MATLAB基瑞矩阵与数组运算 Example 3 ln(1+x)=x- +写-+(-1-1+…（←1<x≤1） x2x3 n 。利用内嵌函数直接计算In2的值； 。利用原始解法前20项的和近似计算In2的值； 。利用下式前3项的和近似计算In2的值。 1+× In 1-x 4口，4+4立4要，三)及0 Xi Chen (chenxi0109@bfsu.edu.cn) MATLAB在经济与管理研究中的应用 11/139
MATLAB ƒ: › ÜÍ|$é Example 3 ln (1 + x) = x − x 2 2 + x 3 3 − . . . + (−1)n−1 x n n + . . . (−1 < x ≤ 1) |^SiºÍÜOéln 2ä¶ |^©){c20ë⁄CqOéln 2ä¶ |^e™c3ë⁄CqOéln 2ä" ln 1 + x 1 − x Xi Chen (chenxi0109@bfsu.edu.cn) MATLAB 3²LÜ+nÔƒ•A^ 11 / 139
MATLAB基瑞矩阵与数组运算 Example 4 arctanx=x- +写7++(-1 x3 x5 x7 2n-1+…（-1≤x≤1) 。利用前20项之和近似计算π值； 。若利用欧拉公式来计算π值，说明计算效率是否有提高； 牙=arctan+arctan 1 3 。若利用下列逆推式再计算π值，说明计算效率是否有提高。 nt1=van+1/van bn+1=van(1+bn) 2 an+bn Pn+1= Pnbn+1(1+an+1) a0=V2,b0=0,p0=2+V2 1+bn+1 行 Xi Chen (chenxi0109@bfsu.edu.cn) MATLAB在经济与管理研究中的应用 12/139
MATLAB ƒ: › ÜÍ|$é Example 4 arctan x = x − x 3 3 + x 5 5 − x 7 7 + . . . + (−1)n−1 x 2n−1 2n − 1 + . . . (−1 ≤ x ≤ 1) |^c20ëÉ⁄CqOéπä¶ e|^Ó.˙™5Oéπäß`²Oé«¥ƒkJp¶ π 4 = arctan 1 2 + arctan 1 3 e|^e_Ì™2Oéπäß`²Oé«¥ƒkJp" an+1 = √ an + 1/ √ an 2 , bn+1 = √ an(1 + bn) an + bn pn+1 = pnbn+1(1 + an+1) 1 + bn+1 , a0 = √ 2, b0 = 0, p0 = 2 + √ 2 Xi Chen (chenxi0109@bfsu.edu.cn) MATLAB 3²LÜ+nÔƒ•A^ 12 / 139
MATLAB基矩阵与数组运算 Iterative methods for solving a square linear system We seek an iteration of the form x+1=F(x).where an initial guess xoRn is given and F is simple to compute. Jacobi Iteration Ax=b (U+L+D)x=b Dx=-(U+L)x+b x=-D-1(U+L)x+D-1b Gauss-Seidel Iteration Ax=b (U+L+D)x=b (L+D)x=-Ux+b x=-(L+D)-1Ux+(L+D)-1b Xi Chen (chenxi0109@bfsu.edu.cn) MATLAB在经济与管理研究中的应用 13/139
MATLAB ƒ: › ÜÍ|$é Iterative methods for solving a square linear system We seek an iteration of the form x k+1 = F(x k ), where an initial guess x 0 ∈ R n is given and F is simple to compute. Jacobi Iteration Ax = b (U + L + D)x = b Dx = −(U + L)x + b x = −D −1 (U + L)x + D −1 b Gauss-Seidel Iteration Ax = b (U + L + D)x = b (L + D)x = −Ux + b x = −(L + D) −1Ux + (L + D) −1 b Xi Chen (chenxi0109@bfsu.edu.cn) MATLAB 3²LÜ+nÔƒ•A^ 13 / 139
MATLAB基暗矩阵与数组运算 Example 5 Using Jacobi and Guass-Seidel iteration to solve the following linear equation system and show their computational efficiency. 7X1 9+2x3 =10 +8x9+ 2x3=8 2X1 +2x为+ 9x3 =6 Theorem 6 (Weierstrass Approximation Theorem) If f(x)is a continuous function in [a,b,then for every e>0,there exists a polynomial p(x)such that If(x)-p(x)I≤e for every x∈[a,bl. Xi Chen (chenxi0109@bfsu.edu.cn) MATLAB在经济与管理研究中的应用 14/139
MATLAB ƒ: › ÜÍ|$é Example 5 Using Jacobi and Guass-Seidel iteration to solve the following linear equation system and show their computational efficiency. 7x1 + x2 + 2x3 = 10 x1 + 8x2 + 2x3 = 8 2x1 + 2x2 + 9x3 = 6 Theorem 6 (Weierstrass Approximation Theorem) If f (x) is a continuous function in [a, b], then for every > 0, there exists a polynomial p(x) such that |f (x) − p(x)| ≤ for every x ∈ [a, b]. Xi Chen (chenxi0109@bfsu.edu.cn) MATLAB 3²LÜ+nÔƒ•A^ 14 / 139
MATLAB基瑞常用数学面数 MATLAB基础 。矩阵与数组运算 ·常用数学函数 。符号运算 。MATLAB编程 。二维与三维绘图 ②最优化问题求解 。 线性规划 o 整数规划 。二次规划 4口，4辱+4之，至，三Q0 灯Chem【chenxi0I09@bfsu.edu:MATLAB在经济与管理研究中的应用 15/139
MATLAB ƒ: ~^ÍÆºÍ 1 MATLAB ƒ: › ÜÍ|$é ~^ÍÆºÍ Œ“$é MATLAB ?ß ëÜn뱄 2 Å`zØK¶) Ç55y Í5y g5y Xi Chen (chenxi0109@bfsu.edu.cn) MATLAB 3²LÜ+nÔƒ•A^ 15 / 139