gmres源程序matlab

合集下载

GM(1,1)模型中的MATLAB程序

GM （1,1）模型中的MATLAB 程序一、GM （1,1）模型的建立：（1）、一次累加生成序列的MA TLAB 命令：>> X0=[142 340 200 500 900 800 490 980 463 1100];>> X1(1)=X0(1)X1 =142>> for k=2:10X1(k)=X1(k-1)+X0(k)endX1 =142 482 682 1182 20822882 3372 4352 4815 5915（2）、由一次累加生成序列紧邻均值生成的)1(Z 的MA TLAB 命令：>> X0=[142 340 200 500 900 800 490 980 463 1100];>> X1(1)=X0(1);>> for k=2:10X1(k)=X1(k-1)+X0(k)Z(k)=(1/2)*(X1(k)+X1(k-1))EndX1 =142 482 682 1182 20822882 3372 4352 4815 5915Z =1.0e+003 *0 0.3120 0.5820 0.9320 1.6320 2.4820 3.12703.86204.58355.3650（3）、GM （1，1）的灰微分方程模型为：b k aZ k X =+)()()1()0(。

设∧α为待估计参数向量，⎥⎦⎤⎢⎣⎡=∧b a α。

利用最小二乘法得到Y B B B ')'(1-∧=α，MA TLAB 程序如下：>> B=[-Z(2:10)',ones(9,1)];>> Y=(X0(2:10))';>> alfa=inv(B'*B)*B'*Yalfa =-0.1062371.6018（4）、GM （1，1）的灰微分方程模型 b k aZ k X =+)()()1()0(的时间相应序列为：ab e a b X k X ak +⋅-=+-∧))1(()1()0()1( 由.6018.371,1062.0=-=b a 令.)1(,)0(u X v a b u -== 计算得到1.3499-=u ， 1.3641=v 。

gmres源程序 matlab

function [x，flag,relres,iter,resvec] = gmres(A,b,restart,tol,maxit,M1，M2,x,varargin)%GMRES Generalized Minimum Residual Method。

％X = GMRES(A,B) attempts to solve the system of linear equations A*X = B% for X。

The N-by—N coefficient matrix A must be square and the right％hand side column vector B must have length N。

This uses the unrestarted% method with MIN(N,10) total iterations.%％X = GMRES（AFUN,B）accepts a function handle AFUN instead of the matrix% A。

AFUN(X）accepts a vector input X and returns the matrix-vector% product A*X. In all of the following syntaxes, you can replace A by% AFUN.％％X = GMRES(A，B,RESTART）restarts the method every RESTART iterations.％If RESTART is N or [］then GMRES uses the unrestarted method as above.％% X = GMRES(A，B,RESTART,TOL）specifies the tolerance of the method。

If% TOL is ［] then GMRES uses the default, 1e-6.％% X = GMRES（A，B,RESTART，TOL，MAXIT）specifies the maximum number of outer % iterations。

matlab经典源程序带有注释(详细经典)

2.1set 与get 函数 (1)2.2callback函数 (2)2.3元胞数组 (4)2.4结构数组 (6)2.5矩阵操作 (9)2.6字符串操作 (13)2.7判断函数使用大全 (16)2.11打开外部程序 (21)2.11程序运行时间 (22)2.14动画 (23)2.12动画 (24)2.23显示多行内容 (26)2.24 uitable 使用 (26)2.27鼠标操作 (27)2.28键盘操作 (27)2.32粘贴板 (28)2.1set 与get 函数set(edit_handle,'String','my value!'); %String为Edit控件的属性%%%2.1-1%创建figure对象hfig=figure(1);%创建坐标轴对象，指定其父对象为figure 1haxes1=axes('parent',hfig);prop.Color='b';prop.FontSize=12;set(haxes1,prop);%%%2.1-2hfig=figure(1);%查询其Units属性值get(hfig,'units')%其Units属性值为pixels（像素）% ans=% pixels%%%2.1-3%figure的Pointer属性标识了鼠标指针的形状set(gcf,'pointer');% 返回值为：[ crosshair | fullcrosshair | {arrow} | ibeam | watch | topl | topr | botl | botr | left | top | right | bottom | circle | cross | fleur | custom | hand ]%%%2.1-4%首先取得标识电脑屏幕大小的度量单位get(0,'units')% ans =% pixels%取得屏幕的尺寸get(0,'screensize')% ans =% 1 1 1280 8002.2callback函数%定义M文件的主函数名称为DefineCallback，不带输入和输出参数function DefineCallbackhFig= figure('units','normalize',...'position',[0.4 0.4 0.3 0.2]);%在窗口中创建按钮控件，并定义其Callback属性uicontrol('parent',hFig,...'style','pushbutton',...'String','Execute Callback',...'units','normalize',...'position',[0.4 0.4 0.3 0.2],...'callback',['figure;',...'x = 0:pi/20:2*pi;',...'y = sin(x);',...'plot(x,y);']);%定义M文件的主函数名称为DefineCallback，不带输入和输出参数function DefineCallback%创建界面窗口hFig= figure('units','normalize',...'position',[0.4 0.4 0.3 0.2]);%在窗口中创建按钮控件hpush=uicontrol('parent',hFig,...'style','pushbutton',...'String','Execute Callback',...'units','normalize',...'position',[0.4 0.4 0.3 0.2]);%设置按钮的Callback属性set(hpush,'callback',@mycallback);%定义回调函数为子函数function mycallback(hobj,event)figure;x = 0:pi/20:2*pi;y = sin(x);plot(x,y);2.3元胞数组a={'hello' [1 2 3;4 5 6];1 {'1''2'}}a ='hello' [2x3 double][ 1] {1x2 cell }%示例2：将元胞数组a中的元胞逐一赋值>> a{1,1}='hello';a{1,2}=[1 2 3;4 5 6];a{2,1}=1;a{2,2}={'1' '2'};>> aa ='hello' [2x3 double][ 1] {1x2 cell }%示例3：使用cell函数来创建元胞数组%生成2x3的元素为空数组的元胞数组>> a=cell(2,3)a =[] [] [][] [] []%示例4：判断数组A是否为元胞数组%定义一个元胞数组A>> A={1 2 3};%判断A是否为元胞数组，如果为元胞数组，则函数>> tf = iscell(A)tf =1%示例5：显示元胞数组C中的内容>> clear>> C={'Smith' [1 2;3 4] [12]};%直接显示元胞数组C中的内容>> celldisp(C)C{1} =SmithC{2} =1 23 4C{3} =12%显示元胞数组C中的内容，数组的名称用cellcontent代替>> celldisp(C,'cellcontent')cellcontent{1} =Smithcellcontent{2} =1 23 4cellcontent{3} =12%示例6：将字符数组转换为元胞数组>> S = ['abc '; 'defg'; 'hi m'];>> cellstr(S)ans ='abc'%原先abc后面的空格被清除'defg''hi m'%i和m之间的空格仍然保留%示例7：显示元胞数组S中的内容（包括空格和字符）>> S = {'abc ', 'defg','hi m'};>> cellplot(S)%示例8：将数字数组A按行或按列转换为元胞数组%A是4x3的数组>> A=[1 2 3;4 5 6;7 8 9;10 11 12];%把A的每一列转换为一个元胞，得到的C是1×3的元胞数组>> C=num2cell(A,1)C =[4x1 double] [4x1 double] [4x1 double] %把A的每一行转换为一个元胞，得到的C是4×1的元胞数组>> C=num2cell(A,2)C =[1x3 double][1x3 double][1x3 double][1x3 double]2.4结构数组%示例1：使用直接法来创建结构数组>> A(1).name = 'Pat';A(1).number = 176554;A(2).name = 'Tony';A(2).number = 901325;>> AA =1x2 struct array with fields:namenumber%示例2：利用struct函数来创建结构数组>> A(1)=struct('name','Pat','number',176554);A(2)=struct('name','Tony','number',901325);>> AA =1x2 struct array with fields:namenumber%示例3：使用deal函数来得到结构体中各结构域的值%定义结构数组A>> = 'Pat'; A.number = 176554;A(2).name = 'Tony';A(2).number = 901325;%得到结构数组中所有name结构域的数据>>[name1,name2] = deal(A(:).name)name1 =Patname2 =Tony%示例4：使用getfield函数来取得结构体中结构域的值%定义mystr结构数组>> mystr(1,1).name = 'alice';mystr(1,1).ID = 0;mystr(2,1).name = 'gertrude';mystr(2,1).ID = 1;%取得mystr(2,1)的结构域name的值>> f = getfield(mystr, {2,1}, 'name')f =gertrude%示例5：删除结构数组中的指定结构域%定义结构数组s>> s.field1=[1 2 3];s.field2='string';s.field3={1 2 3;4 5 6}; %删除结构域field1>> s=rmfield(s,'field1')s =field2: 'string'field3: {2x3 cell}%删除结构域'field2','field3'>> s=rmfield(s,{'field2','field3'})s =field1: [1 2 3]%示例6：%定义结构数组s>> s.field1=[1 2 3];s.field2='string';s.field3={1 2 3;4 5 6}; >> ss =field1: [1 2 3]field2: 'string'field3: {2x3 cell}%将结构数组转换为元胞数组>> c=struct2cell(s)c =[1x3 double]'string'{2x3 cell }%示例7：>>c = {'birch', 'betula', 65; 'maple', 'acer', 50}c ='birch''betula' [65]'maple''acer' [50]>>fields = {'name', 'genus', 'height'}; %fields包含struct中的结构域名>>s = cell2struct(c, fields, 2); %dim=2表示把c中的各行转换为struct数组s =2x1 struct array with fields:namegenusheight>> s(1)ans =name: 'birch'genus: 'betula'height: 65>> s(2)ans =name: 'maple'genus: 'acer'height: 50>> fields = {'field1', 'field2'};>> s = cell2struct(c, fields, 1); %dim=1表示把c中的各列转换为struct数组>> s(1)ans =field1: 'birch'field2: 'maple'>> s(2)ans =field1: 'betula'field2: 'acer'>> s(3)ans =field1:65field2:502.5矩阵操作%示例1： find函数的使用方法。

MATLAB源程序

1、实验目的（1）学习MATLAB的各种二维绘图；（2）学习MATLAB的三维绘图；（3）M ATLAB的绘图修饰（多种、图形注释、绘图颜色、色图矩阵）。

2、实验内容（1）基本二维绘图1）向量绘图x=0:2*pi/100:2*pi;y1=sin(2*x);y2=cos(2*x);plot(x,y1)plot(x,y2)plot(x,y1,x,y2)plot(x,y1);hold on;plot(x,y2);hold off;plot(x',[y1',y2'])设定颜色与线型plot(x,y1,'c:',x,y2,'wo')多窗口绘图figure(1);plot(x,y1);figure(2);plot(x,y2);子图绘图subplot(221);plot(x,y1);subplot(222);plot(x,y2);subplot(223);plot(x,y1,x,y1+y2);subplot(224);plot(x,y2,x,y1-y2);复变函数绘图w=0.01:0.01:10;G=1./(1+2*w*i);subplot(121);plot(G);subplot(122);plot(real(G),imag(G));插值绘图x=0:2*pi/8:2*pi;y=sin(x); plot(x,y,'o');hold on;xi=0:2*pi/100:2*pi;yi=spline(x,y,xi);plot(xi,yi,'m');反白绘图与绘图背景颜色设定whitebgwhitebg('b')whitebg('k')2）函数绘图fplot('sin',[0 4*pi])f='sin(x)';fplot(f,[0 4*pi])fplot('sin(1/x)',[0.01 0.1],1e-3)fplot('[tan(x),sin(x),cos(x)]',[-2*pi,2*pi,-2*pi,2*pi])3)符号函数快捷绘图syms xf=exp(-0.5*x)*sin(x);ezplot(f,[0,10])（2）多种二维绘图1）半对数绘图（频率特性绘图）w=logspace(-1,1);g=20*log10(1./(1+2*w*i));p=angle(1./(1+2*w*i))*180/pi;subplot(211);semilogx(w,g);grid;subplot(212);semilogx(w,p);grid;2）极坐标绘图t=0:2*pi/180:2*pi;mo=cos(2*t);polar(t,mo);t=0:2*pi/8:2*pi; y=sin(t);bar(t,y);t=0:2*pi/8:2*pi; y=sin(t);stem(t,y);t=0:2*pi/8:2*pi; y=sin(t);stairs(t,y);t=-pi:pi/200:pi;comet(t,tan(sin(t))-sin(tan(t)));（3）图形注释y1=dsolve('D2u+2*Du+10*u=0','Du(0)=1,u(0)=0','x');y2=dsolve('D2u+2*Du+10*u=1','Du(0)=1,u(0)=0','x');ezplot(y1,[0,5]);hold on;ezplot(y2,[0,5]);axis([0,5,-0.1,0.2]);title('二阶系统时间响应');axis([0,5,-0.1,0.2]);title('二阶系统时间响应');xlabel('时间t');xlabel('时间t');title('二阶系统时间响应');xlabel('时间t');ylabel('响应幅值y');gtext('零输入响应');gtext('零状态响应');grid;hold off;（4）三维绘图1）三维线图t=0:pi/50:10*pi;plot3(sin(t),cos(t),t);comet3(sin(t),cos(t),t);Z2=[1 1;1 -1];Z4=[Z2 Z2;Z2 -Z2]; Z8=[Z4 Z4;Z4 -Z4]; mesh(Z8);x=-4:0.5:4;y=x; [X,Y]=meshgrid(x,y); Z=X.^2-Y.^2;mesh(X,Y,Z);4)圆锥面网线图t1=0:0.1:0.9;t2=1:0.1:2;r=[t1 -t2+2];[X,Y,Z]=cylinder(r,40);mesh(X,Y,Z);5）视角修饰t1=0:0.1:0.9;t2=1:0.1:2;r=[t1 -t2+2];[X,Y,Z]=cylinder(r,30);mesh(X,Y,Z);subplot(2,2,1);mesh(X,Y,Z);view(0,0);subplot(2,2,2);mesh(X,Y,Z);view(-20,20);subplot(2,2,3);mesh(X,Y,Z);view(-30,30);subplot(2,2,4);mesh(X,Y,Z);view(-40,40);6）MATLAB暖色（hot）函数色图peaks(20);z = 3*(1-x).^2.*exp(-(x.^2) - (y+1).^2) ...- 10*(x/5 - x.^3 - y.^5).*exp(-x.^2-y.^2) ...- 1/3*exp(-(x+1).^2 - y.^2)axis('off')colormap(hot);colorbar('horiz');7）光照修饰surfl(peaks(20));colormap(hot);shading interp;8）表面修饰subplot(131);peaks(20);shading flat;z = 3*(1-x).^2.*exp(-(x.^2) - (y+1).^2) ...- 10*(x/5 - x.^3 - y.^5).*exp(-x.^2-y.^2) ...- 1/3*exp(-(x+1).^2 - y.^2)subplot(132);peaks(20);shading interp;z = 3*(1-x).^2.*exp(-(x.^2) - (y+1).^2) ... - 10*(x/5 - x.^3 - y.^5).*exp(-x.^2-y.^2) ... - 1/3*exp(-(x+1).^2 - y.^2)subplot(133);peaks(20);shading faceted;z = 3*(1-x).^2.*exp(-(x.^2) - (y+1).^2) ... - 10*(x/5 - x.^3 - y.^5).*exp(-x.^2-y.^2) ... - 1/3*exp(-(x+1).^2 - y.^2)9）透视与消隐P=peaks(30);subplot(121);mesh(P);hidden off;subplot(122);mesh(P);hidden on;10）裁剪修饰P=peaks(30);P(20:23,9:15)=NaN*ones(4,7);subplot(121);meshz(P);P=peaks(30);subplot(121);mesh(P);hidden off;subplot(122);mesh(P);hidden on;P=peaks(30);P(20:23,9:15)=NaN*ones(4,7);subplot(121);meshz(P);subplot(122);meshc(P);11）水线修饰waterfall(peaks(20))colormap([1 0 0])12)等高线修饰contour(peaks(20),6)contour3(peaks(20),10)clabel( contour(peaks(20),4))clabel( contour3(peaks(20),3))。

以下是matlab源程序

以下是matlab源程序=========================================== %%%%%%%%%%%%%%人造地震波程序%%%%%%%%%%%%%% 本程序利用反应谱转换为功率谱方法模拟人造地震波% 变量说明：% s0: 谱强度因子dt: 地震波采样时间间隔% kao: 地震波总时间长度% Ts: 平稳地震动的持续时间% t1,t2,C: 控制主震段首末时间和衰减快慢的参数% keceg,omigag: 地表覆盖层的阻尼比和卓越频率% omigah: 反映基岩特性的谱参数%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%xg(1:1500,1)=zeros(1500,1);dt=0.02;kao=21.9;t1=1.5;t2=15;c=0.18;Domiga=2*pi*(25-1/6)/100;Tg=0.25;kece=0.05;alfa=0.9;for i=1:100omiga=(2*pi/6)+Domiga*(i-0.5);T=2*pi/omiga;if T<0.1r=5.5*alfa*T+0.45*alfa;elseif (T>=0.1&T<=Tg)r=alfa;elser=alfa*(Tg/T)^0.9;endgama=omiga/pi;M1=(t2-0.3*t1)/kao;M2=(exp(-2*kece*omiga*t2)-exp(-2*kece*omiga*t1))/(2*kece*omiga*kao);M3=(1-exp(-c*(kao+0.5*t1-t2)))/(c*kao);M=M1+M2+M3;Gomiga=4*kece*r^2/(pi*omiga*M*(sqrt(2*log(gama*kao))+0.5772/sqrt(2*log(gama*kao)))^2); ak=2*sqrt(Gomiga*Domiga);qrand=rand(1)*2*pi;for j=1:1500xg(j)=xg(j)+ak*sin(omiga*j*dt+qrand);endendfor i=1:1500if i*dt<t1fai=(i*dt/t1)^2;elseif (i*dt>=t1&i*dt<t2)fai=1;elsefai=exp(-c*(i*dt-t2));endxg(i)=fai*xg(i);endsubplot(2,1,1);plot(xg);%xg=xg./(max(abs(xg)));save ren xg -ascii;fans=fft(xg);fans=fans.*conj(fans);subplot(2,1,2);plot(fans);。

Matlab源程序代码

正弦波的源程序：(一)，用到的函数1,f2t函数function x=f2t(X)global dt df t f T N%x=f2t(X)%x为时域的取样值矢量%X为x的傅氏变换%X与x长度相同并为2的整幂%本函数需要一个全局变量dt(时域取样间隔) X=[X(N/2+1:N),X(1:N/2)];x=ifft(X)/dt;end2,t2f函数。

function X=t2f(x)global dt df N t f T%X=t2f(x)%x为时域的取样值矢量%X为x的傅氏变换%X与x长度相同,并为2的整幂。

%本函数需要一个全局变量dt(时域取样间隔) H=fft(x);X=[H(N/2+1:N),H(1:N/2)]*dt;end(二),主程序。

1，%（1）绘出正弦信号波形及频谱global dt df t f Nclose allk=input('取样点数＝2^k, k取10左右');if isempty(k), k=10; endf0=input('f0=取1(kz)左右');if isempty(f0), f0=1; endN=2^k;dt=0.01; %msdf=1/(N*dt); %KHzT=N*dt; %截短时间Bs=N*df/2; %系统带宽f=[-Bs+df/2:df:Bs]; %频域横坐标t=[-T/2+dt/2:dt:T/2]; %时域横坐标s=sin(2*pi*f0*t); %输入的正弦信号S=t2f(s); %S是s的傅氏变换a=f2t(S); %a是S的傅氏反变换a=real(a);as=abs(S);subplot(2,1,1) %输出的频谱plot(f,as,'b');gridaxis([-2*f0,+2*f0,min(as),max(as)])xlabel('f (KHz)')ylabel('|S(f)| (V/KHz)') %figure(2)subplot(2,1,2)plot(t,a,'black') %输出信号波形画图gridaxis([-2/f0,+2/f0,-1.5,1.5])xlabel('t(ms)')ylabel('a(t)(V)')gtext('频谱图')最佳基带系统的源程序：(一)，用到的函数f2t函数和t2f函数。

高等数值分析第二章答案

第二章习题参考答案1.解：由于20Ax b−≥，极小化2b Ax −与极小化22Ax b −是等价的。

令22()(,)(,)2(,)x Ax b Ax Ax b b Ax b ϕ=−=+−，对于任意的n R y x ∈,和实数α，)()(),()()(,*222*2****x Ay a x Ay Ay a x ay x b Ax x ϕϕϕϕ≥+=+=+=则有满足若这表示处达到极小值。

在*)(x x ϕ反之，若必有处达到极小，则对任意在nR y x ay x ∈+*)(ϕ0),(2),(2),(20)(**0*=−=+−=+=Ay b Ax Ay Ay a Ay b Ax daay x d a 即ϕ故有 b Ax =*成立。

以上证明了求解,22b Ax b Ax −=等价于极小化即。

等价于极小化2b Ax b Ax −= 推导最速下降法过程如下：),/(),(0),(),(,0),,2)(222)()(11k T k T k T k k T k T k T k k T k k k T k k kT k T k T T x x k r AA r AA r AA r a r AA r AA a r AA r r aA x da dx a r aA x x r A Ax b A Ax A b A x grad x x k==+−=++==−=−=−++=最终得到得出（由取得极小值。

使求出取的负梯度方向，且下降最快的方向是该点在ϕϕϕ给出的算法如下：1）)(000Ax b A r A R x T T n −=∈，计算给定； 2）L ,2,1,0=k 对于）转到否则数。

为一事先给定的停机常则停止；其中若2),/(),(10,11kT k k k k T k k k k k k k k k r A p Ax b r r A a x x Ap Ap p p a k k r =−=+==+=>≤−−εε2.证明 1）正定性由对称正定矩阵的性质，(),0x Ax ≥（当且仅当x ＝0时取等号），所以 ()12,0Axx Ax =≥（当且仅当x ＝0时取等号）2）齐次性()()()121122,(),,AA xx A x x Ax x Ax x αααααα⎡⎤====⎣⎦3）o1方法（一）A 是对称正定矩阵，得到(,())0x y A x y λλ++≥，把它展开如下2(,)(,)(,)(,)0y Ay x Ay y Ax x Ax λλλ+++≥考虑到(,)(,)(,)x Ay Ax y y Ax ==，把上式看成关于λ的一元二次方程，则式子等价于24(,)4(,)(,)0x Ay x Ax y Ay ∆=−≤因此1/21/2(,)(,)(,)x Ay x Ax y Ay ≤所以1/21/221/21/2((,)(,))(,)(,)2(,)(,)(,)(,)2(,)(,)(,)(,)(,)((),())x Ax y Ay x Ax y Ay x Ax y Ay x Ax y Ay x Ay x Ax y Ay x Ay y Ax x y A x y +=++≥++=+++=++两边开平方即可得到AA A x yx y +≤+因此，1/2(,)A x Ax x =是一种向量范数。

GMRES_matlab

gmresGeneralized minimum residual method (with restarts)expand all in page Syntaxx = gmres(A,b)gmres(A,b,restart)gmres(A,b,restart,tol)gmres(A,b,restart,tol,maxit)gmres(A,b,restart,tol,maxit,M)gmres(A,b,restart,tol,maxit,M1,M2)gmres(A,b,restart,tol,maxit,M1,M2,x0)[x,flag] = gmres(A,b,...)[x,flag,relres] = gmres(A,b,...)[x,flag,relres,iter] = gmres(A,b,...)[x,flag,relres,iter,resvec] = gmres(A,b,...)Descriptionx = gmres(A,b) attempts to solve the system of linear equations A*x = b for x. The n-by-n coefficient matrix A must be square and should be large and sparse. The column vector b must have length n. A can be a function handle, afun, such that afun(x) returns A*x. For this syntax, gmres does not restart; the maximum number of iterations is min(n,10).Parameterizing Functions explains how to provide additional parameters to the function afun, as well as the preconditioner function mfun described below, if necessary.If gmres converges, a message to that effect is displayed. If gmres fails to converge after the maximum number of iterations or halts for any reason, a warning message is printed displaying the relative residual norm(b-A*x)/norm(b) and the iteration number at which the method stopped or failed. gmres(A,b,restart) restarts the method every restart inner iterations. The maximum number of outer iterations is min(n/restart,10). The maximum number of total iterationsis restart*min(n/restart,10). If restart is n or [], then gmres does not restart and the maximum number of total iterations is min(n,10).gmres(A,b,restart,tol) specifies the tolerance of the method. If tol is [], then gmres uses the default, 1e-6.gmres(A,b,restart,tol,maxit) specifies the maximum number of outer iterations, i.e., the total number of iterations does not exceed restart*maxit. If maxit is [] then gmres uses the default, min(n/restart,10). If restart is n or [], then the maximum number of total iterationsis maxit (instead of restart*maxit).gmres(A,b,restart,tol,maxit,M) and gmres(A,b,restart,tol,maxit,M1,M2) use preconditioner M or M = M1*M2and effectively solve the system inv(M)*A*x = inv(M)*b for x.If M is [] then gmres applies no preconditioner. M can be a function handle mfun suchthat mfun(x) returns M\x.gmres(A,b,restart,tol,maxit,M1,M2,x0) specifies the first initial guess. If x0 is [],then gmres uses the default, an all-zero vector.[x,flag] = gmres(A,b,...) also returns a convergence flag:flag = 0gmres converged to the desired tolerance tol within maxit outer iterations.flag = 1gmres iterated maxit times but did not converge.flag = 2Preconditioner M was ill-conditioned.flag = 3gmres stagnated. (Two consecutive iterates were the same.)Whenever flag is not 0, the solution x returned is that with minimal norm residual computed over all the iterations. No messages are displayed if the flag output is specified.[x,flag,relres] = gmres(A,b,...) also returns the relative residual norm(b-A*x)/norm(b).If flag is 0, relres <= tol. The third output, relres, is the relative residual of the preconditioned system.[x,flag,relres,iter] = gmres(A,b,...) also returns both the outer and inner iterationnumbers at which x was computed, where 0 <= iter(1) <= maxit and 0 <= iter(2) <= restart.[x,flag,relres,iter,resvec] = gmres(A,b,...) also returns a vector of the residual norms at each inner iteration. These are the residual norms for the preconditioned system.ExamplesUsing gmres with a Matrix InputA = gallery('wilk',21);b = sum(A,2);tol = 1e-12;maxit = 15;M1 = diag([10:-1:1 1 1:10]);x = gmres(A,b,10,tol,maxit,M1);displays the following message:gmres(10) converged at outer iteration 2 (inner iteration 9) toa solution with relative residual 3.3e-013Using gmres with a Function HandleThis example replaces the matrix A in the previous example with a handle to a matrix-vector productfunction afun, and the preconditioner M1 with a handle to a backsolve function mfun. The example iscontained in a function run_gmres that∙Calls gmres with the function handle @afun as its first argument.∙Contains afun and mfun as nested functions, so that all variables in run_gmres are available to afun and mfun.The following shows the code for run_gmres:function x1 = run_gmresn = 21;b = afun(ones(n,1));tol = 1e-12; maxit = 15;x1 = gmres(@afun,b,10,tol,maxit,@mfun);function y = afun(x)y = [0; x(1:n-1)] + ...[((n-1)/2:-1:0)'; (1:(n-1)/2)'].*x + ...[x(2:n); 0];endfunction y = mfun(r)y = r ./ [((n-1)/2:-1:1)'; 1; (1:(n-1)/2)'];endendWhen you enterx1 = run_gmres;MATLAB® software displays the messagegmres(10) converged at outer iteration 2 (inner iteration 10) to a solution with relative residual 1.1e-013.Using a Preconditioner without RestartThis example demonstrates the use of a preconditioner without restarting gmres.Load west0479, a real 479-by-479 nonsymmetric sparse matrix.load west0479;A = west0479;Set the tolerance and maximum number of iterations.tol = 1e-12; maxit = 20;Define b so that the true solution is a vector of all ones.b = full(sum(A,2));[x0,fl0,rr0,it0,rv0] = gmres(A,b,[],tol,maxit);fl0 is 1 because gmres does not converge to the requested tolerance 1e-12 within the requested 20 iterations. The best approximate solution that gmres returns is the last one (as indicated by it0(2) = 20). MATLAB stores the residual history in rv0.Plot the behavior of gmres.semilogy(0:maxit,rv0/norm(b),'-o');xlabel('Iteration number');ylabel('Relative residual');The plot shows that the solution converges slowly. A preconditioner may improve the outcome.Use ilu to form the preconditioner, since A is nonsymmetric.[L,U] = ilu(A,struct('type','ilutp','droptol',1e-5));Error using iluThere is a pivot equal to zero. Consider decreasingthe drop tolerance or consider using the 'udiag' option.Note MATLAB cannot construct the incomplete LU as it would result in a singular factor, which is useless as a preconditioner.As indicated by the error message, try again with a reduced drop tolerance.[L,U] = ilu(A,struct('type','ilutp','droptol',1e-6));[x1,fl1,rr1,it1,rv1] = gmres(A,b,[],tol,maxit,L,U);fl1 is 0 because gmres drives the relative residual to 9.5436e-14 (the value of rr1). The relative residual is less than the prescribed tolerance of 1e-12 at the sixth iteration (the value of it1(2)) when preconditioned by the incomplete LU factorization with a drop tolerance of 1e-6. The output, rv1(1), is norm(M\b), where M = L*U. The output, rv1(7), is norm(U\(L\(b-A*x1))).Follow the progress of gmres by plotting the relative residuals at each iteration starting from the initial estimate (iterate number 0).semilogy(0:it1(2),rv1/norm(b),'-o');xlabel('Iteration number');ylabel('Relative residual');Using a Preconditioner with RestartThis example demonstrates the use of a preconditioner with restarted gmres.Load west0479, a real 479-by-479 nonsymmetric sparse matrix.load west0479;A = west0479;Define b so that the true solution is a vector of all ones.b = full(sum(A,2));Construct an incomplete LU preconditioner as in the previous example.[L,U] = ilu(A,struct('type','ilutp','droptol',1e-6));The benefit to using restarted gmres is to limit the amount of memory required to execute the method. Without restart, gmres requires maxit vectors of storage to keep the basis of the Krylov subspace. Also,gmres must orthogonalize against all of the previous vectors at each step. Restarting limits the amount of workspace used and the amount of work done per outer iteration. Note that even though preconditioned gmres converged in six iterations above, the algorithm allowed for as many as twenty basis vectors and therefore, allocated all of that space up front.Execute gmres(3), gmres(4), and gmres(5)tol = 1e-12; maxit = 20;re3 = 3;[x3,fl3,rr3,it3,rv3] = gmres(A,b,re3,tol,maxit,L,U);re4 = 4;[x4,fl4,rr4,it4,rv4] = gmres(A,b,re4,tol,maxit,L,U);re5 = 5;[x5,fl5,rr5,it5,rv5] = gmres(A,b,re5,tol,maxit,L,U);fl3, fl4, and fl5 are all 0 because in each case restarted gmres drives the relative residual to less than the prescribed tolerance of 1e-12.The following plots show the convergence histories of eachrestarted gmres method. gmres(3) converges at outer iteration 5, inner iteration 3 (it3 = [5, 3]) which would be the same as outer iteration 6, inner iteration 0, hence the marking of 6 on the final tick mark.figuresemilogy(1:1/3:6,rv3/norm(b),'-o');set(gca,'XTick',[1:1/3:6],...'XTickLabel',...['1';' ';' ';'2';' ';' ';'3';' ';' ';'4';' ';' ';'5';' ';' ';'6';])title('gmres(3)')xlabel('Iteration number');ylabel('Relative residual');figuresemilogy(1:1/4:3,rv4/norm(b),'-o');set(gca,'XTick',[1:1/4:3],...'XTickLabel',['1';' ';' ';' ';'2';' ';' ';' ';'3'])title('gmres(4)')xlabel('Iteration number');ylabel('Relative residual');figuresemilogy(1:1/5:2.8,rv5/norm(b),'-o');set(gca,'XTick',[1:1/5:2.8],...'XTickLabel',['1';' ';' ';' ';' ';'2';' ';' ';' ';' ']) title('gmres(5)')xlabel('Iteration number');ylabel('Relative residual');ReferencesBarrett, R., M. Berry, T. F. Chan, et al., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia, 1994.Saad, Youcef and Martin H. Schultz, "GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems," SIAM J. Sci. Stat. Comput., July 1986, Vol. 7, No. 3, pp. 856-869.。

GMRES_matlab_ch

GMRES在matlab中的描述参考文献：MATLAB函数库查询辞典徐东艳孟晓刚编著[函数描述]X=gmres(A,b)视图求解线性方程组A*x=b的解x。

nXn的稀疏矩阵A必须是方程且应是大型稀疏矩阵。

列向量b的长度必须为n。

参数A可以是一个函数afun以使得afun(x)返回A*x，对于这一语法格式，gmres并不重新启动，迭代的最大次数为min(n,10)。

如果gmres收敛，则显示这一结果的信息。

如果gmres在最大迭代步后没有收敛或者由于某种原因而停止，则打印警告信息，显示相对残差范数norm(b-A*x)/norm(b)并且显示该方法停止或者失效时的迭代步数。

Gmres(A,b,restart)在每一次重新开始内部迭代时才重新开始方法。

外部迭代的最大数目为min(n/restart,10),总迭代的最大步数为restart*min(n/restart,10).如果restart为n或者[]，则gmres并不重新开始，则总迭代的最大数目为min(n，10)Gmres(A,b,restart,tol)指定方法的误差。

如果tol为[]，则函数gmres将使用默认值1e-6.Gmres(A,b,restart,tol,maxit)指定外部迭代的最大次数，也就是说，迭代的总次数不超过restart*matrix。

如果参数maxit为[]，则矩阵gmres使用默认值min(n/restart,10);如果参数restart为n或者[],则总迭代的最大次数为maxit（而非restart*matxit）。

Gmres(A,b,restart,tol,maxit,M)和Gmres(A,b,restart,tol,maxit,M1,M2)使用预处理矩阵M或者M=M1*M2有效地求解方程组inv(M)*A*x=inv(M)*b.如果M为[],则函数gmres不适应预处理矩阵，M可以是一个函数，它返回M\xGmres(A,b,restart,tol,maxit,M1,M2,x0)指定初始的猜测值。

GMRES在电力系统潮流计算中的应用研究

GMRES在电力系统潮流计算中的应用研究作者：高成来源：《科学与技术》2018年第23期摘要：本文阐述一种基于matlab的GMRES实现方法，主要针对于大规模跨区域电力系统分析中潮流计算的修正方程，在求解这个高维数超稀疏的线性方程组中避免了采用直接法进行求解，满足大型电力系统潮流计算的需要。

该方法采用基于Krylov子空间的Arnoldi过程生成单位正交化基和系数矩陣，然后采用基于Given变换的回代法求解最小二乘问题求解向量，结合重启动技术，极大地提高计算的效率和收敛性。

该方法具有算法稳定、简单及通用性强等特点，能够为后续电力工作者应用于潮流计算和电力系统分析中进一步扩展应用。

算例结果验证文中实现的正确性和有效性。

关键词：Matlab GMRES方法潮流计算修正方程大型稀疏线性方程组引言所谓电力系统潮流，是指系统中所有运行参数的总体，包括各个母线电压的大小和相位、各个发电机和负荷的功率及电流，以及各个变压器和线路等元件所通过的功率、电流和其损耗。

潮流问题是电力系统分析的基础和核心。

随着我国经济的发展，电力工业发展很快，而且向着大电网、智能化的方向发展，根据我国制定“西电东送、南北互供、全国联网”的战略构想，已基本上实现除西藏、新疆、海南、台湾外全国联网。

Krylov子空间法是二十世纪十大算法之一，具有存储量少，计算量小且易于并行等优点，非常适合并行求解高维稀疏线性方程组。

根据不同的极小化方法Krylov子空间法主要有CG （Conjugate Gradient）法、MINRES（Minimal Residual）法、GMRES 法（Generalised Minimal Residual）、BiCG 法（Bi-Conjugate Gradient）、CGS法（Conjugate Gradient Squared）。

其中，GMRES适合于求解大型非对称线性方程组，结合预条件处理技术的GMRES法具有良好的收敛特性和较高的数值稳定性，同时采用重启动技术可以有效的降低对内存占有的需求。

gmres源程序matlab

function [x,flag,relres,iter,resvec] = gmres(A,b,restart,tol,maxit,M1,M2,x,varargin)%GMRES Generalized Minimum Residual Method.% X = GMRES(A,B) attempts to solve the system of linear equations A*X = B% for X. The N-by-N coefficient matrix A must be square and the right% hand side column vector B must have length N. This uses the unrestarted% method with MIN(N,10) total iterations.%% X = GMRES(AFUN,B) accepts a function handle AFUN instead of the matrix% A. AFUN(X) accepts a vector input X and returns the matrix-vector% product A*X. In all of the following syntaxes, you can replace A by% AFUN.%% X = GMRES(A,B,RESTART) restarts the method every RESTART iterations.% If RESTART is N or [] then GMRES uses the unrestarted method as above.%% X = GMRES(A,B,RESTART,TOL) specifies the tolerance of the method. If% TOL is [] then GMRES uses the default, 1e-6.%% X = GMRES(A,B,RESTART,TOL,MAXIT) specifies the maximum number of outer% iterations. Note: the total number of iterations is RESTART*MAXIT. If% MAXIT is [] then GMRES uses the default, MIN(N/RESTART,10). If RESTART% is N or [] then the total number of iterations is MAXIT.%% X = GMRES(A,B,RESTART,TOL,MAXIT,M) and% X = GMRES(A,B,RESTART,TOL,MAXIT,M1,M2) use preconditioner M or M=M1*M2% and effectively solve the system inv(M)*A*X = inv(M)*B for X. If M is% [] then a preconditioner is not applied. M may be a function handle% returning M\X.%% X = GMRES(A,B,RESTART,TOL,MAXIT,M1,M2,X0) specifies the first initial% guess. If X0 is [] then GMRES uses the default, an all zero vector.%% [X,FLAG] = GMRES(A,B,...) also returns a convergence FLAG:% 0 GMRES converged to the desired tolerance TOL within MAXIT iterations.% 1 GMRES iterated MAXIT times but did not converge.% 2 preconditioner M was ill-conditioned.% 3 GMRES stagnated (two consecutive iterates were the same).%% [X,FLAG,RELRES] = GMRES(A,B,...) also returns the relative residual% NORM(B-A*X)/NORM(B). If FLAG is 0, then RELRES <= TOL. Note with% preconditioners M1,M2, the residual is NORM(M2\(M1\(B-A*X))).%% [X,FLAG,RELRES,ITER] = GMRES(A,B,...) also returns both the outer and% inner iteration numbers at which X was computed: 0 <= ITER(1) <= MAXIT % and 0 <= ITER(2) <= RESTART.%% [X,FLAG,RELRES,ITER,RESVEC] = GMRES(A,B,...) also returns a vector of% the residual norms at each inner iteration, including NORM(B-A*X0).% Note with preconditioners M1,M2, the residual is NORM(M2\(M1\(B-A*X))). %% Example:% n = 21; A = gallery('wilk',n); b = sum(A,2);% tol = 1e-12; maxit = 15; M = diag([10:-1:1 1 1:10]);% x = gmres(A,b,10,tol,maxit,M);% Or, use this matrix-vector product function% %-----------------------------------------------------------------% % function y = afun(x,n)% y = [0; x(1:n-1)] + [((n-1)/2:-1:0)'; (1:(n-1)/2)'].*x+[x(2:n); 0]; % %-----------------------------------------------------------------% % and this preconditioner backsolve function% %------------------------------------------%% function y = mfun(r,n)% y = r ./ [((n-1)/2:-1:1)'; 1; (1:(n-1)/2)'];% %------------------------------------------%% as inputs to GMRES:% x1 = gmres(@(x)afun(x,n),b,10,tol,maxit,@(x)mfun(x,n));%% Class support for inputs A,B,M1,M2,X0 and the output of AFUN:% float: double%% See also BICG, BICGSTAB, BICGSTABL, CGS, LSQR, MINRES, PCG, QMR, SYMMLQ, % TFQMR, ILU, FUNCTION_HANDLE.% References% H.F. Walker, "Implementation of the GMRES Method Using Householder% Transformations", SIAM J. Sci. Comp. Vol 9. No 1. January 1988.% Copyright 1984-2010 The MathWorks, Inc.% $Revision: 1.21.4.13 $ $Date: 2010/02/25 08:11:48 $if (nargin < 2)error('MATLAB:gmres:NumInputs','Not enough input arguments.');end% Determine whether A is a matrix or a function.[atype,afun,afcnstr] = iterchk(A);if strcmp(atype,'matrix')% Check matrix and right hand side vector inputs have appropriate sizes[m,n] = size(A);if (m ~= n)error('MATLAB:gmres:SquareMatrix','Matrix must be square.');endif ~isequal(size(b),[m,1])error('MATLAB:gmres:VectorSize', '%s %d %s', ...'Right hand side must be a column vector of length', m,...'to match the coefficient matrix.');endelsem = size(b,1);n = m;if ~iscolumn(b)error('MATLAB:gmres:Vector','Right hand side must be a column vector.'); endend% Assign default values to unspecified parametersif (nargin < 3) || isempty(restart) || (restart == n)restarted = false;elserestarted = true;endif (nargin < 4) || isempty(tol)tol = 1e-6;endwarned = 0;if tol < epswarning('MATLAB:gmres:tooSmallTolerance', ...strcat('Input tol is smaller than eps and may not be achieved',...' by GMRES\n',' Try to use a bigger tolerance'));warned = 1;tol = eps;elseif tol >= 1warning('MATLAB:gmres:tooBigTolerance', ...strcat('Input tol is bigger than 1 \n',...' Try to use a smaller tolerance'));warned = 1;tol = 1-eps;endif (nargin < 5) || isempty(maxit)if restartedmaxit = min(ceil(n/restart),10);elsemaxit = min(n,10);endendif restartedouter = maxit;if restart > nwarning('MATLAB:gmres:tooManyInnerIts', 'RESTART is %d %s\n%s %d.', ... restart, 'but it should be bounded by SIZE(A,1).', ...' Setting RESTART to', n);restart = n;endinner = restart;elseouter = 1;if maxit > nwarning('MATLAB:gmres:tooManyInnerIts', 'MAXIT is %d %s\n%s %d.', ... maxit, 'but it should be bounded by SIZE(A,1).', ...' Setting MAXIT to', n);maxit = n;endinner = maxit;end% Check for all zero right hand side vector => all zero solutionn2b = norm(b); % Norm of rhs vector, bif (n2b == 0) % if rhs vector is all zerosx = zeros(n,1); % then solution is all zerosflag = 0; % a valid solution has been obtainedrelres = 0; % the relative residual is actually 0/0iter = [0 0]; % no iterations need be performedresvec = 0; % resvec(1) = norm(b-A*x) = norm(0)if (nargout < 2)itermsg('gmres',tol,maxit,0,flag,iter,NaN);endreturnendif ((nargin >= 6) && ~isempty(M1))existM1 = 1;[m1type,m1fun,m1fcnstr] = iterchk(M1);if strcmp(m1type,'matrix')if ~isequal(size(M1),[m,m])error('MATLAB:gmres:PreConditioner1Size', '%s %d %s',...'Preconditioner must be a square matrix of size', m, ... 'to match the problem size.');endendelseexistM1 = 0;m1type = 'matrix';endif ((nargin >= 7) && ~isempty(M2))existM2 = 1;[m2type,m2fun,m2fcnstr] = iterchk(M2);if strcmp(m2type,'matrix')if ~isequal(size(M2),[m,m])error('MATLAB:gmres:PreConditioner2Size', '%s %d %s', ...'Preconditioner must be a square matrix of size', m, ... 'to match the problem size.');endendelseexistM2 = 0;m2type = 'matrix';endif ((nargin >= 8) && ~isempty(x))if ~isequal(size(x),[n,1])error('MATLAB:gmres:XoSize', '%s %d %s', ...'Initial guess must be a column vector of length', n, ...'to match the problem size.');endelsex = zeros(n,1);endif ((nargin > 8) && strcmp(atype,'matrix') && ...strcmp(m1type,'matrix') && strcmp(m2type,'matrix'))error('MATLAB:gmres:TooManyInputs','Too many input arguments.');end% Set up for the methodflag = 1;xmin = x; % Iterate which has minimal residual so far imin = 0; % "Outer" iteration at which xmin was computed jmin = 0; % "Inner" iteration at which xmin was computed tolb = tol * n2b; % Relative toleranceevalxm = 0;stag = 0;moresteps = 0;maxmsteps = min([floor(n/50),5,n-maxit]);maxstagsteps = 3;minupdated = 0;x0iszero = (norm(x) == 0);r = b - iterapp('mtimes',afun,atype,afcnstr,x,varargin{:});normr = norm(r); % Norm of initial residualif (normr <= tolb) % Initial guess is a good enough solutionflag = 0;relres = normr / n2b;iter = [0 0];resvec = normr;if (nargout < 2)itermsg('gmres',tol,maxit,[0 0],flag,iter,relres);endreturnendminv_b = b;if existM1r = iterapp('mldivide',m1fun,m1type,m1fcnstr,r,varargin{:});if ~x0iszerominv_b = iterapp('mldivide',m1fun,m1type,m1fcnstr,b,varargin{:});elseminv_b = r;endif ~all(isfinite(r)) || ~all(isfinite(minv_b))flag = 2;x = xmin;relres = normr / n2b;iter = [0 0];resvec = normr;returnendendif existM2r = iterapp('mldivide',m2fun,m2type,m2fcnstr,r,varargin{:});if ~x0iszerominv_b = iterapp('mldivide',m2fun,m2type,m2fcnstr,minv_b,varargin{:}); elseminv_b = r;endif ~all(isfinite(r)) || ~all(isfinite(minv_b))flag = 2;x = xmin;relres = normr / n2b;iter = [0 0];resvec = normr;returnendendnormr = norm(r); % norm of the preconditioned residualn2minv_b = norm(minv_b); % norm of the preconditioned rhsclear minv_b;tolb = tol * n2minv_b;if (normr <= tolb) % Initial guess is a good enough solutionflag = 0;relres = normr / n2minv_b;iter = [0 0];resvec = n2minv_b;if (nargout < 2)itermsg('gmres',tol,maxit,[0 0],flag,iter,relres);endreturnendresvec = zeros(inner*outer+1,1); % Preallocate vector for norm of residuals resvec(1) = normr; % resvec(1) = norm(b-A*x0)normrmin = normr; % Norm of residual from xmin% Preallocate J to hold the Given's rotation constants.J = zeros(2,inner);U = zeros(n,inner);R = zeros(inner,inner);w = zeros(inner+1,1);for outiter = 1 : outer% Construct u for Householder reflector.% u = r + sign(r(1))*||r||*e1u = r;normr = norm(r);beta = scalarsign(r(1))*normr;u(1) = u(1) + beta;u = u / norm(u);U(:,1) = u;% Apply Householder projection to r.% w = r - 2*u*u'*r;w(1) = -beta;for initer = 1 : inner% Form P1*P2*P3...Pj*ej.% v = Pj*ej = ej - 2*u*u'*ejv = -2*(u(initer)')*u;v(initer) = v(initer) + 1;% v = P1*P2*...Pjm1*(Pj*ej)for k = (initer-1):-1:1v = v - U(:,k)*(2*(U(:,k)'*v));end% Explicitly normalize v to reduce the effects of round-off.v = v/norm(v);% Apply A to v.v = iterapp('mtimes',afun,atype,afcnstr,v,varargin{:});% Apply Preconditioner.if existM1v = iterapp('mldivide',m1fun,m1type,m1fcnstr,v,varargin{:}); if ~all(isfinite(v))flag = 2;breakendendif existM2v = iterapp('mldivide',m2fun,m2type,m2fcnstr,v,varargin{:}); if ~all(isfinite(v))flag = 2;breakendend% Form Pj*Pj-1*...P1*Av.for k = 1:initerv = v - U(:,k)*(2*(U(:,k)'*v));end% Determine Pj+1.if (initer ~= length(v))% Construct u for Householder reflector Pj+1.u = [zeros(initer,1); v(initer+1:end)];alpha = norm(u);if (alpha ~= 0)alpha = scalarsign(v(initer+1))*alpha;% u = v(initer+1:end) +% sign(v(initer+1))*||v(initer+1:end)||*e_{initer+1) u(initer+1) = u(initer+1) + alpha;u = u / norm(u);U(:,initer+1) = u;% Apply Pj+1 to v.% v = v - 2*u*(u'*v);v(initer+2:end) = 0;v(initer+1) = -alpha;endend% Apply Given's rotations to the newly formed v.for colJ = 1:initer-1tmpv = v(colJ);v(colJ) = conj(J(1,colJ))*v(colJ) + conj(J(2,colJ))*v(colJ+1); v(colJ+1) = -J(2,colJ)*tmpv + J(1,colJ)*v(colJ+1);end% Compute Given's rotation Jm.if ~(initer==length(v))rho = norm(v(initer:initer+1));J(:,initer) = v(initer:initer+1)./rho;w(initer+1) = -J(2,initer).*w(initer);w(initer) = conj(J(1,initer)).*w(initer);v(initer) = rho;v(initer+1) = 0;endR(:,initer) = v(1:inner);normr = abs(w(initer+1));resvec((outiter-1)*inner+initer+1) = normr;normr_act = normr;if (normr <= tolb || stag >= maxstagsteps || moresteps)if evalxm == 0ytmp = R(1:initer,1:initer) \ w(1:initer);additive = U(:,initer)*(-2*ytmp(initer)*conj(U(initer,initer))); additive(initer) = additive(initer) + ytmp(initer);for k = initer-1 : -1 : 1additive(k) = additive(k) + ytmp(k);additive = additive - U(:,k)*(2*(U(:,k)'*additive));endif norm(additive) < eps*norm(x)stag = stag + 1;elsestag = 0;endxm = x + additive;evalxm = 1;elseif evalxm == 1addvc = [-(R(1:initer-1,1:initer-1)\R(1:initer-1,initer))*... (w(initer)/R(initer,initer)); w(initer)/R(initer,initer)]; if norm(addvc) < eps*norm(xm)stag = stag + 1;elsestag = 0;endadditive = U(:,initer)*(-2*addvc(initer)*conj(U(initer,initer)));additive(initer) = additive(initer) + addvc(initer);for k = initer-1 : -1 : 1additive(k) = additive(k) + addvc(k);additive = additive - U(:,k)*(2*(U(:,k)'*additive));endxm = xm + additive;endr = b - iterapp('mtimes',afun,atype,afcnstr,xm,varargin{:});if norm(r) <= tol*n2bx = xm;flag = 0;iter = [outiter, initer];breakendminv_r = r;minv_r = iterapp('mldivide',m1fun,m1type,m1fcnstr,r,varargin{:});if ~all(isfinite(minv_r))flag = 2;breakendendif existM2minv_r = iterapp('mldivide',m2fun,m2type,m2fcnstr,minv_r,varargin{:});if ~all(isfinite(minv_r))flag = 2;breakendendnormr_act = norm(minv_r);resvec((outiter-1)*inner+initer+1) = normr_act;if normr_act <= normrminnormrmin = normr_act;imin = outiter;jmin = initer;xmin = xm;minupdated = 1;endif normr_act <= tolbx = xm;flag = 0;iter = [outiter, initer];breakelseif stag >= maxstagsteps && moresteps == 0stag = 0;endmoresteps = moresteps + 1;if moresteps >= maxmstepsif ~warnedwarning('MATLAB:gmres:tooSmallTolerance', ...strcat('Input tol might be obviously smaller than',... ' eps*cond(A) and may not be achieved by GMRES\n',... ' Try to use a bigger tolerance'));flag = 3;iter = [outiter, initer];break;endendendif normr_act <= normrminnormrmin = normr_act;imin = outiter;jmin = initer;minupdated = 1;endif stag >= maxstagstepsflag = 3;break;endend % ends inner loopevalxm = 0;if flag ~= 0if minupdatedidx = jmin;elseidx = initer;endy = R(1:idx,1:idx) \ w(1:idx);additive = U(:,idx)*(-2*y(idx)*conj(U(idx,idx)));additive(idx) = additive(idx) + y(idx);for k = idx-1 : -1 : 1additive(k) = additive(k) + y(k);additive = additive - U(:,k)*(2*(U(:,k)'*additive));endx = x + additive;xmin = x;r = b - iterapp('mtimes',afun,atype,afcnstr,x,varargin{:});minv_r = r;if existM1minv_r = iterapp('mldivide',m1fun,m1type,m1fcnstr,r,varargin{:}); if ~all(isfinite(minv_r))flag = 2;endendif existM2minv_r = iterapp('mldivide',m2fun,m2type,m2fcnstr,minv_r,varargin{:});if ~all(isfinite(minv_r))flag = 2;breakendendnormr_act = norm(minv_r);r = minv_r;endif normr_act <= normrminxmin = x;normrmin = normr_act;imin = outiter;jmin = initer;endif flag == 3break;endif normr_act <= tolbflag = 0;iter = [outiter, initer];break;endminupdated = 0;end % ends outer loop% returned solution is that with minimum residualif flag == 0relres = normr_act / n2minv_b;elsex = xmin;iter = [imin jmin];relres = normr_act / n2minv_b;end% truncate the zeros from resvecif flag <= 1 || flag == 3resvec = resvec(1:(outiter-1)*inner+initer+1);indices = resvec==0;resvec = resvec(~indices);elseif initer == 0resvec = resvec(1:(outiter-1)*inner+1);elseresvec = resvec(1:(outiter-1)*inner+initer);endend% only display a message if the output flag is not usedif nargout < 2if restarteditermsg(sprintf('gmres(%d)',restart),tol,maxit,[outiter initer],flag,iter,relres);elseitermsg(sprintf('gmres'),tol,maxit,initer,flag,iter(2),relres); endendfunction sgn = scalarsign(d)sgn = sign(d);if (sgn == 0)sgn = 1;end。

matlab计算方程组

matlab计算方程组Matlab作为一款试用范围广泛的科学计算软件，其计算方程组的能力也是非常强大的。

在Matlab中，可以通过多种方式计算方程组，比如使用直接法、迭代法、线性方程组求解器等等。

下面将分步骤阐述使用Matlab计算方程组的方法。

一、使用直接法求解直接法是一种将系数矩阵直接求逆再与常数向量相乘的方法，通常在方程组的规模较小时使用。

下面是使用Matlab求解线性方程组的示例代码：```matlab% 定义系数矩阵和常数向量A = [1 2 3; 4 5 6; 7 8 9];b = [3; 6; 9];% 求解方程组x = A\b;disp(x);```这段代码首先定义了一个3x3的系数矩阵A和一个3x1的常数向量b，然后使用反斜线符号来求解方程组。

该符号将A的逆矩阵乘上b，得到解向量x。

二、使用迭代法求解当方程组的规模较大时，直接法的计算量可能会非常大，在这种情况下可以使用迭代法来求解方程组。

迭代法的主要思想是通过反复迭代求解来逼近方程组的解。

常见的迭代法有Jacobi迭代法、Gauss-Seidel迭代法等。

以Jacobi迭代法为例，下面是使用Matlab求解线性方程组的示例代码：```matlab% 定义系数矩阵和常数向量A = [1 2 3; 4 5 6; 7 8 9];b = [3; 6; 9];% 定义Jacobi迭代法函数function [x, k] = jacobi(A, b, x0, tol, max_iter)D = diag(diag(A));L = -tril(A, -1);U = -triu(A, 1);x = x0;for k = 1:max_iterx = inv(D)*(b + L*x + U*x);if norm(A*x - b) < tolreturnendendend% 求解方程组x0 = [0; 0; 0];tol = 1e-6;max_iter = 1000;[x, k] = jacobi(A, b, x0, tol, max_iter);disp(x);```这段代码首先定义了一个3x3的系数矩阵A和一个3x1的常数向量b，然后定义了一个Jacobi迭代法的函数来求解方程组。

Matlab源程序代码

function X=t2f(x)global dt df N t f T%X=t2f(x)%x为时域的取样值矢量%X为x的傅氏变换%X与x长度相同,并为2的整幂。

%本函数需要一个全局变量dt(时域取样间隔) H=fft(x);X=[H(N/2+1:N),H(1:N/2)]*dt;end(二),主程序。

1，%（1）绘出正弦信号波形及频谱global dt df t f Nclose allk=input('取样点数＝2^k, k取10摆布');if isempty(k), k=10; endf0=input('f0=取1(kz)摆布');if isempty(f0), f0=1; endN=2^k;dt=0.01; %msdf=1/(N*dt); %KHzT=N*dt; %截短期Bs=N*df/2; %系统带宽f=[-Bs+df/2:df:Bs]; %频域横坐标t=[-T/2+dt/2:dt:T/2]; %时域横坐标s=sin(2*pi*f0*t); %输入的正弦信号S=t2f(s); %S是s的傅氏变换a=f2t(S); %a是S的傅氏反变换a=real(a);as=abs(S);subplot(2,1,1) %输出的频谱plot(f,as,'b');gridaxis([-2*f0,+2*f0,min(as),max(as)])xlabel('f (KHz)')ylabel('|S(f)| (V/KHz)') %figure(2)subplot(2,1,2)plot(t,a,'black') %输出信号波形画图gridaxis([-2/f0,+2/f0,-1.5,1.5])xlabel('t(ms)')ylabel('a(t)(V)')gtext('频谱图')最佳基带系统的源程序：(一)，用到的函数f2t函数和t2f函数。

GM(1,1)模型的MATLAB源代码2

%px0为残差预测数列，ab为求得的系数,rel为平均相对误差(为百分比)
%默认的number参数为原数组大小
if nargin==1
number=max(size(x0));
end
n=max(size(x0)); %数组大小..
[px0,ab,rel]=gm11(x0,number);
Gm（1，1）
function [px0,ab,rel]=gm11(x0,number)
%[px0,ab,rel]=gm11(x0,number)
%px0为预测数列，rel为平均相对误差,rel为平均相对误差(为百分比)
%默认的number参数为原数组大小
if nargin==1 %对输入矩阵进行判断，如不是一维列矩阵，进行转置变换
px0;
while last<number
begin=last-step+1;
temp=px0(begin:last);
temp=gm11(temp,step+1);
last=last+1;
px0(last)=temp(step+1);
end
end
end
z=zeros(size(x0));
for k=2:n
z(k)=0.5*(x1(k)+x1(k-1)); %计算数据矩阵B的第一列数据
end
y=x0';
y(1)=[];
b(:,1)=-z';
b(:,2)=1;
b(1,:)=[];
ab=inv(b'*b)*b'*y; %计算参数矩阵
temp=abs(temp);

基于加权GMRES的多项式预处理广义极小残差法

基于加权GMRES的多项式预处理广义极小残差法关朋燕;李春光;景何仿【摘要】Essai对求解非对称线性方程组的广义极小残差法(GMRES)进行改进，提出加权GMRES方法(WGMRES)。

该方法具有良好的收敛效果，但计算量较大。

本文利用加权GMRES本身构造出一种有效的多项式预处理并与加权GMRES结合，构造了一种新算法。

该算法能够显著地减少加权GMRES迭代次数，并能减少运算量和储存量。

数值算例表明，相对于加权GMRES方法来说，新算法减少了运算时间和迭代次数。

%Essai presented a weighted GMRES method (GMRES) by improving GMRES method to solve nonsymmetric linear systems. The method has good convergence, but the computation cost is increasing. In this paper, we propose an efficient polynomial precondi-tioner based on WGMRES, and obtain a new algorithm. Numerical experiments indicate that the new algorithm can considerably reduce the iterative steps and computation cost.【期刊名称】《工程数学学报》【年(卷),期】2014(000)005【总页数】10页(P697-706)【关键词】多项式预处理;加权Arnoldi算法;加权GMRES算法;迭代法;平行板突扩管【作者】关朋燕;李春光;景何仿【作者单位】北方民族大学信息与计算科学学院，银川 750021;北方民族大学数值计算与工程应用研究所，银川 750021;北方民族大学数值计算与工程应用研究所，银川 750021【正文语种】中文1 引言许多科学计算问题经过离散可归结为解线性方程组其中A∈CN×N非奇异，b是列向量．解这类方程组的一类方法就是基于最小残差法其中qn−1(x)为多项式且degqn−1≤n−1，使得‖rn‖最小，或rn=b−Axn表示残差，Kn(r0,A)=span{r0,Ar0,···,An−1r0}是Krylov子空间．令pn=b−zqn−1(z)，此为残差多项式，于是rn=pn(A)r0．在这类方法中应用最广的是GMRES方法[1]，GMRES是在Krylov子空间上进行迭代，该算法的收敛性由残差模的单调性得出，而且具有良好的数值稳定性，但具有较长的迭代公式，因此每步迭代的计算量与储存量都线性增长，同时难以有效的并行运算．Saad采用循环GMRES法，但GMRES(m)[2]会失去超线性收敛，产生停滞．1998年，Essai对GMRES方法进行改进，提出加权GMRES方法[3]，其优点在于具有良好的收敛性，但也增加了计算量．Nachtigal等[4-11]讨论了GMRES和其他方法相结合策略．文献[12–17]是将加权思想与其他方法相结合文章．本文采用多项式预处理加权GMRES(m)法及多项式预处理加权GMRES(m)法．其主要问题是找到有效的低阶多项式pn(x)，迭代解应用到pn(A)Ax=pn(A)b中，这会产生一个Krylov子空间它是Krylov子空间K(n+1)(m−1)+1(A;r0)的子空间．本文直接利用加权GMRES 方法本身来构造预处理多项式，多项式的预处理和混合迭代[4]极为相似，其思想就是尽可能减少迭代次数．随着迭代次数的减少，存储量和运算量也会相应减少．2 加权Arnoldi算法先来定义D-内积及D-范数．如果D=diag{d1,d2,···,dn}(di＞0,i=1,2,···,n)是一个对角矩阵，u,v∈Rn是两个向量，那么它们的D-内积就定义为显然，当d1=d2= ···=dn=1时，(u,v)D=(u,v)．如果(u,v)D=0，就称u和v关于D相互正交，也可以表示为u⊥Dv．同样可以定义D-范数求解方程组(1)的加权Arnoldi算法[2]：步骤1 构造v1，进行第2至第6步，对于j=1,2,···,k；步骤2 计算w=Avj；步骤3 计算hij=(w,vi)D,w=w−hijvi,i=1,2,···,j；步骤4 计算hj+1,j=‖w‖D；步骤5 如果hj+1,j=0，则停止；步骤6 计算vj+1=w/hj+1,j；步骤7 结束．3 预处理多项式的构造加权GMRES方法的主要思想是Krylov子空间Kn(r0,A)找到一个向量zn，使其D-范数极小化zn∈Kn(r0,A)可写作zn=Vnyn，其中Vn=(v1,v2,···,vn)，使得是加权ArnoldiD 正交过程形成的Krylov子空间的一组向量．所以有其中现在利用残差多项式来构造出多项式预处理因子，Kn表示N×n矩阵，其列为Krylov子空间Kn=span{r0,Ar0,···,An−1r0}的基向量．由加权GMRES方法，有以下迭代公式因为Vn的列和Kn的列张成的是同一个空间，我们有其中Cn为上三角矩阵由(4)式表示出向量vj和vj+1，并代入(3)式，比较两边关于Ajv0的系数，可得到再用加权GMRES方法解Hessenberg矩阵最小二乘问题，得xn=x0+Vny．因为Vny=KnCny，可得出向量Cny为多项式qn−1(z)的系数pn(z)=1−zqn−1(z)即为残差多项式．可得令k=n−1，由此式可得qk(A)≈ A−1，可参见文献[17]，所以可用qk(A)作预处理矩阵．4 多项式预处理加权GMRES(m)算法根据上面的分析，给出以下多项式加权GMRES(m)算法，简记为PWGMRES(m)算法：1) 输入：近似解精度ε，Krylov子空间的维数m，所需计算的预处理多项式qn−1(z)的次数k，初始值x0；2) 计算3) 选择向量d，使得4) 利用加权Arnoldi过程计算Hk,Vk；5) 对于j=1,2,···,k；6) 根据(6)式计算cj+1；7) 计算8) 结束j循环；9) 由QR分解计算最小二乘问题得到的解为ym，并令xk=x0+Vky；10) 计算预处理多项式qk(z)；11) 对qk(A)Ax=qk(A)b用加权GMRES(m)算法；12) 结束．5 数值算例为了说明PWGMRES(m)算法的有效性，给出数值算例．关于加权di的选择，Essai在文献[3]中WGMRES算法里建议取其中r0的任意分量不能为零．选取向量b使得方程Ax=b的精确解为其相关残差需要说明的是本文所有算法的程序是用Matlab在4CPU,3.20GHz，内存为1.00GB的或残差电脑上编写的，且每个算例中r0的任意分量不为零．算例1 本例中矩阵nos2−sm取自矩阵市场(/MatrixMarket/)，条件数为6.3E+09，非零元素个数为2547，阶数为957×957,ε=10−15,η≤ 10−8．其k=10,m=30预处理后的矩阵与没作预处理的矩阵的残差与迭代次数和残差与迭代时间的关系，如图1所示．图1: 算例1中PWGMRES(30)和WGMRES(30)比较算例2 本例中矩阵orsirr−1−sm取自矩阵市场(/MatrixMar ket/)，条件数为1E+02，非零元素个数为6858，阶数为1030×1030,ε=10−8,η≤10−7．其k=2,m=40预处理后的矩阵与没作预处理的矩阵的残差与迭代次数和残差与迭代时间的关系，如图2所示．图2: 算例2中PWGMRES(40)和WGMRES(40)比较算例3 本例中矩阵orsreg−1−sm取自矩阵市场(/MatrixMar ket/)，条件数为1E+02，非零元素个数为14133，阶数为2205×2205,ε=10−6,η≤10−6．其k=1,m=30预处理后的矩阵与没作预处理的矩阵的残差与迭代次数和残差与迭代时间的关系，如图3所示．图3: 算例3中PWGMRES(30)和WGMRES(30)比较算例4 本例中矩阵为Grcar，阶数为1000×1000,ε=10−12,η≤ 10−8．其k=10,m=30预处理后的矩阵与没作预处理的矩阵的残差与迭代次数和残差与迭代时间的关系，如图4所示．图4: 算例4中PWGMRES(30)和WGMRES(30)比较算例5 平行板组成的突扩管道，如图5所示，流动为层流，进口为均匀流速，根据文献[18]中提出的圆图扩管内流动雷诺数的定义Re=ud/v，导出相应的进口速度uin=Re∗v/d，管道壁面为光滑的无滑移边界，密度保持恒定不变，为ρ=1.0kg/m3，进口宽度为2d=2，整个管道的宽为2D=4，其上下管道都足够长(这里取30)，出口边界采用充分发展条件，试确定管道中流体的速度分布与压力分布．图5: 算例5二维图扩管示意图边界条件：对称线上入口处：u,v为给定值；固体边界线上：u=v=0；雷诺数：Re=200．将求解区域进行剖分，网格数为15×10，方程离散后的系数矩阵呈五对角状，非零元素的个数为1500，ε=10−6,η≤10−6．其k=5,m=30．解得算例5预处理后的矩阵和没作预处理的矩阵，其最后一个压力残差与迭代次数和残差与迭代时间的关系，如图6所示．图6: 算例5中PWGMRES(30)和WGMRES(30)比较为了说明计算的效率，表1给出上述五个算例所用的CPU时间及迭代的次数；表2给出了算例5所用外迭代次数与CPU时间．表1:所用的CPU时间及迭代次数算例算法迭代次数CPU(s)算例1 PWGMRES(m)340 16.7739 WGMRES(m)3121 77.4142算例2 PWGMRES(m)776 62.7500 WGMRES(m)2490 175.6090算例3 PWGMRES(m)142 71.8600 WGMRES(m)274 86.6250算例4PWGMRES(m)406 61.4060 SIMPLEC 1041 69.5150 0.3590一个压力矩阵算例5中最后PWGMRES(m)18 WGMRES(m)112 1.5150表2: 平行板突扩管网格数为15×10时算法的CPU比较算法1外迭代次数CPU(s)PWGMRES(m)94 191.3807 WGMRES(m)94 223.7347从以上五个数值算例的迭代次数和迭代时间分别与残差的关系图及CPU时间表比较可看出，PWGMRES(m)算法收敛所需要的迭代次数以及CPU时间均比WGMRES(m)算法的少，从而说明了所构造的算法的有效性．大量的例子表明：通常解大型稀疏对称矩阵或块三对角矩阵时，预处理多项式qk(z)次数k的选择可以随意选择，但当取k为5–15时收敛效果相对较好；解其它大型稀疏类型的矩阵时k值不宜过大，这样会增加计算预处理子的时间与运算量，有可能还会导致矩阵不收敛．6 结论本文我们利用多项式预处理技术对WGMRES(m)进行改进，得到PWGMRES(m)较之WGMRES(m)有较好的收敛速度，同时也大大减少了运算量与运算时间，但如何选取预处理多项式qk(z)次数k与权值d使PWGMRES(m)收敛速度达到最优，目前我们建议的值只是经验值．参考文献：[1]Saad Y,Schultz M H.GMRES:a generalized minimal residual algorithm for solving nonsymmetric linear systems[J].Computational and Applied Mathematics,1986,7(3):856-869[2]蔡大用，白峰杉.高等数值分析[M].北京：清华大学出版社，1997 Cai D Y,Bai F S.Advanced Numerical Analysis[M].Beijing:Tsinghua University Press,1997 [3]Essai A.Weighted FOM and GMRES for solving nonsymmtric linear systems[J].Numerical Algorithms,1998,89(3-4):277-292[4]Nachtigal N M,et al.A hybrid GMRES algorithm for nonsymmetric linear systems[J].Society for Industrial and Applied Mathematics Journal on Matrix Analysis and Applications,1992,13(3):765-825[5]Saad Y.Least squares polynomials in the complex plane and their ues for solving any nonlinear systems[J].Society for Industrial and Applied Mathematics Journal on Numerical Analysis,1987,24(1):155-169[6]An H B,Bai Z Z.NGLM:a globally convergent Newton-GMRESmethod[J].Mathematica Numerica Sinica,2005,27(2):151-174[7]An H B,Bai Z Z.On efficient variants and global convergence of the Newton-GMRES method[J].Journal of Numerical Methods and Computer Applications,2005,26(4):291-300[8]徐钟济.蒙特卡罗方法[M].上海：上海科学技术出版社，1985 Xu Z J.Monte Carlo Method[M].Shanghai:Shanghai Science and Technology Press,1985 [9]全忠，向淑晃.基于GMRES的多项式预处理广义极小残差法[J].计算数学，2006,28(4):365-376 Quan Z,Xiang S H.A GMRES method based on polynomial preconditioning algorithm[J].Mathematica Numerica Sinica,2006,28(4):365-376[10]Niu Q,et al.Accelerate weighted GMRES by augmenting error approximations[J].International Journal of ComputerMathematics,2010,87(9):2101-2112[11]杨圣炜，卢琳璋.一种加权的Simpler GMRES算法[J].厦门大学学报(自然科学版)，2008,47(4):484-488 Yang S W,Lu L Z.A weighted SimpleGMRES[J].Journal of Xiamen University(Natural Science),2008,47(4):484-488[12]Cao Z H,Yu X Y.A note on weighted FOM and GMRES for solving nonsymmetric linear systems[J].Applied Mathematics and Computation,2004,151(3):719-727[13]Jing Y F,Huang T Z.Restarted weighted full orthogonalization method for shifted linear systems[J].Computers and Mathematics with Applications,2009,57(9):1583-1591[14]Najaf iH S,Zareamoghaddam H.A new computational GMRESmethod[J].Applied Mathematics and Computation,2008,199(2):527-534 [15]Najaf iH S,Ghazvini H.Weighted restarting method in the weighted Arnoldi algorithm for computing the eigenvalues of a nonsymmetric matrix[J].Applied Mathematics and Computation,2006,175(2):1276-1287[16]Meng B.A weighted GMRES method augmented with eigenvectors[J].Far East Journal of Mathematical Sciences,2009,34(2):165-176[17]Salyor P E,Smolarski D C.Implementation of an adaptive algorithm for Richardson’s method[J].Linear Algebra and its Applications,1991,154-156:615-646[18]罗奇，等.计算流体力学[M].北京：科学出版社，1983 Roach,etputational Fluid Mechanics[M].Beijing:Science Press,1983。

Matlab实现GM（1,1）模型（源代码）

Matlab实现GM（1,1）模型（源代码）关于这个模型的介绍不想多说了，只是⼀个娱乐⽽已。

下⾯是所有的代码，直接粘到你的M⽂件⾥⾯，然后跑就是了。

⼀分钱不收。

function [ simulation,params] = GM( org )n=length(org);%⼀次累加for i=1:nacc(i)=sum(org(1:i));end%计算背景值for i=1:(n-1)zk(i)=0.5*(acc(i)+acc(i+1));end%求解参数params=polyfit(zk,org(2:end),1);%计算模拟值for i=1:nif i==1simulation(i)=org(1);elsesimulation(i)=(org(1)+params(2)/params(1))*(1-exp(-params(1)))*exp(params(1)*(i-1));endendplot(1:n,org,'-o',1:n,simulation,'-*');legend('原始序列','拟合序列');细⼼的朋友应该会注意到⼀件事情：灰⾊预测模型⾥⾯不是有那么多矩阵运算吗，在这⾥怎么没有？如果你跑了程序⼜会发现，这个程序计算出来的结果，跟任何⼀本书上写的结果完全⼀样。

这是为啥嘞？如果再细⼼点，会看到我这⾥获取参数的代码： params=polyfit(zk,org(2:end),1);。

没错，就是线性拟合。

任何⼀本讲灰⾊理论的书上都会告诉你，参数估计是由这个式⼦加上最⼩⼆乘法得到的。

这事情的确没有错，但如果你仔细点，把移到等式的右边去，你发现了什么。

没错，这TM不就是⼀个离散直线⽅程么。

那么这⾥只需要把看成是直线⽅程⾥的y,⽽把看成是x。

那这个参数估计，就只是个简单的线性回归⽽已。

这样⼀来，只要是能实现线性回归的函数，完全都可以实现它所谓的参数估计。

gml matlab代码

GML MATLAB代码什么是GML？GML（GameMaker Language）是一种专门用于游戏开发的脚本语言，它是由YoYo Games开发的GameMaker软件所使用的语言。

GML是一种基于C语言的脚本语言，具有简洁、易于学习和使用的特点。

GML提供了丰富的功能和库，方便开发者创建2D游戏。

什么是MATLAB？MATLAB是一种高级的数值计算和科学编程语言，由MathWorks开发。

它具有强大的数值分析和数据可视化功能，广泛应用于工程、科学和金融领域。

MATLAB提供了丰富的函数和工具箱，方便开发者进行数据处理、图像处理、信号处理等各种科学计算。

GML和MATLAB之间的联系尽管GML和MATLAB是两种不同的编程语言，但它们在某些方面存在联系。

首先，它们都是面向特定领域的编程语言。

GML专注于游戏开发，而MATLAB则专注于数值计算和科学工程。

其次，它们都提供了丰富的函数和库，方便开发者进行快速开发。

在某些情况下，开发者可能需要将GML和MATLAB结合起来使用。

例如，在游戏开发中，可能需要使用MATLAB进行复杂的数学计算或图像处理，然后将结果应用到游戏中。

为了实现这一目标，可以使用以下两种方法：方法一：通过GML调用MATLABGML提供了一些函数和方法，可以直接调用MATLAB的函数。

开发者可以使用GML 的外部脚本功能，将MATLAB代码封装到一个外部脚本中，然后通过GML调用该脚本。

这样就可以在GML中使用MATLAB的功能。

以下是一个简单的例子，演示了如何在GML中调用MATLAB的函数：var result;result = external_call("matlab_script", "add_numbers", 2, 3);show_message("The result is: " + string(result));上述代码中，external_call函数用于调用MATLAB脚本中的add_numbers函数，将参数2和3传递给该函数，并将结果存储在result变量中。