代碼資源精通matlab科學(xué)計(jì)算063rd

6編程來(lái)實(shí)現(xiàn)這些算法。函數(shù)函數(shù)在區(qū)間[a,b上已知一連續(xù)函數(shù)f(x，如果f(x的表達(dá)式太過(guò)復(fù)雜不利于用計(jì)算機(jī)來(lái)進(jìn)行計(jì)算，自然而然地想到用一簡(jiǎn)單函數(shù)去近似f(x)，這就是函數(shù)近問(wèn)題。如果f(x的表達(dá)式未知，只知道描述f(x切￥ ￥f(x)fnTn其中Tn(x)為n次切多項(xiàng)式，具體表達(dá)式可通過(guò)遞推得出T0(x)1,T1(x)Tn1(x)2xTn(x)1Tn(x)Tm(x)dx

nnm112

nmf1f0f1fn21Tn(x)f11在中編程實(shí)現(xiàn)的切近法函數(shù)為：Chebyshev。調(diào)用格式：f=Chebyshevy,k)fChebyshev其中，yk為近已知函數(shù)所需項(xiàng)數(shù)x0是近點(diǎn)的x坐標(biāo)f是求得的切近多項(xiàng)式或是在x0處的近值。在中實(shí)現(xiàn)切近的代碼如下：functionf=%近點(diǎn)的x坐標(biāo)%求得的切近多項(xiàng)式或在x0處的近值symst;T(1:k+1)=t;T(1)=T(2)=t;c(1:k+1)=0.0;f=c(1)+c(2)*t;forT(i)=2*t*T(i-1)-T(i-f=f+c(i)*T(i);f=if(nargin==f=subs(f,'t',x0);f=

【例6-1】 （取6項(xiàng) 近函數(shù) ，2求當(dāng)x0.5解：在命令窗口中輸入以下命令>>f=Chebyshev('1/(2-x)',6)f=>>f=Chebyshev('1/(2-x)',6,0.5)f=0.6293事實(shí)上函數(shù)的準(zhǔn)確值為1(20.5)0.6667，近結(jié)果比較接近準(zhǔn)確值勒讓德P0(x)1,P1(x)(n1)Pn1(x)(2n1)xPn(x)1P1

(x)dx

n1

n

2nf2n11P(x)f 1在中編程實(shí)現(xiàn)的勒讓德近法函數(shù)為：Legendre。調(diào)用格式：fLegendrey,k)fLegendre其中，yk為近已知函數(shù)所需項(xiàng)數(shù)x0為近點(diǎn)的x坐標(biāo)f為求得的勒讓德近多項(xiàng)式或在x0處的近值。在中實(shí)現(xiàn)勒讓德近的代碼如下：functionf=%近點(diǎn)的x坐標(biāo)%求得的勒讓德近多項(xiàng)式或在x0處的近值symst;P(1:k+1)=t;P(1)=P(2)=t;c(1:k+1)=0.0;f=c(1)+c(2)*t;forP(i)=((2*i-3)*P(i-1)*t-(i-2)*P(i-2))/(i-c(i)=int(subs(y,findsym(sym(y)),t)*P(i),t,-1,1)/2;f=f+c(i)*P(i);if(nargin==f=subs(f,'t',x0);f=

【例6-2】勒讓 （取6項(xiàng) 2

x0.5解：在命令窗口中輸入以下命令>>f=Legendre('1/(2-f>>f=Legendre('1/(2-x)',6,0.5)f=從近結(jié)果看，函數(shù)的準(zhǔn)確值為1(20.5)0.6667，與上例相比，可看出勒讓德級(jí)數(shù) LLpkMf(x) M1qk首先，求解線性方程組Aqb，得到(q1,q2 ,qn)的值。其

qn an1

an1 qn1 an2A ,q ,b

a2n1

a2n 1dnfa0f(0),ann!然后通過(guò)下式求出(p0, ,pn)的值p0a0,q0

nnpn在中編程實(shí)現(xiàn)的近法函數(shù)為：Pade。調(diào)用格式：f=Pade(yn)fPadeyn其中，yx0為近點(diǎn)的x坐標(biāo)f為求得的有理分式或在x0處的近值。在中實(shí)現(xiàn)函數(shù)的近的代碼如下：functionf=%近點(diǎn)的x坐標(biāo)%求得的有理分式或在x0處的近值symsA=q=zeros(n,1);p=zeros(n+1,1);b=zeros(n,1);yy=0;a(1:2*n)=yy=a(i)=subs(sym(yy),findsym(sym(yy)),0.0)/factorial(i);b(i,1)=-a(n+i);q=p(1)=subs(sym(y),findsym(sym(y)),0.0);p(i+1)=a(n)+q(i)*subs(sym(y),findsym(sym(y)),0.0);f_1=f_2=1;f_1=f_1+p(i)*(t^(i-f_2=f_2+if(nargin==3)f=f=subs(f,'t',x0);f=f_1/f_2;f=vpa(f,6); （取4項(xiàng)）近函數(shù)1

，并求當(dāng)x解：在命令窗口中輸入以下命令>>f=Pade('1/(1-f>>f=Pade('1/(1-f從近結(jié)果看，函數(shù)的準(zhǔn)確值為1(10.5)2，而用4次有理分式的近已經(jīng)達(dá) 在中編程實(shí)現(xiàn)的連續(xù)周期函數(shù)的近法函數(shù)為：FZZ。調(diào)用格式：[A0,ABFZZfuncTn)。其中，func為已知函數(shù)；T為已知函數(shù)的周期；n為展開(kāi)級(jí)數(shù)的項(xiàng)數(shù)；A0AB在中實(shí)現(xiàn)連續(xù)周期函數(shù)的近的代碼如下function[A0,A,B]=FZZ(func,T,symsfunc=subs(sym(func),findsym(sym(func)),sym('t'));A(k)=int(func*cos(2*pi*k*t/T),t,-T/2,T/2)*2/T;B(k)=int(func*sin(2*pi*k*t/T),t,-T/2,T/2)*2/T;【例6-4】近應(yīng)用實(shí)例。用級(jí)數(shù)（取5項(xiàng)）近函數(shù)yx，輸出系解：在命令窗口中輸入以下命令>>[A0,A,B]=FZZ('x',2*pi,5)A0=0A[0.,0.,0.,0.,B -1.,.6667,-.5000,結(jié)果表明，如果取5項(xiàng)近函數(shù)yx，有下面的式子y2sin

sin

0.5sin

yx0

N1NN

ykiknck

fne

N(k ,n在中編程實(shí)現(xiàn)的離散周期數(shù)據(jù)點(diǎn)近法函數(shù)為：DFF。調(diào)用格式：cDFFfN)其中，fNc為離散近系數(shù)在中實(shí)現(xiàn)離散周期函數(shù)的近的代碼如下functionc=%近系數(shù)【例6-5】離散近應(yīng)用實(shí)例。對(duì)下列數(shù)據(jù)點(diǎn)進(jìn)行離散變換N123456y000-0-0-0解：在命令窗口中輸入以下命令>>y=[0.84150.90930.1411-0.7568-0.9589->>c=cColumns1through-0.0926-0.5003i-0.0260-0.0194i-0.0251+Columns4through-0.0260+0.0194i-0.0926+0.5003i-0.0172-6.2對(duì)給定的試驗(yàn)數(shù)據(jù)點(diǎn)(xi,yi)(i1, ,N)，可構(gòu)造m次多項(xiàng)式P(x)a0 [ajxijyi

(mNi1j cma0 b0

cm1a1

a b

Nkckxi,(k iibk

NN

yxk,(k 在中編程實(shí)現(xiàn)的多項(xiàng)式曲線擬合函數(shù)為：multifit。調(diào)用格式：AmultifitX,Y,m)其中，Xx坐標(biāo)向量；Yy坐標(biāo)向量；m為擬合多項(xiàng)式的次數(shù)；A在中實(shí)現(xiàn)多項(xiàng)式曲線擬合的代碼如下function%試驗(yàn)數(shù)據(jù)點(diǎn)的x%試驗(yàn)數(shù)據(jù)點(diǎn)的yif(N~=M)forj=1:(2*m+1) %求出c和bfork=1:N

fors=2:(m+1) 6-6】x123y25解：在命令窗口中輸入以下命令>>x=>>y=[25>>A=A y0.12820.3235x0.8718x2實(shí)驗(yàn)數(shù)據(jù)(xiyiyf(x，擬合的標(biāo)準(zhǔn)是使（f(xiyi）的平方取極小, yf(x)ax, (xi,yi

i0,1,n

F(a,b)(axibyi解出a與bpolyfit

a

bP=polyfit(X,Y,N次多項(xiàng)式擬合數(shù)據(jù)點(diǎn)[xiyi6-7】polyfitx11111y11111解：在命令窗口中輸入以下命令>>x=>>y=[11.0954 1.2649>>p=P-0.01040.0854-0.31210.9086即擬合的多項(xiàng)式為：y0.0104x40.0854x30.3121x20.9086x0.3285在中編程實(shí)現(xiàn)的線性最小二乘擬合函數(shù)為：LZXEC 試驗(yàn)數(shù)據(jù)點(diǎn)的x坐標(biāo)向量； 試驗(yàn)數(shù)據(jù)點(diǎn)的y坐標(biāo)向量； function%試驗(yàn)數(shù)據(jù)點(diǎn)的x%試驗(yàn)數(shù)據(jù)點(diǎn)的yif(length(xlength(y))ndisp('x和y的維數(shù)不相等！'); A(2,2)zeros(2,1);fori1:nA(1,1)A(1,1)A(1,2)A(1,2)B(1,1)B(1,1)B(2,1)B(2,1)A(2,1)sab6-8】x12345y11435解：在命令窗口中輸入以下命令>>x>>y[1.51.843.4>>[a,b]LZXECa b yx正交多項(xiàng)式最小二乘擬合是選取一組在給定點(diǎn)上正交的多項(xiàng)式函數(shù)系Bi(x)作為基p(x)b0B0(x)b1B1(x) bmBmnnyiBj(xinbj n

B2(x

xkBi(xkik B0(x)1,B1(x)x Bi1(x)(xi)Bi(x)i

Bi(xk B2(x ik Bi1(xk 對(duì)給定的試驗(yàn)數(shù)據(jù)點(diǎn)(xi,yi)(i1,2, ,N)，構(gòu)造m次正交多項(xiàng)式最小二乘擬合的多項(xiàng)令B0(x)1，根據(jù)遞推有： b0(yk)/(n1),0(xk)/(n a0b0

k

k

ykB1(x ykB1(xkxB2(xk 11B1(x)c0c1x由遞推有c0-0,c11，b1k0 ,1k0 , 11n1B2(xkn1

k

B2(xk

k

B2(xkn1k n00k

B2(xk更新近多項(xiàng)式的系數(shù)：a0a0b1c0,a1對(duì)于t2,3, ,m，設(shè)Bt(x)r0r1x rtxt，Bt1(x)s0s1x st1xt1Bt2(x)w0w1x wt2xt2，由遞推rtstrt1t1st1st,trit1sisi1t1wi(i,takakbtatbt

k0,1,,t在中編程實(shí)現(xiàn)的正交多項(xiàng)式最小二乘擬合函數(shù)為,t其中，x：試驗(yàn)數(shù)據(jù)點(diǎn)的x坐標(biāo)向量；y：試驗(yàn)數(shù)據(jù)點(diǎn)的y坐標(biāo)向量；m：擬合多項(xiàng)式的次數(shù)；a：擬合多項(xiàng)式的系數(shù)向量。function%試驗(yàn)數(shù)據(jù)點(diǎn)的x%試驗(yàn)數(shù)據(jù)點(diǎn)的yif(length(xlength(y))nlength(x);disp('x和y symsdzeros(1,m1);qalphazeros(1,m1);fork0:m B2d(1)n;forq(1)q(1)alpha(1)alpha(1)q(1)alpha(1)a(1)q(1); B1[alpha(1)1];for

d(2)d(2)q(2)q(2)alpha(2)alpha(2)q(2)alpha(2)a(1 a(2)q(2); betad(2)/d(1);forBB(i)B(i1)alpha(i1)*B1(i1)B1(i2);forj2:i2B(j)alpha(i1)*B1(j)B1(j1)beta*B2(j);B(1)BFB*transpose(px(1:i));forl1:nQxd(i)d(i)q(i)q(i)alpha(i)alpha(i)alpha(i)q(i)