浙江工业大学化材学院 jackdong - ScienceNet.cnimage.sciencenet.cn/olddata/kexue.com.cn/bbs/upload/...浙江工业大学化材学院 jackdong 2 1.2. Lagrange 插值多项式

浙江工业大学化材学院 jackdong

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数值计算方法

算法设计及其 MATLAB 实现

浙江工业大学

Jackdong

QQ: 7214478

Email:[email protected]

2006 年 12 月 17 日 ======================================================================


A

目录

第一章插值方法............................................................1

1.1. Lagrange插值 ...................................................................................................................1 1.2. Lagrange 插值多项式......................................................................................................2 1.3. Newton多项式 ..................................................................................................................3 1.4. 切比雪夫逼近...................................................................................................................4 1.5. 逐步插值...........................................................................................................................5 1.6. 分段三次Hermite插值......................................................................................................6 1.7. 分段三次样条插值...........................................................................................................7

第二章数值积分..........................................................10

2.1. 复化Simpson公式...........................................................................................................10 2.2. 变步长梯形法.................................................................................................................12 2.3. Romberg加速法..............................................................................................................14 2.4. 三点Gauss公式 ...............................................................................................................16

第三章常微分方程的差分解法..................................17

3.1. 改进的Euler方法 ............................................................................................................17 3.2. Heun方法 ........................................................................................................................18 3.3. 四次Taylor方法 ..............................................................................................................19 3.4. 四阶Runge-Kutta法 ........................................................................................................19 3.5. Runge-Kutta-Fehlbrg法 ..................................................................................................21 3.6. 二阶Adams预报校正系统 .............................................................................................24 3.7. 改进的四阶Adams预报校正系统 .................................................................................25 3.8. Milne-Simpson方法........................................................................................................27 3.9. Hamming方法.................................................................................................................29 3.10. 微分方程组四阶Runge-Kutta解法 ..............................................................................30 3.11. 线性打靶法...................................................................................................................31 3.12. 求解三对方程组的程序...............................................................................................32 3.13. 有限差分法...................................................................................................................33 附：三阶、四阶、五阶Rungkuta法 .....................................................................................35

第四章方程求根..........................................................40

4.1. 二分法.............................................................................................................................40 4.2. 开方法.............................................................................................................................41 4.3. Newton 下山法 ..............................................................................................................42 4.4. 快速弦截法.....................................................................................................................43 4.5. 不动点迭代法.................................................................................................................44 4.6. 试值法或试位法.............................................................................................................45 4.7. Steffensen加速法 ............................................................................................................46 4.8. Muller法..........................................................................................................................48


B

第五章线性方程组的迭代法....................................51

5.1. Jacobi迭代法 .................................................................................................................51 5.2. Gauss-Seidel迭代 ...........................................................................................................53 5.3. 非线性Seidel迭代...........................................................................................................56 5.4. 牛顿-拉夫森法 ...............................................................................................................57 5.5. 超松弛迭代.....................................................................................................................59 5.6. 对超松弛迭代.................................................................................................................60

第六章线性方程组的直接法......................................63

6.1. 追赶法.............................................................................................................................63 6.2. Cholesky方法...................................................................................................................64 6.3. 矩阵分解方法.................................................................................................................66 6.4. 消去法.............................................................................................................................69

第七章数值优化..........................................................71

7.1. 黄金分割法求极小值.....................................................................................................71 7.2. 斐波那契法求极小值.....................................................................................................73 7.3. 用 2 次插值求局部最小值.............................................................................................76 7.4. 内德-米德法求最小值 ...................................................................................................81 7.5. 最速下降法或梯度法.....................................................................................................86


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第一章插值方法

1.1. Lagrange 插值

功能：计算多项式在 x=x0 处的值 ---------------------------------------------------------------------------------------------------------------------- function [y0,N]=Lagrange_eval(X,Y,x0) % X,Y :已知的插值点坐标 % x0 :插值点 % y0 :Lagrange 插值多项式在 x0 处的值 % N :Lagrange 插值函数的权系数 m=length(X) N=zeros(m,1) y0=0; for i=1:m N(i)=1; for j=1:m if j~=i N(i)=N(i)*(x0-X(j))/(X(i)-X(j)); end end y0=y0+Y(i)*N(i); end ----------------------------------------------------------------------------------------------------------------------


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1.2. Lagrange 插值多项式

功能：求基于 N+1 个点的拉格朗日多项式 ----------------------------------------- function [C,L]=lagran(X,Y) % input - X is a vector that contains a list of abscissas % - Y is a vector that contains a list of ordinates % output - C is a matrix that contains the coefficients of the lagrange interpolatory polynomial - L is a matrix that contains the lagrange coefficients polynomial w=length(X); n=w-1; L=zeros(w,w); for k=1:n+1 V=1; for j=1:n+1 if k~=j V=conv(V,poly(X(j)))/(X(k)-X(j)); end end L(k,:)=V; end C=Y*L; ----------------------------------------------------------------------------------------------------------------------


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1.3. Newton 多项式

功能：构造和计算通过(xk,yk)=(xk,f(xk)),k=0....N 的次数小于等于 N 的牛顿多项式 ----------------------------------------- function [C,D]=newpoly(X,Y) % input - X is a vector that contains a list of abscissas % - Y is a vector that contains a list of ordinates % output - C is a matrix that contains the coefficients of the newton interpolatory polynomial - D is the divided-difference table n=length(X); D=zeros(n,n); D(:,1)=Y'; for j=2:n for k=j:n D(k,j)=(D(k,j-1)-D(k-1,j-1))/(X(k)-X(k-j-1)); end end C=D(n,n); for k=(n-1):-1:1 C=conv(c,poly(X(k))); m=length(C); C(m)=C(m)+D(k.k); end ----------------------------------------------------------------------------------------------------------------------


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1.4. 切比雪夫逼近

功能：构造并计算[-1，1]上的 N 次切比雪夫多项式 ----------------------------------------- function [C,X,Y]=cheby(fun,n,a,b) % input - fun is the string function to be approximated % -a and b are the left and right endpoints % - n is the degree of the chebyshev interpolating polynomial % output - C is the coefficient list for the polynomial - X contains the abscissas - Y contains the ordinates if nargin ==2,a=1;b=1;end d=pi/(2*n+2); C=zeros(1,n+1); for k=1:n+1 X(k)=cos((2*k-1)*d); end X=(b-a)*X/2+(a+b)/2; x=X; Y=eval(fun); for k=1:n+1 z=(2*k-1)*d; for j=1:n+1 C(j)=C(j)+Y(k)*cos((j-1)*z);


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end end C=2*C/(n+1); C(1)=C(1)/2; ----------------------------------------------------------------------------------------------------------------------

1.5. 逐步插值

功能：计算逐步插值多项式在 x0 处的值 ----------------------------------------- function y0=neville_eval(X,Y,x0) % X,Y :已知的插值点坐标 % x0 :插值点 % y0 :neville 逐步插值多项式在 x0 处的值 m=length(X) P=zeros(m,1) P1=zeros(m,1) P=Y; for i=1:m P1=P; k=1; for j=i+1:m k=k+1; P(j)=P(j-1)+(P1(j)-P1(j-1))*(x0-X(k-1))/(X(j)-X(k-1));


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end if abs(P(m)-P(m-1))<10^-6 y0=P(m); return; end end y0=P(m); ----------------------------------------------------------------------------------------------------------------------

1.6. 分段三次 Hermite 插值

功能：利于分段三次 Hermite 插值计算插值点处的函数近似值 ----------------------------------------- function y0=Hermite_inter(X,Y,DY,x0) % X,Y :已知的插值点坐标向量 % x0 :插值点 % y0 :Hermite 插值多项式在 x0 处的值 % DY :插值点处的导数值 N＝length(X) for i=1:N if x0 >=X(i) & x0 <=X(i+1) k=i;break; end end a1=x0-X(k+1);


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a2=x0-X(k); a3=X(k)-X(k+1); y0=(a1/a3)^2*(1-2*a2/a3)*Y(k)+(-a2/a3)^2*(1+2*a1/a3)*Y(k+1)+(a1/a3)^2*a2*DY(k)+(-a2/a3)^2*a1*DY(k+1); ----------------------------------------------------------------------------------------------------------------------

1.7. 分段三次样条插值

功能：计算在插值点处的函数值，并拟合曲线 ----------------------------------------- function [y0,C]=Spline_interp(X,Y,s0,sN,x0) % X,Y :已知的插值点坐标向量 % s0,sN:两个端点的一次导数值 % x0 :插值点 % y0 :三次样条函数在 x0 处的值 % C :分段三次样条函数的系数 N=length(X) C=zeros(4,N-1); h=zeros(1,N-1); mu=zeros(1,N-1); lmt=zeros(1,N-1);d=zeros(1,N); h=X(1,2:N)-X(1,1:N-1); mu(1,N-1)=1; lmt(1,1)=1; mu(1,2:N-2)=h(1,1:N-2)/(h(1,1:N-2)+h(1,1:N-1)); lmt(1,2:N-1)=h(1,2:N-1)/(h(1,1:N-2)+h(1,2:N-1)); d(1,1)=6*(Y(1,2)-Y(1,1))/h(1,1)-s0/h(1,1); d(1,N)=6*(sN-(Y(1,N)-Y(1,N-1))/h(1,N-1)/h(1,N-1);


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d(1,2:N-1)=6*((Y(1,3:N)-Y(1,2:N-1))/h(1,2:N-1)-(Y(1,2:N-1)-Y(1,1:N-2))/h(1,1:N-2)/h(1,1:N-2)+h(1,2:N-1); %追赶法解三对角方程组 bit=zeros(1,N-1); bit(1,1)=lmt(1,1)/2; for i=2:N-1 bit(1,i)=lmt(1,i)/(2-mu(1,i-1)*bit(1,i-1)); end y=zeros(1,N); y(1,1)=d(1,1)/2; for i=2:N y(1,i)=(d(1,i)-mu(1,i-1)*y(1,i-1))/(2-mu(1,i-1)*bit(1,i-1)); end x=zeros(1,N); x(1,N)=y(1,N); for i=N-1:-1:1 x(1,i)=y(1,i)-bit(1,i)*x(1,i+1); end v=zeros(1,N-1); v(1,1:N-1)=(Y(1,2:N)-Y(1,1:N-1))/h(1,1:N-1); C(4,:)=Y(1,1:N-1); C(3,:)=v-h*(2*x(1,N-1)+x(1,2:N))/6;


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C(2,:)=x(1,1:N-1)/2; C(1,:)=(x(1,2:N)-x(1,1:N-1))/(6*h); if nargin<5 y0=0; else for j=1:N-1 if x0>=X(1,j) & x0<X(1,j+1) omg=x0-X(1,j); y0=(C(4,:)*omg+C(3,:)*omg+C(2,:)*omg)+C(1,j); end end end ======================================================================


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第二章数值积分

2.1. 复化 Simpson 公式

功能：利用复化 Simpson 公式计算被积函数 f(x)在给定区间上的积分值 ----------------------------------------- function S=FSimpson(f,a,b,n) % f:被积函数句柄 % a,b:积分区间的两个端点 % n:子区间个数 % S:用复化 Simpson 法求得的积分值 h=(b-a)/n; fa=feval(f,a); fb=feval(f,b); S=fa+fb; x=a; for i=1:N x=x+h/2; fx=feval(f,x); S=S+4*fx; x=x+h/2; fx=feval(f,x); S=S+2*fx; end


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S=h*S/6; ---------------------------------------------------------------------------------------------------------------------- 附：函数值为向量形式的 simpson 求积法 function I=simpson_h(f,h) %调用格式 I＝simpson(f,h) %f 为一向量，指定已知节点处的函数值 %h 为步长 n=length(f)-1; if n==1 fprintf('Data has only one interval'),return; end; if n==2 I=(h/3)*(f(1)+4*f(2)+f(3));return; end; if n==3 I=(3*h/8)*(f(1)+3*f(2)+3*f(3)+f(4));return; end; I=0; if 2*floor(n)~=n % floor is a function round towards -inf I=3*(h/8)*(f(n-2)+3*f(n-1)+3*f(n)+f(n+1)); m=n-3; else m=n;


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end; I=I+(h/3)*(f(1)+4*sum(f(2:2:m))+f(m+1)); if m>2 I=I+(h/3)*2*sum(f(3:2:m)); end; ---------------------------------------------------------------------------------------------------------------------- 附：函数值为向量形式的复合 simpson 求积法 function I=simpson_n(fname,a,b,n) %调用格式：I=simpson_n('fname',a,b,n) %其中 a,b 为积分区间两个端点，n 为子区间数目 h=(b-a)/n; x=a+(0:n)*h; f=feval(fname,x); I=simpson_h(f,h) % 调用上面编译好的 simpson_h 函数 ----------------------------------------------------------------------------------------------------------------------

2.2. 变步长梯形法

功能：利用变步长梯形法计算函数 f(x)在给定区间的积分值 ----------------------------------------------------------- function [T,n]=bbct(f,a,b,eps) % f:被积函数句柄 % a,b:积分区间的两个端点 % eps:精度 % n:二分区间的次数


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% T:用变步长梯形法求得的积分值 h=b-a; fa=feval(f,a); fb=feval(f,b); T1=h*(fa+fb)/2; T2=T1/2+h*feval(f,a+h/2)/2; n=1; %按变步长梯形法求积分值 while abs(T2-T1)>=eps h=h/2; T1=T2; S=0; x=x+h/2; while x<b fx=feval(f,x); S=S+fx; x=x+h; end T2=T1/2+S*h/2; n=n+1; end T=T2;


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---------------------------------------------------------------------------------------------------------------------- 附：函数值为向量形式的梯形求积法 function I=trapez_h(f,h) %调用格式 I=trapez_h(f,h) %其中 f 为一向量，即给定的一系列纵坐标值 %h 为步长 I=h*(sum(f)-(f(1)+f(length(f)))/2); ---------------------------------------------------------------------------------------------------------------------- 附：函数值为向量形式的复合梯形求积法 function I=trapez_n(fname,a,b,n) %调用格式 trapez_n('fname',a,b,n) %a，b 为区间的两个端点，n 为子区间个数 %其中需要调用另外一个函数：trapez_h,是梯形求积法的函数，如上。 h=(b-a)/n; x=a+(0:n)*h; f=feval(fname,x); I=trapez_h(f,h) ----------------------------------------------------------------------------------------------------------------------

2.3. Romberg 加速法

功能：利用 Romberg 加速算法计算被积函数 f(x)在给定区间的积分值 ----------------------------------------- funvtion [quad,R]=Romberg(f,a,b,eps) % f:被积函数句柄


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% a,b:积分区间的两个端点 % eps:精度 % quad:用 Romberg 加速算法求得的积分值 h=b-a; R(1,1)=h*(feval(f,a)+feval(f,b))/2; M=1; J=0; err=1; while err>eps J=J+1; h=h/2; S=0; for p=1:M x=a+h*(2*p-1); S=S+feval(f,x); end R(J+1,1)=R(J,1)/2+h*S; M=2*M; for k=1:J R(J+1,k+1)=R(J+1,k)+(R(J+1,k)-R(J,k))/(4^k-1); end err=abs(R(J+1,J)-R(J+1,J+1)); end quad=R(J+1,J+1);


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2.4. 三点 Gauss 公式

功能：利用三点 Gauss 公式计算被积函数 f(x)在给定区间的积分值 ----------------------------------------- function G=TGauss(f,a,b) % f:被积函数句柄 % a,b:积分区间的两个端点 % G:用三点 Gauss 公式法求得的积分值 x1=(a+b)/2-sqrt(3/5)*(b-a)/2; x2=(a+b)/2+sqrt(3/5)*(b-a)/2; G=(b-a)*(5*feval(f,x1)/9+8*feval(f,(a+b)/2)/9+5*feval(f,x2)/9)/2; ======================================================================


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第三章常微分方程的差分解法

3.1. 改进的 Euler 方法

功能：用改进 Euler 法求解常微分方程 ----------------------------------------- function E=MendEuler(f,a,b,n,ya) % f:微分方程右端函数句柄 % a,b:自变量取值区间的两个端点 % n:区间等分的个数 % ya:函数初值 y(a) % E=[x',y']:自变量 X 和解 Y 所组成的矩阵 h=(b-a)/n; y=zeros(1,n+1); x=zeros(1,n+1); y(1)=y(a); x=a:h:b; for i=1:n y1=y(i)+h*feval(f,x(i),y(i)); y2=y(i)+h*feval(f,x(i+1),y1); y(i+1)=(y1+y2)/2; end E=[x',y']; ----------------------------------------------------------------------------------------------------------------------


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3.2. Heun 方法

功能：通过计算 yk+1=yk+(h/2)*(f(tk,yk)+f(tk+1,yk+h*f(tk,yk)))求 [a,b]上的初值问题

y'=f(t,y),y(a)=y0,的近似解，其中 k=0,1,...M-1 ---------------------------------------------------------- function H=henu(f,a,b,ya,M) % input - f is the function entered as a string 'f' - a and b are the left and right end points - ya is the initial condition y(a) - M is the number of steps % output - H=[T' Y'] where T is the vector of abscissas and Y is the vector of ordinates h=(b-a)/M; T=zeros(1,M+1); Y=zeros(1,M+1); T=a:h:b; Y(1)=ya; for i=1:M k1=feval(f,T(i),Y(i)); k2==feval(f,T(i+1),Y(i)+h*k1); Y(i+1)=Y(i)+(h/2)*(k1+k2); end H=[T' Y']; ----------------------------------------------------------------------------------------------------------------------


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3.3. 四次 Taylor 方法

功能：通过计算 y',y'',y'''和 y''''，并在每一步中适用泰勒多项式，求[a,b]上的初值问题

y'=f(t,y),y(a)=y0 的近似解 -------------------------------------------------- function T4=taylor(df,a,b,ya,M) % input - df=[y' y'' y''' y''''] entered as a string 'df' ,where y'=f(t,y) % - a and b are the left and right end points % - ya is the initial condition y(a) % - M is the number of steps % output - T4=[T' Y'] where T is the vector of abscissas and Y is the vector of ordinates h=(b-a)/M; T=zeros(1,M+1); Y=zeros(1,M+1); T=a:h:b; Y(1)=ya; for i=1:M D=feval(df,T(i),Y(i)); Y(i+1)=Y(i)+h*(D(1)+h*(D(2)/2+h*(D(3)/6+h*D(4)/24))); end T4=[T' Y']; ----------------------------------------------------------------------------------------------------------------------

3.4. 四阶 Runge-Kutta 法


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功能：用四阶 Runge-Kutta 法求解常微分方程 --------------------------------------------- function R=Rungkuta4(f, a, b, n, ya) % f:微分方程右端函数句柄 % a,b:自变量取值区间的两个端点 % n:区间等分的个数 % ya:函数初值 y(a) % R=[x',y']:自变量 X 和解 Y 所组成的矩阵 h=(b-a)/n; x=zeros(,n+1); y=zeros(1,n+1); x=a:h:b; y(1)=ya; for i=1:n k1=h*feval(f,x(i),y(i)); k2=h*feval(f,x(i)+h/2,y(i)+k1/2); k3=h*feval(f,x(i)+h/2,y(i)+k2/2); k4=h*feval(f,x(i)+h,y(i)+k3); y(i+1)=y(i)+(k1+2*k2+2*k3+k4)/6; end R=[x',y']; ----------------------------------------------------------------------------------------------------------------------


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3.5. Runge-Kutta-Fehlbrg 法

功能：用误差控制和步长方法求解[a,b]上的初值问题 y'=f(t,y),y(a)=y0 的近似解 ------------------------------------------------------------------ function R=rkf45(f,a,b,ya,M,tol) % input - f is the function entered as a string 'f' % - a and b are the left and right end points % - ya is the initial condition y(a) % - M is the number of steps % -tol is the tolerance % output - R=[T' Y'] where T is the vector of abscissas and Y is the vector of ordinates a2=1/4;a3=3/8;a4=12/13;a5=1;a6=1/2; b2=1/4;b3=3/32;b4=1932/2197;b5=439/216;b6=-8/27; c3=9/32;c4=-7200/2197;c5=-8;c6=2; d4=7296/2197;d5=3680/513;d6=-3544/2565; e5=-845/4104;e6=1859/4104; f6=-11/40; r1=1/360;r3=-128/4275;r4=-2197/75240;r5=1/50;r6=2/55; n1=25/216;n3=1408/2565;n4=2197/4104;n5=-1/5; big=1e15; h=(b-a)/M; hmin=h/64;hmax=64*h;max1=200; Y(1)=ya;T(1)=a; j=1;


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br=b-0.00001*abs(b); while (T(j)<b) if ((T(j)+h)>br) h=b-T(j); end k1=h*feval(f,T(j),Y(j)); y2=Y(j)+b2*k1; if big<abs(y2),break;end k2=h*feval(f,T(j)+a2*h,y2); y3=Y(j)+b3*k1+c3*k2; if big<abs(y3),break;end k3=h*feval(f,T(j)+a3*h,y3); y4=Y(j)+b4*k1+c4*k2+d4*k3; if big<abs(y4),break;end k4=h*feval(f,T(j)+a4*h,y4); y5=Y(j)+b5*k1+c5*k2+d5*k3+e5*k4; if big<abs(y5),break;end k5=h*feval(f,T(j)+a5*h,y5); y6=Y(j)+b6*k1+c6*k2+d6*k3+e6*k4+f6*k5; if big<abs(y6),break;end k6=h*feval(f,T(j)+a6*h,y6); err=abs(r1*k1+r3*k3+r4*k4+r5*k5+r6*k6);


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ynew=Y(j)+n1*k1+n3*k3+n4*k4+n5*k5; if ((err<tol) | (h<2*hmin) Y(j+1)=ynew; if ((T(j)+h)>br) T(j+1)=b; else T(j+1)=T(j)+h; end j=j+1; end if (err==0) s=0; else s=0.84*(tol*h/err)^0.25; end if ((s<0.75) & (h>2*hmin)) h=h/2; end if ((s>1.50) & (2*h<hmax)) h=2*h; end if ((big<abs(Y(j))) | (max1==j)),break;end


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M=j; if (b>T(j)) M=j+1; else M=j; end end R=[T' Y']; ----------------------------------------------------------------------------------------------------------------------

3.6. 二阶 Adams 预报校正系统

功能：用二阶 Adams 预报校正系统求解常微分方程 ---------------------------------------------------- function A=Adams(f,a,b,n,ya) % f:微分方程右端函数句柄 % a,b:自变量取值区间的两个端点 % n:区间等分的个数 % ya:函数初值 y(a) % A=[x',y']:自变量 X 和解 Y 所组成的矩阵 h=(b-a)/n; x=zeros(1,n+1); y=zeros(1,n+1); x=a:h:b;


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y(1)=ya; for i=1:n if i==1 y1=y(i)+h*feval(f,x(i),y(i)); y2=y(i)+h*feval(f,x(i+1),y1); y(i+1)=(y1+y2)/2; dy1=feval(f,x(i),y(i)); dy2=feval(f,x(i+1),y(i+1); else y(i+1)=y(i)+h*(3*dy2-dy1)/2; P=feval(f,x(i+1),y(i+1)); y(i+1)=y(i)+h*(P+dy2)/2; dy1=dy2; dy2=feval(f,x(i+1),y(i+1)); end end A=[x',y']; ----------------------------------------------------------------------------------------------------------------------

3.7. 改进的四阶 Adams 预报校正系统

功能：用改进的 Adams 预报校正系统求解常微分方程 --------------------------------------------------- function A=CAdams4PC(f,a,b,n,ya)


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% f:微分方程右端函数句柄 % a,b:自变量取值区间的两个端点 % n:区间等分的个数 % ya:函数初值 y(a) % A=[x',y']:自变量 X 和解 Y 所组成的矩阵 if n<4 break: end h=(b-a)/n; x=zeros(1,n+1); y=zeros(1,n+1); x=a:h:b; y(1)=ya; F=zero(1,4); for i=1:n if i<4 k1=feval(f,x(i),y(i)); k2=feval(f,x(i)+h/2,y(i)+(h/2)*k1); k3=feval(f,x(i)+h/2,y(i)+(h/2)*k2); k4=feval(f,x(i)+h,y(i)+h*k3); y(i+1)=y(i)+(h/6)*(k1+2*k2+2*k3+k4); elseif i==4


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F=feval(f,x(i-3:i),y(i-3:i)); py=y(i)+(h/24)*(F*[-9,37,-59,55]'); p=feval(f,x(i+1),py); F=[F(2) F(3) F(4) p]; y(i+1)=y(i)+(h/24)*(F*[1,5,-19,9]'); p=py;c=y(i+1); else F=feval(f,x(i-3:i),y(i-3:i)); py=y(i)+(h/24)*(F*[-9,37,-59,55]'); my=py-251*(p-c)/270; m=feval(f,x(i+1),my); F=[F(2) F(3) F(4) m]; cy=y(i)+(h/24)*(F*[1,5,-19,9]'); y(i+1)=cy+19*(py-cy)/270; p=py;c=cy; end end A=[x',y']; ----------------------------------------------------------------------------------------------------------------------

3.8. Milne-Simpson 方法

功能：用预报子：p(k+1)=y(k-3)+(4h/3)*(2f(k-2)-f(k-1)+2*f(k)) 和校正子：y(k+1)=y(k-1)+(h/3)*(f(k-1)+4f(k)+f(k+1)),求区间[a,b]上的初值问题


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y'=f(t,y),y(a)=y0 的近似解 -------------------------------------------------- function M=miline(f,T,Y) % input - f is the function entered as a string 'f' % - T is the vector of abscissas % - Y is the vector of ordinates % Remark: The first four coordinates of T and Y must have starting values obtained with RK4 % output - M=[T' Y'] where T is the vector of abscissas and Y is the vector of ordinates n=length(T); if n<5,break;end F=zeros(1,4); F=feval(f,T(1:4),Y(1:4)); h=T(2)-T(1); pold=0; yold=0; for k=4:n-1 pnew=Y(k-3)+(4*h/3)*(F(2:4)*[2 -1 2]'); pmod=pnew+28*(yold-pold)/29; T(k+1)=T(1)+h*k; F=[F(2) F(3) F(4) feval(f,T(k+1),pmod)]; Y(k+1)=Y(k-1)+(h/3)*(F(2:4)*[1 4 1]'); pold=pnew; yold=Y(k+1);


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F(4)=feval(f,T(k+1),Y(k+1)); end M=[T' Y']; ----------------------------------------------------------------------------------------------------------------------

3.9. Hamming 方法

功能：用预报子： p(k+1)=y(k-3)+(4h/3)*(2f(k-2)-f(k-1)+2*f(k)) 和校正子：

y(k+1)=(y(k)-y(k-2))/8+(3h/8)*(-f(k-1)+2f(k)+f(k+1)), 求区间 [a,b] 上的初值问题

y'=f(t,y),y(a)=y0 的近似解 ---------------------------------------------------------------- function H=hamming(f,T,Y) % input - f is the function entered as a string 'f' % - T is the vector of abscissas % - Y is the vector of ordinates % Remark: The first four coordinates of T and Y must have starting values obtained with RK4 % output - H=[T' Y'] where T is the vector of abscissas and Y is the vector of ordinates n=length(T); if n<5,break;end F=zeros(1,4); F=feval(f,T(1:4),Y(1:4)); h=T(2)-T(1); pold=0; cold=0; for k=4:n-1


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pnew=Y(k-3)+(4*h/3)*(F(2:4)*[2 -1 2]'); pmod=pnew+112*(cold-pold)/121; T(k+1)=T(1)+h*k; F=[F(2) F(3) F(4) feval(f,T(k+1),pmod)]; cnew=(9*Y(k)-Y(k-2)+3*h*(F(2:4)*[-1 2 1]'))/8; Y(k+1)=cnew+9*(pnew-cnew)/121; pold=pnew; cold=cnew; F(4)=feval(f,T(k+1),Y(k+1)); end H=[T' Y']; ----------------------------------------------------------------------------------------------------------------------

3.10. 微分方程组四阶 Runge-Kutta 解法

功能：求区间[a,b]上的微分方程组 x1'(t)=f1(t,x1(t),......xn(t)) . . . . . . . . . xn'(t)=fn(t,x1(t),......xn(t)) --------------------------------------------- function [T,Z]=rks4(F,a,b,Za,M) % input - F is the function entered as a string 'f' % - a and b are the left and right end points % - Za=[x(a) y(a)] are the initial conditions


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% - M is the number of steps % output - T is the vector of steps - Z=[x1(t) ......xn(t)] where xk(t) is the approximation to the kth dependent variable h=(b-a)/M; T=zeros(1,M+1); Z=zeros(M+1,length(Za)); T=a:h:b; Z(1,:)=Za; for i=1:M k1=h*feval(F,T(i),Z(i)); k2=h*feval(F,T(i)+h/2,Z(i,:)+k1/2); k3=h*feval(F,T(i)+h/2,Z(i,:)+k2/2); k4=h*feval(F,T(i)+h,Z(i,:)+k3); Z(i+1,:)=Z(i,:)+(k1+2*k2+2*k3+k4)/6; end ----------------------------------------------------------------------------------------------------------------------

3.11. 线性打靶法

功能: 求区间[a,b]上的边值问题 x''=p(t)*x'(t)+q(t)*x(t)+r(t),x(a)=alpha,x(b)=beta 的近似解，

调用四阶 RK 方法 -------------------------------------------------- function L=linsht(F1,F2,a,b,alpha,beta,M) % input - F1 and F2 are the system of first-order equations representing the I.V.P,'s (9)and (10),respectively,input as


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% strings 'F1' 'F2' % - a and b are the left and right end points % - alpha=x(a) and beta=x(b) :boundary conditions % - M is the number of steps % output - L=[T' x] where T' is the (M+1)*1 vector of ordinates %solve the system F1 Za=[alpha,0]; [T,Z]=rks4(F1,a,b,Za,M); U=Z(:,1); %solve the system F2 Za=[0,1]; [T,Z]=rks4(F2,a,b,Za,M); V=Z(:,1); %calculate the solution to the boundary value problem X=U+(beta-U(M+1)*V)/V(M+1); L=[T' X]; ----------------------------------------------------------------------------------------------------------------------

3.12. 求解三对方程组的程序

功能：求解三对角方程组 CX=B,其中 C 为三对角矩阵 --------------------------------------------------- function X=trisys(A,D,C,B) %Input - A is the subdiagnonal of the coefficient matrix


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% - D is the main diagnonal of the coefficient matrix % - C is the superdiagnonal of the coefficient matrix % - B is the constant vector of the linear system %output - X is the solution vector N=length(B); for k=2:N mult=A(k-1)/D(k-1); D(k)=D(k)-mult*C(k-1); B(k)=B(k)-mult*B(k-1); end X(N)=B(N)/D(N); for k=N-1:-1:1 X(k)=(B(k)-C(k)*X(k+1))/D(k); end ----------------------------------------------------------------------------------------------------------------------

3.13. 有限差分法

功能：用 O(h^2) 阶的有限差分方法求区间 [a ， b] 上的边值问题

x''=p(t)*x'(t)+q(t)*x(t)+r(t),x(a)=alpha,x(b)=beta 的逼近。 -------------------------------------------------------- function F=linsht(p,q,r,a,b,alpha,beta,N) % input - p,q and r are the coefficient function,input as strings 'p' ,'q' , 'r' % - a and b are the left and right end points % - alpha=x(a) and beta=x(b) :boundary conditions % - N is the number of steps


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% output - F=[T' X'] where T' is the 1*N vector of abscissas and X'is the 1*N vector of ordinates T=zeros(1,N+1); X=zeros(1,N+1); Va=zeros(1,N-2); Vb=zeros(1,N-1); Vc=zeros(1,N-2); Vd=zeros(1,N-1); h=(b-a)/N; Vt=a+h:a+h(N-1); Vb=-h^2*feval(r,Vt); Vb(1)=Vb(1)+(1+h/2*feval(p,Vt(1)))*alpha; Vb(N-1)=Vb(N-1)+(1-h/2*feval(p,Vt(N-1)))*beta; Vd=2+h^2*feval(q,Vt); Vta=Vt(1,2:N-1); Va=-1-h/2*feval(p,Vta); Vtc=Vt(1,1:N-2); Vc=-1+h/2*feval(p,Vtc); X=trisys(Va,Vd,Vc,Vb); T=[a,Vt,b]; X=[alpha,X,beta]; F=[T' X];


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附：三阶、四阶、五阶 Rungkuta 法

三阶 Rungkuta 法 function Y = Rungkuta3(odefun,tspan,y0,varargin) % Rungkuta3 Solve differential equations with a non-adaptive method of order 3. % Y = Rungkuta3 (ODEFUN,TSPAN,Y0) with TSPAN = [T1, T2, T3, ... TN] integrates % the system of differential equations y' = f(t,y) by stepping from T0 to % T1 to TN. Function ODEFUN(T,Y) must return f(t,y) in a column vector. % The vector Y0 is the initial conditions at T0. Each row in the solution % array Y corresponds to a time specified in TSPAN. % Y = Rungkuta3 (ODEFUN,TSPAN,Y0,P1,P2...) passes the additional parameters % P1,P2... to the derivative function as ODEFUN(T,Y,P1,P2...). % This is a non-adaptive solver. The step sequence is determined by TSPAN % but the derivative function ODEFUN is evaluated multiple times per step. % The solver implements the Bogacki-Shampine Runge-Kutta method of order 3. % Example % tspan = 0:0.1:20; % y = Rungkuta3 (@vdp1,tspan,[2 0]); % plot (tspan,y(:,1)); % solves the system y' = vdp1 (t,y) with a constant step size of 0.1, % and plots the first component of the solution. if ~isnumeric(tspan) error('TSPAN should be a vector of integration steps.'); end if ~isnumeric(y0) error ('Y0 should be a vector of initial conditions.'); end h = diff (tspan); if any(sign(h(1))*h <= 0) error('Entries of TSPAN are not in order.') end try f0 = feval(odefun,tspan(1),y0,varargin{:}); catch msg = ['Unable to evaluate the ODEFUN at t0,y0. ',lasterr]; error(msg); end


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y0 = y0(:); % Make a column vector. if ~isequal(size(y0),size(f0)) error('Inconsistent sizes of Y0 and f(t0,y0).'); end neq = length(y0); N = length(tspan); Y = zeros(neq,N); F = zeros(neq,3); Y(:,1) = y0; for i = 2:N ti = tspan(i-1); hi = h(i-1); yi = Y(:,i-1); F(:,1) = feval(odefun,ti,yi,varargin{:}); F(:,2) = feval(odefun,ti+0.5*hi,yi+0.5*hi*F(:,1),varargin{:}); F(:,3) = feval(odefun,ti+0.75*hi,yi+0.75*hi*F(:,2),varargin{:}); Y(:,i) = yi + (hi/9)*(2*F(:,1) + 3*F(:,2) + 4*F(:,3)); end Y = Y.'; ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 四阶龙格－库塔法 function Y =Rungkuta4(odefun,tspan,y0,varargin) % Rungkuta4 Solve differential equations with a non-adaptive method of order 4. % Y = Rungkuta4 (ODEFUN,TSPAN,Y0) with TSPAN = [T1, T2, T3, ... TN] integrates % the system of differential equations y' = f(t,y) by stepping from T0 to % T1 to TN. Function ODEFUN(T,Y) must return f(t,y) in a column vector. % The vector Y0 is the initial conditions at T0. Each row in the solution % array Y corresponds to a time specified in TSPAN. % Y = Rungkuta4 (ODEFUN, TSPAN, Y0, P1, P2...) passes the additional parameters % P1, P2... to the derivative function as ODEFUN (T, Y, P1, P2...). % This is a non-adaptive solver. The step sequence is determined by TSPAN % but the derivative function ODEFUN is evaluated multiple times per step. % The solver implements the classical Runge-Kutta method of order 4. % Example % tspan = 0:0.1:20; % y = Rungkuta4 (@vdp1,tspan,[2 0]); % plot(tspan,y(:,1)); % solves the system y' = vdp1(t,y) with a constant step size of 0.1, % and plots the first component of the solution.


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if ~isnumeric(tspan) error('TSPAN should be a vector of integration steps.'); end if ~isnumeric(y0) error('Y0 should be a vector of initial conditions.'); end h = diff(tspan); if any(sign(h(1))*h <= 0) error('Entries of TSPAN are not in order.') end try f0 = feval(odefun,tspan(1),y0,varargin{:}); catch msg = ['Unable to evaluate the ODEFUN at t0,y0. ',lasterr]; error(msg); end y0 = y0(:); % Make a column vector. if ~isequal(size(y0),size(f0)) error('Inconsistent sizes of Y0 and f(t0,y0).'); end neq = length(y0); N = length(tspan); Y = zeros(neq,N); F = zeros(neq,4); Y(:,1) = y0; for i = 2:N ti = tspan(i-1); hi = h(i-1); yi = Y(:,i-1); F(:,1) = feval(odefun,ti,yi,varargin{:}); F(:,2) = feval(odefun,ti+0.5*hi,yi+0.5*hi*F(:,1),varargin{:}); F(:,3) = feval(odefun,ti+0.5*hi,yi+0.5*hi*F(:,2),varargin{:}); F(:,4) = feval(odefun,tspan(i),yi+hi*F(:,3),varargin{:}); Y(:,i) = yi + (hi/6)*(F(:,1) + 2*F(:,2) + 2*F(:,3) + F(:,4)); end Y = Y.'; ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


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五阶龙格－库塔法 function Y = Rungkuta5(odefun,tspan,y0,varargin) % Rungkuta5Solve differential equations with a non-adaptive method of order 5. % Y = Rungkuta5 (ODEFUN, TSPAN, Y0) with TSPAN = [T1, T2, T3 ... TN] integrates % the system of differential equations y' = f(t,y) by stepping from T0 to % T1 to TN. Function ODEFUN(T,Y) must return f(t,y) in a column vector. % The vector Y0 is the initial conditions at T0. Each row in the solution % array Y corresponds to a time specified in TSPAN. % Y = Rungkuta5 (ODEFUN,TSPAN,Y0,P1,P2...) passes the additional parameters % P1,P2... to the derivative function as ODEFUN(T,Y,P1,P2...). % This is a non-adaptive solver. The step sequence is determined by TSPAN % but the derivative function ODEFUN is evaluated multiple times per step. % The solver implements the Dormand-Prince method of order 5 in a general % framework of explicit Runge-Kutta methods. % Example % tspan = 0:0.1:20; % y = Rungkuta5 (@vdp1,tspan,[2 0]); % plot(tspan,y(:,1)); % solves the system y' = vdp1(t,y) with a constant step size of 0.1, % and plots the first component of the solution. if ~isnumeric(tspan) error('TSPAN should be a vector of integration steps.'); end if ~isnumeric(y0) error('Y0 should be a vector of initial conditions.'); end h = diff(tspan); if any(sign(h(1))*h <= 0) error('Entries of TSPAN are not in order.') end try f0 = feval(odefun,tspan(1),y0,varargin{:}); catch msg = ['Unable to evaluate the ODEFUN at t0,y0. ',lasterr]; error(msg); end y0 = y0 (:); % Make a column vector.


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if ~isequal(size(y0),size(f0)) error('Inconsistent sizes of Y0 and f(t0,y0).'); end neq = length(y0); N = length (tspan); Y = zeros (neq,N); % Method coefficients -- Butcher's tableau % C | A % --+--- % | B C = [1/5; 3/10; 4/5; 8/9; 1]; A = [ 1/5, 0, 0, 0, 0;3/40, 9/40, 0, 0, 0;44/45 -56/15, 32/9, 0, 0;... 19372/6561, -25360/2187, 64448/6561, -212/729, 0; 9017/3168, -355/33, 46732/5247, 49/176, -5103/18656]; B = [35/384, 0, 500/1113, 125/192, -2187/6784, 11/84]; % More convenient storage A = A.'; B = B(:); nstages = length(B); F = zeros (neq,nstages); Y(:,1) = y0; for i = 2:N ti = tspan(i-1); hi = h(i-1); yi = Y(:,i-1); % General explicit Runge-Kutta framework F(:,1) = feval(odefun,ti,yi,varargin{:}); for stage = 2:nstages tstage = ti + C(stage-1)*hi; ystage = yi + F(:,1:stage-1)*(hi*A(1:stage-1,stage-1)); F(:,stage) = feval(odefun,tstage,ystage,varargin{:}); end Y(:,i) = yi + F*(hi*B); end Y = Y.'; ======================================================================


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第四章方程求根

4.1. 二分法

功能：用二分法求解非线性方程 f(x)=0 在区间[a,b]内的根 ------------------------------------------------------- function [x,k]=demimethod(a,b,f,emg) % a,b :求解区间的两个端点 % f :所求方程的函数名 % emg ：精度指标 % x：所求近似解 % k: 循环次数 fa=feval(f,a); fab=feval(f,(a+b)/2); k=0; while abs(b-a)>emg if fab==0 x=(a+b)/2; return; elsif fa*fab<0 b=(a+b)/2; else a=(a+b)/2;


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end fa=feval(f,a); fab=feval(f,(a+b)/2); k=k+1; end x=(a+b)/2; ----------------------------------------------------------------------------------------------------------------------

4.2. 开方法

功能：求实数的开方运算 ---------------------------- function y=Kaifang(a,eps,x0) % a：被开放数 % eps :精度指标 % x0 :初值 % y:a 的开放 x(1)=x0; x(2)=(x(1)+a/x(1))/2; k=2; while abs(x(k)-x(k-1))>eps x(k+1)=(x(k)+a/x(k))/2; k=k+1; end y=x';


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----------------------------------------------------------------------------------------------------------------------

4.3. Newton 下山法

功能：用牛顿下山法求解非线性方程 f(x)=0 的根 ----------------------------------------------- function [x,k]=Mendnewton(f,x0,emg) % f ：非线性方程 % x0 :迭代初值 % emg :精度指标 % k,u : 迭代次数和下山因子 [f1,d1]=feval(f,x0); k=1; x(1)=x0; x(2)=x(1)-f1/d1; while abs(f1)>emg u=1; k=k+1; [f1,d1]=feval(f,x(k)); x(k+1)=x(k)-u*f1/d1; [f2,d2]=feval(f,x(k+1)); while abs(f2)>abs(f1) u=u/2; x(k+1)=x(k)-u*f1/d1;


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[f2,d2]=feval(f,x(k+1)); end end ------------------------------------------------------------------------------------------------------------------

4.4. 快速弦截法

功能：有快速弦截法求解非线性方程 f(x)=0 的根 ---------------------------------------------------- function [x,k]=Fast_chord(f,x1,x2,emg) % f:非线性方程函数 % x1,x2 : 迭代初值 % emg :精度指标 % k :循环次数 k=1; y1=feval(f,x1); y2=feval(f,x2); x(k)=x2-(x2-x1)*y2/(y2-y1); y(k)=feval(f,x(k)); k=k+1; x(k)=x(k-1)-(x(k-1)-x2)*y(k-1)/(y(k-1)-y2); while abs(x(k)-x(k-1))>emg y(k)=feval(f,x(k)); x(k+1)=x(k)-(x(k)-x(k-1))*y(k)/(y(k)-y(k-1)); k=k+1; end ----------------------------------------------------------------------------------------------------------------------


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4.5. 不动点迭代法

功能：求方程 x=g(x)的近似值，起始值为 p0，迭代式：p(n+1)=g(p(n)) ---------------------------------------------------------------------------------------------------------------------- function [k,p,err,P]=fixpt(g,p0,tol,max1) % input - g is the iteration function input as a string 'g' % - p0 is the initial guess for the fixed point % - tol is the tolerance % - max1 is the maximum number of interations %outpt - k is the number of iterations that were carried out % - p is the approximation to the fixed point % - err is the error in the approximation % - P contains the sequence {pn} P(1)=p0; for k=2:max1 P(k)=feval(g,P(k-1)); err=abs(P(k)-P(k-1)); relerr=err/(abs(P(k))+eps); p=P(k); if (err<tol) | (relerr<tol),break;end end if k==max1 disp('maximum number of iterations exceeded') end P=P'; ----------------------------------------------------------------------------------------------------------------------


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4.6. 试值法或试位法

功能：求解方程 f(x)=0 在[a,b]内的根，前提是 f(x)连续，且 f(a)*f(b)<0 ------------------------------------------------------- function [c,err,yc]=regula(f,a,b,delta,epsilon,max1) % input - f is the iteration function input as a string 'f' % - a and b are the left and right points % - delta is the tolerance for the zero % - epsilon is the tolerance for the value of f at the zero % - max1 is the maximum number of interations %outpt - c is the zero % - err is the error estimate for c % - yc=f(c) ya=feval(f,a); yb=feval(f,b); if ya*yb>0 disp('Note f(a)*f(b)>0'), break; end for k=1:max1 dx=yb*(b-a)/(yb-a); c=b-dx; ac=c-a; yc=feval(f,c);


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if yc==0 ,break; elseif yb*yc<0 b=c; else a=c; ya=yc; end dx=min(abs(dx),ac); if abs(dx)<delta,break;end if abs(yc)<epsilon,break;end end c； err=abs(b-a)/2; yc=feval(f,c); ---------------------------------------------------------------------------------------------------------------------

4.7. Steffensen 加速法

功能：给定初始近似值 p0，快速寻找不动点方程 x=g(x)的解，假设 g(x)和 g'(x)是连续的，

且|'g(x)|<1,通常的不动点迭代缓慢线性收敛到 p ------------------------------------------ function [P,Q]=steff(f,df,p0,delta,epsilon,max1) % input - f is the object function input as a string 'f' % - df is the derivative of f input as a string 'f' % - delta is the tolerance for p0 % - epsilon is the tolerance for the function value of y % - max1 is the maximum number of interations %outpt - P is the Steffensen approximation to the zero % - Q is the matrix containing the Steffensen sequence


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R=zeros(max1,3); R(1,1)=p0; for k=1:max1 for j=2:3 nrdenom=feval(df,R(k,j-1)); if nrdenom==0 disp('division by zero in Newton method'); break; else R(k,j)=R(k,j-1)-feval(f,R(k,j-1))/nrdenom; end aadenom=R(k,3)-2*R(k,2)+R(k,1); if aadenom==0 disp('division by zero in Aitken's Acceleration); break; else R(k+1,)=R(k,1)-(R(k,2)-R(k,1))^2/aadenom; end end if (nrdenom==0) | (aadenom==0) break; end


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err=abs(R(k,1)-R(k+1,1)); relerr=err/(abs(R(k+1,1))+delta); y=feval(f,R(k+1,1)); if (err<delta) | (relerr<delta) | (y<epsilon) P=R(k+1,1); Q=R(1:k+1,:); break; end end ----------------------------------------------------------------------------------------------------------------------

4.8. Muller 法

功能：给定 3 个初始近似值 p0，p1,p2，求方程 f(x)=0 的根 ----------------------------------------------------------------- function [p,y,err]=muller(f,p0,p1,p2,delta,epsilon,max1) % input - f is the object function input as a string 'f' % - p0,p1,p2 are the initial approximations % - delta is the tolerance for p0,p1,p2 % - epsilon is the tolerance for the function values y % - max1 is the maximum number of interations %outpt - p is the muller approximation to the zero of f % - y is the function value y=f(p) % - err is the error in the approximation of p P=[p0 p1 p2];


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Y=feval(f,P); for k=1:max1 h0=P(1)-P(3); h1=P(2)-P(3); e0=Y(1)-Y(3); e1=Y(2)-Y(3); c=Y(3); denom=h1*h0^2-h0*h1^2; a=(e0*h1-e1*h0)/denom; b=(e1*h0^2-e0*h1^2)/denom; if b^2-4*a*c>0 disc=sqrt(b^2-4*a*c); else disc=0; end if b<0 disc=-disc; end z=-2*c/(b+disc); p=P(3)+z; if abs(p-P(2))<abs(p-P(1)) Q=[P(2) P(1) P(3)];


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P=Q; Y=feval(f,P); end if abs(p-P(3))<abs(p-P(2)) R=[P(1) P(3) P(1)]; P=R; Y=feval(f,P); end P(3)=P; Y(3)=feval(f,P(3)); y=Y(3); err=abs(z); relerr=err/(abs(p)+dela); if (err<delta) | (relerr<delta) |(abs(y)<epsilon) break; end end


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第五章线性方程组的迭代法

5.1. Jacobi 迭代法

功能：用 Jacobi 迭代法求解线性方程组 A*x=b ----------------------------------------------- function [x,k]=Jacobimethod(A,b,x0,N,emg) % A:线性方程组左端矩阵 % b:线性方程组右端向量 % x0:迭代初值 % N:迭代次数上界，若迭代次数大于 n，则迭代失败 % emg:精度指标 % k:迭代次数 % x:用迭代法求得的线性方程组的近似解 n=length(A); x1=zeros(n,1);x2=zeros(n,1); x1=x0; k=0; r=max(abs(b-A*x1)); while r>emg for i=1:n sum=0; for j=1:n if i~=j


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sum=sum+A(i,j)*x1(j); end end x2(i)=(b(i)-sum)/A(i,i); end r=max(abs(x2-x1)); x1=x2; k=k+1; if k>N disp('迭代失败，返回')； return; end end ---------------------------------------------------- 附：Jacobi 迭代程序的另一种写法 function X=jacobi(A,B,P,delta,max1) % input - A is an N*N nonsingular matrix - B is an N*1 matrix - P is an N*1 matrix :the initial guess - delta is the tolerance for P - max1 is the maximum number of iterations % output - X is an N*1 matrix : the jacobi approximation to the solution of AX=B


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N=length(B); for k=1:max1 for j=1:N X(j)=(B(j)-A(j,[1;j-1,j+1:N])*P([1:j+1,j+1:N]))/A(j,j); end err=abs(norm(X'-P)); relerr=err/(norm(X)+eps); P=X'; if (err<delta) |(relerr<delta) break; end end X=X'; ----------------------------------------------------------------------------------------------------------------------

5.2. Gauss-Seidel 迭代

功能：用 Gauss-Seidel 迭代求解线性方程组 A*x=b ---------------------------------------------------------------------------------------------------------------------- function [x,k]=Gaussmethod(A,b,x0,N,emg) % A:线性方程组左端矩阵 % b:线性方程组右端向量 % x0:迭代初值 % N:迭代次数上界，若迭代次数大于 n，则迭代失败 % emg:精度指标


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% k:迭代次数 % x:用迭代法求得的线性方程组的近似解 n=length(A); x1=zeros(n,1);x2=zeros(n,1); x1=x0; r=max(abs(b-A*x1)); k=0; while r>emg for i=1:n sum=0; for j=1:n if j>i sum=sum+A(i,j)*x1(j); elseif j<i sum=sum+A(i,j)*x2(j); end end x2(i)=(b(i)-sum)/A(i,i); end r=max(abs(x2-x1)); x1=x2; k=k+1;


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if k>N disp('迭代失败，返回')； return; end end x=x1; ---------------------------------------------------------------- 附：Gauss-Seidel 迭代程序的另一种写法 function X=gseid(A,B,P,delta,max1) % input - A is an N*N nonsingular matrix - B is an N*1 matrix - P is an N*1 matrix :the initial guess - delta is the tolerance for P - max1 is the maximum number of iterations % output - X is an N*1 matrix : the jacobi approximation to the solution of AX=B N=length(B); for k=1:max1 for j=1:N if j==1 X(1)=(B(1)-A(1,2:N)*P(2:N))/A(1,1); elseif j==N X(N)=(B(N)-A(N,1:N-1)*(X(1:N))')/A(N,N); else


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X(j)=(B(j)-A(j,1:j-1)*X(1:j-1)'-A(j,j+1:N)*P(j+1:N)/A(j,j); end err=abs(norm(X'-P)); relerr=err/(norm(X)+eps); P=X'; if (err<delta) |(relerr<delta) break; end end X=X'; ----------------------------------------------------------------------------------------------------------------------

5.3.非线性 Seidel 迭代

功能：求解非线性不动点方程组 X=G(X),给定初始近似值 P0,并生成序列{Pk},收敛到解 P ----------------------------------------------- function [P,iter]=seidel(G,P,delta,max1) % input - G is an N*N nonlinear matrix % - P is the initial guess at solution % - delta is the error bound % - max1 is the maximum number of iterations % output - P is the seidel approimation to the solution % - iter is the number of iterations required


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N=length(P); for k=1:max1 X=P; for j=1:N A=feval('G',X); X(j)=A(j); end err=abs(norm(X-P)); relerr=err/(norm(X)+eps); P=X; iter=k; if (err<delta) |(relerr<delta) break; end end ----------------------------------------------------------------------------------------------------------------------

5.4. 牛顿-拉夫森法

功能：求解非线性方程组 F(X)=0，给定初始近似值 P0，并生成序列{Pk},收敛到解 P ----------------------------------------------- function [P,iter,err]=newdim(F,JF,P,delta,epsilon,max1) % input - F is the system saved as the M-file F.m % - JF is the Jacobian of F saved as the M-file JF.m


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% - P is the initial approximation to the solution % - delta is the tolerance for P % - max1 is the maximum number of iterations % output - P is the approimation to the solution % - iter is the number of iterations required % - err is the error estimate for P Y=feval(F,P); for k=1:max1 J=feval(JF,P); Q=P-(J\Y); Z=feval(F,Q); err=norm(Q-P); err=abs(norm(X-P)); relerr=err/(norm(Q)+eps); P=Q; Y=Z; iter=k; if (err<delta) |(relerr<delta) |(abs(Y)<epsilon) break; end end ----------------------------------------------------------------------------------------------------------------------


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5.5. 超松弛迭代

功能：用超松弛迭代法求解线性方程组 --------------------------------------------- function [x,k]=SORmethod(A,b,x0,N,emg,w) % A:线性方程组左端矩阵 % b:线性方程组右端向量 % x0:迭代初值 % N:迭代次数上界，若迭代次数大于 n，则迭代失败 % emg:精度指标 % w:松弛因子 % x:用迭代法求得的线性方程组的近似解 n=length(A); x1=zeros(n,1);x2=zeros(n,1); x1=x0; r=max(abs(b-A*x1)); k=0; while r>emg for i=1:n sum=0; for j=1:n if j>i sum=sum+A(i,j)*x1(j);


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elseif j<i sum=sum+A(i,j)*x2(j); end end x2(i)=x1(i)+w*(b(i)-sum)/A(i,i); end r=max(abs(x2-x1)); x1=x2; k=k+1; if k>N disp('迭代失败，返回')； return; end end x=x1; ----------------------------------------------------------------------------------------------------------------------

5.6. 对超松弛迭代

功能：用对超松弛(SSOR)迭代法求解线性方程组 -------------------------------------------------- function [x,k]=SSORmethod(A,b,x0,N,emg,w) % A:线性方程组左端矩阵 % b:线性方程组右端向量 % x0:迭代初值


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% N:迭代次数上界，若迭代次数大于 n，则迭代失败 % emg:精度指标 % w:松弛因子 % x:用迭代法求得的线性方程组的近似解 n=length(A); x1=zeros(n,1);x2=zeros(n,1); x3=zeros(n,1); x1=x0; r=max(abs(b-A*x1)); k=0; while r>emg for i=1:n sum=0; for j=1:n if j>i sum=sum+A(i,j)*x1(j); elseif j<i sum=sum+A(i,j)*x2(j); end end x2(i)=(1-w)*x1(i)+w*(b(i)-sum)/A(i,i); end for i=n:-1:1


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sum=0; for j=1:n if j>i sum=sum+A(i,j)*x3(j); elseif j<i sum=sum+A(i,j)*x2(j); end end x3(i)=(1-w)*x2(i)+w*(b(i)-sum)/A(i,i); end r=max(abs(x3-x1)); x1=x3; k=k+1; if k>N disp('迭代失败，返回')； return; end end x=x1;


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第六章线性方程组的直接法

6.1. 追赶法

功能：用追赶法解三对角线性方程组 A*x=f,A 为三对角矩阵 ------------------------------------------------------- function x=threedia(a,b,c,f) % a:三对角矩阵 A 的下对角线元素,a(1)=0 % b:三对角矩阵 A 的对角线元素 % c:三对角矩阵 A 的上对角线元素,c(N)=0 % f:方程组的右端向量 N=length(f); x=zeros(1,N);y=zeros(1,N); d=zeros(1,N);u=zeros(1,N); d(1)=b(1); for i=1:N-1 u(i)=c(i)/d(i); d(i+1)=b(i+1)-a(i+1)*u(i); end % 追的过程 y(1)=f(1)/d(1); for i=2:N y(i)=(f(i)-a(i)*y(i-1))/d(i); end


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% 赶的过程 x(N)=y(N); for i=N-1:-1:1 x(i)=y(i)-u(i)*x(i+1); end ----------------------------------------------------------------------------------------------------------------------

6.2. Cholesky 方法

功能：用 Cholesky 分解法求解对称方程组 A*x=b ----------------------------------------------------- function x=Chol_decompose(A,b) % A:对称矩阵 % b:方程组的右端向量 % L:单位下三角矩阵 % D:单位上三角矩阵 % 对矩阵 A 进行三角分解：A=LDL' N=length(A); L=zeros(N,N);D=zeros(1,N); for i=1:N L(i,i)=1; end D(1)=A(1,1); for i=2:N


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for j=1:i-1 if j==1 L(i,j)=A(i,j)/D(j); else sum1=0; for k=1:j-1 sum1=sum1+ L(i,k)*D(k)*L(j,k); end L(i,j)=(A(i,j)-sum1)/D(j); end end sum2=0; for k=1:i-1 sum2=sum2+L(i,k)^2*D(k); end D(i)=A(i,i)-sum2; end % 分别求解线性方程组 Ly=b;L'x=y/D y=zeros(1,N); y(1)=b(1); for i=2:N sumi=0;


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for k=1:i-1 sumi=sumi+L(i,k)*y(k); end y(i)=b(i)-sumi; end x=zeros(1,N); x(N)=y(N)/D(N); for i=N-1:-1:1 sumi=0; for k=i+1:N sumi=sumi+L(k,i)*x(k); end x(i)=y(i)/D(i)-sumi; end ----------------------------------------------------------------------------------------------------------------------

6.3. 矩阵分解方法

功能：基于 Gauss 消去法的 LU 分解求解线性方程组 A*x=b ------------------------------------------------------ function x=lu_decompose(A,b) % A:系数矩阵 % b:方程组的右端向量 % L:单位下三角矩阵 n=length(b);


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L=eye(n);U=zeros(n,n); x=zeros(n,1);y=zeros(n,1); for i=1:n U(1,i)=A(1,i); if i==1 L(i,1)=1; else L(i,1)=A(i,1)/U(1,1); end end for i=2:n for j=1:n sum=0; for k=1:i-1 sum=sum+ L(i,k)*U(k,j); end U(i,j)=A(i,j)-sum; if j~=n sum=0; for k=1:i-1 sum=sum+L(j+1,k)*U(k,i); end


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L(j+1,1)=(A(j+1,i)-sum)/U(i,i); end end end % 求解线性方程组 Ly=b y(1)=b(1); for i=2:n sum=0; for j=1:k-1 sum=sumi+L(k,j)*y(j); end y(k)=b(k)-sum; end % 解方程组 Ux=y x(n)=y(n)/U(n,n); for i=n-1:-1:1 sum=0; for j=k+1:n sum=sum+U(k,i)*x(j); end x(k)=(y(k)-sum)/U(k,k); end ----------------------------------------------------------------------------------------------------------------------


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6.4. 消去法

功能：用 Gauss 列主元素消去法求解线性方程组 A*x=b -------------------------------------------------------------- function x=Gauss_pivot(A,b) % A:系数矩阵 % b:方程组的右端向量 n=length(b); x=zeros(n,1);c=zeros(1,n);d1=0; for i=1:n-1 max=abs(A(i,i)); m=i; for j=i+1:n if max<abs(A(j,i)); m=j; end end if m~=1 for k=i:n c(k)=A(i,k); A(i,k)=A(m,k); A(m,k)=c(k); end d1=b(i);


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b(i)=b(m); b(m)=d1; end for k=i+1:n for j=i+1:n A(k,j)=A(k,j)-A(i,j)*A(k,j)/A(i,i); end b(k)=b(k)-b(i)*A(k,j)/A(i,i); A(k,i)=0; end end x(n)=b(n)/A(n,n); for i=n-1:-1:1 sum=0; for j=i+1:n sum=sum+ A(i,j)*x(j); end x(i)=(b(i)-sum)/A(i,i); end ======================================================================


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第七章数值优化

7.1. 黄金分割法求极小值

功能：用黄金分割法求 f(x)在区间[a,b]上的近似极小值。当且仅当 f(x)在[a,b]上为单峰

时此方法适用 ----------------------------------------------------------------- function [S,E,G]=golden(f,a,b,delta,epsilon) % input - f is the object function input as a string 'f' % - a and b are the end points of the interval % - delta is the tolerance for the abscissas % - epsilon is the tolerance for the ordinates % output - S=(p,yp) contains the abscissa p and the ordinate yp of the minimum % - E=(dp,dy) contains the error bounds for p and yp % -G is an n*4 matrix : the kth row contains % -[ak ck dk bk]: the values of a,c,d and b at the kth interation r1= (sqrt(5)-1)/2; r2=r1^2; h=b-a; ya=feval(f,a); yb=feval(f,b); c=a+r2*h; d=a+r1*h; yc=feval(f,c);


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yd=feval(f,d); k=1; A(k)=a;B(k)=b;C(k)=c;D(k)=d; while (abs(yb-ya)>epsilon) | (h>delta) k=k+1; if (yc<yd) b=d; yb=yd; d=c; yd=yc; h=b-a; c=a+r2*h; yc=feval(f,c); else a=c; ya=yc; c=d; yc=yd; h=b-a; d=a+r1*h; ya=feval(f,d); end


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A(k)=a;B(k)=b;C(k)=c;D(k)=d; end dp=abs(b-a); dy=abs(yb-ya); p=a; yp=ya; if (yb<ya) p=b; yp=yb; end; G=[A' C' D' B']; S=[p yp]; E=[dp dy]; ----------------------------------------------------------------------------------------------------------------------

7.2. 斐波那契法求极小值

功能：用斐波那契法求 f(x)在区间[a,b]上的近似极小值。当且仅当 f(x)在[a,b]上为单峰时

此方法适用 ----------------------------------------------------------------- function X=fibonacci(f,a,b,tol,e) % input - f is the object function input as a string 'f' % - a and b are the end points of the interval % - tol : length of uncertainy % - e :distinguishability of constant


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% output - X : x and y coordinates of minimum % note : this function calls the m-file fib.m while had been edited i=1; F=1; while F<=(b-a)/tol F=fib(i); i=i+1; end n=i-1; A=zeros(1,n-2);B=zeros(1,n-2); A(1)=a; B(1)=b; c=A(1)+(fib(n-2)/fib(n))*(B(1)-A(1)); d=A(1)+(fib(n-1)/fib(n))*(B(1)-A(1)); k=1; while k=n-3 if feval(f,c)>feval(f,d) A(k+1)=c; B(k+1)=B(k); c=d; d=A(k+1)+(fib(n-k-1)/fib(n-k))*(B(k+1)-A(k+1));


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else A(k+1)=A(k); B(k+1)=d; d=c; c=A(k+1)+(fib(n-k-2)/fib(n-k))*(B(k+1)-A(k+1)); end k=k+1; end if feval(f,c)>feval(f,d) A(n-2)=c; B(n-2)=B(n-3); c=d; d=A(n-2)+(0.5+e)*(B(n-2)-A(n-2)); else A(n-2)=A(N-3); B(n-2)=d; d=c; c=A(n-2)+(0.5+e)*(B(n-2)-A(n-2)); end if feval(f,c)>feval(f,d) a=c; b=B(n-2);


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else a=A(n-2); b=d; end X=[(a+b)/2 feval(f,(a+b)/2)]; ------------------------------------------------------------- % 下面的程序用来生成斐波那契数 function y=fib(n) fz(1)=1; fz(2)=1; for k=3:n fz(k)=fz(k-1)+fz(k-1); end y=fz(n); ----------------------------------------------------------------------------------------------------------------------

7.3. 用 2 次插值求局部最小值

功能：求 f(x)在区间[a,b]上的局部极小值，从初始近似点p0开始，然后在区间[a,p0]和[p0,b]上进行搜索 ------------------------------------------------------------------- function [p,yp,dp,dy,P]=quadmin(f,a,b,delta,epsilon) % input - f is the object function input as a string 'f' % - a and b are the end points of the interval % - delta is the tolerance for the abscissas % - epsilon is the tolerance for the ordinates


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% output - p is the abscissa of the minimum % - yp is the ordinate of the minimum % - dp is the error bound for p % - dy is the error bound for yp % - P is the vector of interations p0=a; maxj=20; maxk=30; big=1e6; err=1; P(k)=p0; cond=0; h=1; if (abs(p0)>1e4),h=abs(p0)/1e4;end while (k<maxk & err>epsilon & cond~=5) f1=feval(f,p0+0.00001)-feval(f,p0-0.00001)/0.00002; if (f1>0),h=-abs(h);end p1=p0+h; p2=p0+2*h; pmin=p0; y0=feval(f,p0); y1=feval(f,p1);


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y2=feval(f,p2); ymin=y0; cond=0; j=0; while (j<maxj & abs(h)>delta & cond==0) if (y0<=y1) p2=p1; y2=y1; h=h/2; p1=p0+h; y1=feval(f,p1); else if y2<y1 p1=p2; y1=y2; h=2*h; p2=p0+2*h; y2=feval(f,p2); else cond=-1; end end


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j=j+1; if (abs(h)>big | abs(p0)>big), cond=5;end end if cond==5 pmin=p1; ymin=feval(f,p1); else d=4*y1-2*y0-2*y2; if d<0 hmin=h*(4*y1-3*y0-y2)/d; else hmin=h/3; cond=4; end pmin=p0+hmin; ymin=feval(f,pmin); h=abs(h); h0=abs(hmin); h1=abs(hmin-h); h2=abs(hmin-2*h); if (h0<h),h=h0;end if (h1<h),h=h1;end


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if (h2<h),h=h2;end if (h==0),h=hmin;end if (h<delta),cond=1;end if (abs(h)>big) | abs(pmin>big),cond=5;end e0=abs(y0-ymin); e1=abs(y1-ymin); e2=abs(y2-ymin); if (e0~=0 & e0<err),err=e0;end if (e1~=0 & e1<err),err=e1;end if (e2~=0 & e2<err),err=e2;end if (e0~=0 & e1==0 & e2==0),err==0;end if (err<epsilon),cond=2;end p0=pmin; k=k+1; P(k)=P0; end if (cond==2 & h<delta),cond=3;end end p=p0; dp=h; yp=feval(f,p); dy=err; ----------------------------------------------------------------------------------------------------------------------


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7.4. 内德-米德法求最小值

功能：求 f(x1,x2,......xn)的近似最小值，其中 f 是 N 个实量的连续函数，给定 N+1 个初

始点 Vk=(vk1,vk2......vkN),k=0,1,......N --------------------------------------------------- function [V0,y0,dV,dy]=nelder(F,V,min1,max1,epsilon,show) % input - F is the object function input as a string 'F' - V is a 3*n matrix containing staring simplex - min1 & max1 are minimum and maximum number of interations - epsilon is the tolerance - show==1 displays interations (P and Q) % output -V0 is the vertex for the minimum - y0 is the function value F(V0) - dV is the size of the final simplex - dy is the error bound for the minimum - P is a matrix containing the vertex iteration - Q is an array containing the interations for F(P) if nargin ==5 show=0; end [mm n]=size(V); for j=1:n+1 Z=V(j,1:n); Y(j)=feval(F,Z);


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end [mm lo]=min(Y); [mm hi]=max(Y); li=hi; ho=lo; for j=1:n+1 if (j~=lo & j~=ho & j<=Y(li)) li=j; end if (j~=hi & j~=lo & Y(j)>=Y(ho)) ho=j; end end cnt=0; while (Y(hi)>Y(lo)+epsilon & cnt<max1)| cnt<min1 S=zeros(1,n+1); for j=1:n+1 S=S+V(j,1:n); end M=(S-V(hi,1:n))/n; R=2*M-V(hi,1:n); yR=feval(F,R);


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if (yR<Y(ho)) if Y((li)<yR) V(hi,1:n)=R; Y(hi)=yR; else E=2*R-M; yE=feval(F,E); if (yE<Y(li)) V(hi,1:n)=E; Y(hi)=yE; else V(hi,1:n)=R; Y(hi)=yR; end end else if (yR<Y(hi)) V(hi,1:n)=R; Y(hi)=yR; end C=(V(hi,1:n)+M)/2; yC=feval(F,C);


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C2=(M+R)/2; yC2=feval(F,C2); if (yC2<yC) C=C2; yC=yC2; end if (yC<Y(hi)) V(hi,1:n)=C; Y(hi)=yC; else for j=1:n+1 if (j~=lo) V(j,1:n)=(V(j,1:n)+V(lo,1:n))/2; Z=V(j,1:n); Y(j)=feval(F,Z); end end end [mm lo]=min(Y); [mm hi]=max(Y); li=hi; ho=lo;


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for j=1:n+1 if (j~=lo & j~=hi & Y(j)<=Y(li)) li=j; end if (j~=hi & j~=lo & Y(j)>=Y(ho)) ho=j; end end cnt=cnt+1; P(cnt,:)=V(lo,:); Q(cnt)=Y(lo); end snorm=0; for j=1:n+1 s=norm(V(j)-V(lo)); if (s>=snorm) snorm=s; end end Q=Q'; V0=V(lo,1:n) y0=Y(lo);


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dV=snorm; dy=abs(Y(hi)-Y(lo)); if (show==1) disp(P); disp(Q); end ----------------------------------------------------------------------------------------------------------------------

7.5. 最速下降法或梯度法

功能：用数值方法求 f(X)的近似局部极小值，其中 f 为 N 个实数变量的连续函数，

X=(x1,x2,,.....xN),从 P0 开始，适用梯度方法 ---------------------------------------------------------- function [P0,y0,err]=grads(F,G,P0,max1,delta,epsilon,show) % input - F is the object function input as a string 'F' - G=-(i/norm(grad F))*grad F: the search direction input as a string 'G' - P0 is the initial starting point - max1 is the maximum number of interations - delta is the tolerance for hmin in the single parameter minimization in the search direction - epsilon is the tolerance for the err in y0 - show : if show==1 the interations are displayed % output - P0 is the point for the minimum - y0 is the function value F(P0) - err is the error bound for y0


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- P is a vector containing the iterations if nargin==5,show=0;end [mm n]=size(P0); maxj=10; big=1e8 h=1; P=zeros(maxj,n+1); len=norm(P0); y0=feval(F,P0); if (len>1e4),h=len/1e4;end cond=0;err=1;cnt=0; P(cnt+1,:)=[P0 y0]; while (cnt<max1 & cond~=5 & (h>delta | err>epsilon)) S=feval(G,P0); P1=P0+h*S; P2=P0+2*h*S; y1=feval(F,P1); y2=feval(F,P2); cond=0; j=0; while (j<maxj & cond==0) len=norm(P0)


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if (y0<y1) P2=P1; y2=y1; h=h/2; P1=P0+h*S; y1=feval(F,P1); else if y2<y1 P1=P2; y1=y2; h=2*h; P2=P0+2*h*S; y2=feval(F,P2); else cond=-1; end end j=j+1; if (h<delta), cond=1;end if (abs(h)>big | len>big),cond=5;end end if cond==5


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Pmin=P1; ymin=y1; else d=4*y1-2*y0-2*y2; if d<0 hmin=h*(4*y1-3*y0-y2)/d; else cond=4; hmin=h/3; end Pmin=P0+hmin*S; ymin=feval(F,Pmin); h=abs(h); h0=abs(hmin); h1=abs(hmin-h); h2=abs(hmin-2*h); if (h0<h),h=h0;end if (h1<h),h=h1;end if (h2<h),h=h2;end if (h==0),h=hmin;end if (h<delta),cond=1;end e0=abs(y0-ymin);


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e1=abs(y1-ymin); e2=abs(y2-ymin); if (e0~=0 & e0<err),err=e0;end if (e1~=0 & e1<err),err=e1;end if (e2~=0 & e2<err),err=e2;end if (e0==0 & e1==0 & e2==0),err==0;end if (err<epsilon),cond=2;end if (cond==2 & h<delta),cond=3;end end cnt=cnt+1; P(cnt+1,:)=[Pmin ymin]; P0=Pmin; y0=ymin; end if show==1 disp(P)； end ======================================================================

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