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类型用Romberg方法求解积分.doc

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    用Romberg方法求解积分.doc
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    1、1.用 Romberg 方法求解积分 ,要求误差不超过dx8.0 -210=a解:Romberg.m 文件:function I, step = Romberg(f, a, b,EPS)% Romberg.m 是用龙贝格公式求积分% f 为被积函数% EPS 为积分结果精度% a,b 为积分区间的上下限% I 为积分结果;step 为积分的子区间数m = 1k = 0Er = 0.1H =b-aS = zeros(1, 1)S(1, 1) = (H/2) * (subs(sym(f),findsym(sym(f),a)+subs(sym(f),findsym(sym(f),b)while Er

    2、 EPSk = k + 1f1 = 0H = H/2for i = 1:mx = a +H*(2*i-1)f1 = f1 + subs(sym(f),findsym(sym(f),x)endS(k+1, 1) = S(k, 1)/2 + H*f1m = 2 * mfor n = 1:kS(k+1, n+1) = S(k+1, n) + (S(k+1, n)-S(k, n)/(4n-1)endEr = abs(S(k+1, n+1)-S(k, n)endI = S(k+1, k+1)step = k命令:clearclcformat shorta = 0; b = 0.8; EPS = 1e-2

    3、;I, step = Romberg(x(1/2), a, b, EPS)计算结果:m =1k =0Er =0.1000H =0.8000S =0S =0.3578k =1f1 =0H =0.4000x =0.4000f1 =0.6325S =0.35780.4319m =2S =0.3578 00.4319 0.4566Er =0.0988k =2f1 =0H =0.2000x =0.2000f1 =0.4472x =0.6000f1 =1.2218S =0.3578 00.4319 0.45660.4603 0m =4S =0.3578 00.4319 0.45660.4603 0.469

    4、8S =0.3578 0 00.4319 0.4566 00.4603 0.4698 0.4707Er =0.0141k =3f1 =0H =0.1000x =0.1000f1 =0.3162x =0.3000f1 =0.8640x =0.5000f1 =1.5711x =0.7000f1 =2.4077S =0.3578 0 00.4319 0.4566 00.4603 0.4698 0.47070.4709 0 0m =8S =0.3578 0 00.4319 0.4566 00.4603 0.4698 0.47070.4709 0.4745 0S =0.3578 0 00.4319 0.

    5、4566 00.4603 0.4698 0.47070.4709 0.4745 0.4748S =0.3578 0 0 00.4319 0.4566 0 00.4603 0.4698 0.4707 00.4709 0.4745 0.4748 0.4748Er =0.0042I =0.4748step =3I =0.4748step =32.设方程组 试用 Jacobi 迭代法求解此方程, ,当.3102,24521xx Tx0,)0(时终止迭代。5)()(31makiki解:Jacobi.m 文件:function Jacobi(A, b, max, eps) %max 为最大迭代次数,eps

    6、 为容许误差n = length(A); x = zeros(n, 1); x1 = zeros(n, 1); k = 0;while 1x1(1) = ( b(1) - A(1,2:n) * x(2:n,1) )/A(1,1)for i = 2:n-1x1(i) = ( b(i) - A(i,1:i-1) * x(1:i-1,1) - A(i,i+1:n) * x(i+1:n,1)/A(i,i)endx1(n) = ( b(n) - A(n,1:n-1) * x(1:n-1,1) )/A(n,n)k = k + 1if sum(abs(x1 - x) = maxfprintf(The Met

    7、hod is disconvergentn)breakendx = x1endif k maxfor i = 1:nfprintf( x %d = %fn,i,x1(i) )endend命令:clearclcformat shortA = 5 2 1; -1 4 2; 2 -3 10;b = -12 20 3;max = 100;eps = 1e-5Jacobi(A, b, max, eps)计算结果:i =1A =5 2 1-1 4 22 -3 10b =-12203D =5 0 00 4 00 0 10L =0 0 01 0 0-2 3 0U =0 -2 -10 0 -20 0 0D0 =

    8、0.2000 0 00 0.2500 00 0 0.1000x0 =000B =0 -0.4000 -0.20000.2500 0 -0.5000-0.2000 0.3000 0f =-2.40005.00000.3000x =-2.40005.00000.3000x0 =-2.40005.00000.3000i =2x =-4.46004.25002.2800x0 =-4.46004.25002.2800i =3x =-4.55602.74502.4670x0 =-4.55602.74502.4670i =4x =-3.99142.62752.0347x0 =-3.99142.62752.0

    9、347i =5x =-3.85792.98481.8865x0 =-3.85792.98481.8865i =6x =-3.97123.09221.9670x0 =-3.97123.09221.9670i =7x =-4.03033.02372.0219x0 =-4.03033.02372.0219i =8x =-4.01392.98152.0132x0 =-4.01392.98152.0132i =9x =-3.99522.99001.9972x0 =-3.99522.99001.9972i =10x =-3.99543.00261.9960x0 =-3.99543.00261.9960i

    10、=11x =-4.00023.00311.9999x0 =-4.00023.00311.9999i =12x =-4.00123.00002.0010x0 =-4.00123.00002.0010i =13x =-4.00022.99922.0002x0 =-4.00022.99922.0002i =14x =-3.99972.99981.9998x0 =-3.99972.99981.9998i =15x =-3.99993.00021.9999x0 =-3.99993.00021.9999i =16x =-4.00003.00012.0000x0 =-4.00003.00012.0000i =17x =-4.00003.00002.0000x0 =-4.00003.00002.0000i =18x =-4.00003.00002.0000x0 =-4.00003.00002.0000i =19x =-4.00003.00002.0000x0 =-4.00003.00002.0000i =20x =-4.00003.00002.000x =-4.00003.00002.0000i =20

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