\y(1:n) polyval(beta,(2010-1979)/30) s = ( (1950:10:2000)'-1979 )/30; X = [s.^2, s, ones(size(s),1)] y = [ ] [m,n] = size(X); for k = 1:min(m-1,n) i = k:m; u = X(i,k); sigma = norm(u); if sigma ~= 0 if u(1) ~= 0, sigma = sign(u(1))*sigma; end u(1) = u(1) + sigma; rho = 1/( sigma *u(1));

X = -1.1403 0.6945 -1.7987 0 -0.3122 0.4589 0 -0.2279 0.8786 0 0.0342 0.9985 0 0.4743 0.8186 0 1.0923 0.3390 y = 1.0e+005 * -1.4311 0.2787 0.7439 0.9860 1.0148 0.7991 X = -1.1403 0.6945 -1.7987 0 1.2525 0.3587 0 0 0.8640 0 0 1.0007 0 0 0.8490 0 0 0.4090 y = 1.0e+005 * -1.4311 0.9033 0.8349 0.9723 0.8255 0.3630 X = -1.1403 0.6945 -1.7987 0 1.2525 0.3587 0 0 -1.6236 0 0 0 y = 1.0e+005 * -1.4311 0.9033 -1.5667 0.0062 0.0058 -0.0318

X = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 秩亏损 (Rank Deficiency) 接近秩亏损 X\y ans = -7.5000 0 7.8333 pinv(X)*y ans = -7.5556 0.1111 7.7778 X\y-pinv(X)*y ans = 0.0556 -0.1111 0.0556 y = 16 17 18 19 20

dbstop at 14 in xxx %F12 dbquit dbtype xxx.m 12:15 %type,edit error(‘xxxxx’) pause(n) keyboard; %K>>return 调试

Solve 1. Block cyclic reduction (BCR) B. L. Buzbee, G. H. Golub, C.W. Nielson, On direct methods for solving Poisson ’ s equations, SIAM J. Numer. Anal., 7(1970), pp. 627-656. 2. Block orthogonal factorization (BOF) G. Fairweather, I. Gladwell, Algorithms for almost block diagonal linear systems, SIAM Review, 46(2004), pp. 49-58. S. J. Wright, Stable parallel algorithms for two-point boundary value problem, SIAM J. Sci. Statist. Comput., 13(1992), pp. 742-764. 进一步探讨用分块循环约化法于下面方程的求解

Cholesky factorization 1. Basic Cholesky factorization G. H. Golub, C. F. Van Loan, Matrix Computations, 3 rd ed., Johns Hopkins University Press, Baltimore and London, 1996. 1) gaxpy Cholesky 2) outer-product Cholesky 3) 分块点积的 Cholesky 分解 4) 其他实现方法 比较各种方法的效率，并与 Matlab 函数 chol() 比较。

2. Incomplete Cholesky (IC) factorization T. A. Manteuffel, An incomplete factorization technique for positive definite linear systems, Mathematics of Computation, 34, 150(1980), 473-497. J. Meijerink, H. A. Van der Vorst, An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, Math. Comput., 31, 137(1977), pp. 134-155. R. S. Varga, E. B. Saff, V. Mehrmann, Incomplete factorizations of matrices and connections with H-matrices, SIAM J. Numer. Anal., 17(1980), pp. 787-793. 参考 G.H. Golub, C. F. Van Loan 的《 Matrix Computation 》. 中文版 p.620

I. E. Kaporin, High quality preconditioning of a general symmetric positive definite matrix based on its -docomposition, Numer. Linear Algebra Appl., 5(1998), pp. 483-509. 3. Robust Incomplete Cholesky factorization 1) T. A. Manteuffel, An incomplete factorization technique for positive definite linear systems, Mathematics of Computation, 34, 150(1980), 473- 497. M. Ajiz, A. Jennings, A robust incomplete Choleski-conjugate gradient algorithm, Internat. J. Numer. Methods Engrg., 20, 5(1984), pp. 949-966. M. Tismenetsky, A new preconditioning technique for solving large sparse linear systems, Linear Algebra Appl., 154-156(1991), pp. 331-353. 2) 3)

Let A be the matrix whose entries are zero everywhere except for the primes 2, 3, 5, 7, …, 224737 along the main diagonal and the number 1 in all the positions with. What is the (1,1) entry of ?

The Lorenz Strange Attractor The Lorenz system is given by Using the parameters and the initial conditions and (1)apply the Runge-Kutta method using step size to solve the system for and then plot each of the projection z vs. x, y vs. x, and y vs. z. (2)Perturb the initial conditions slightly to x(0)=-8.02, y(0)=7.98 (and same z(0)), solve the new system with the same method, and plot the original x minus the new x versus t on the whole time interval.

Numerical algorithms for solving the linear two-point boundary value problem (BVP) Two approach: 1)multiple shooting technique 2)finite-difference method( 建议用此法 ) S. J. Wright, Stable parallel algorithms for two-point boundary value problems, SIAM J. Sci. Stat. Comput., 13(1992), pp. 742-764.

Download ppt "Least Squares. 给定一组测量或观测值 假设 y 是 n 个函数的线性组合 设 m×n 矩阵 X 定义为 所以 β=X\y."

Similar presentations