计算课程讲义1

1、非线性方程组求解 完成日期09-8-20or whereand迭代法寻找一个数列 使得 迭代步长 使得迭代终止 （收敛） 或者 超过最大迭代次数 （发散）Algorithm given 初始化 , 0，最大迭代次数maxrepeat1. 计算 和 或者 2. if , return .3. 计算或者4. until max牛顿迭代法 n=1 This is called the linearized equation.n1 其中Unconstrained minimization无约束最优问题 convex diffmin , 其中牛顿迭代法 其中 Algorithm given 初始化 ,

2、0，最大迭代次数maxrepeat1. 计算 和 2. if , return .3. 计算4. until maxEquality constrained minimization等式约束最优问题 with ; convex diff, .min subject to 牛顿迭代法, .步长为当为可行性解（）内点法(primal-dual interior-point methods),KKT minimize subject to 其中 convex diff ; with .我们要解如下非线性方程组 所以 其中 The primal-dual search direction线性方程组求解

3、 完成日期09-8-21 LU分解； Cholesky分解（） i.e, QR分解 i.e, QR分解；Cholesky分解Nonsingular matrix The LU factorizationWhere P is an -permutation matrix, L is a unit lower triangular-matrix ( a lower triangular matrix with diagonal elements equal to one),and U is nonsingular upper triangular -matrix.Positive definite

4、 matrix The Cholesky factorizationwhere L is lower triangular with positive diagonal elements matrix with zero nullspace The QR factorization Where Q is -orthogonal matrix and R is an -upper triangular matrix with positive diagonal elements.Triangular matrix：A matrix A is upper triangular if if 。A t

5、riangular matrix is called nonsingular if all the diagonal elements are nonzero. It is singular if one or more of the diagonal elements are zero.Forward/Back substitutionA is a nonsingular lower triangular matrix of order n. are解法Flop count: 1+3+5+(2n-1)=n2 。Forward/Back substitution (Recursive form

6、ulation) then The Cholesky factorization: Positive definite matrix positive definite: A is symmetric and for all nonzero x. , where L is lower triangular with positive diagonal elements therefore and is lower triangular with positive diagonal elements.makes:Therefore , , . In fact, . So, , and that makes positive definite matrix. Schur Complement (of positive definite matrix elementis positive definite) whereis a nonzero n-1 vector then x is a nonzero n vector, so .Therefore is positive definite matrix of order n-1. Solving linear equations by Cholesky factorizatio