正交多項式

這段代碼可以給出常用的4類正交多項式的具體表達式,后續將給出課自定義的任意正交多項式代碼

%%正交多項式%%此函數包括勒讓德正交多項式,切比雪夫正交多項式(兩類),拉蓋爾正交多項式,埃爾米特正交多項式,輸入項數應從1開始%%n是多項式的項數,n>=0,type是類型,分為Legendre、Chebyshev、Laguerre、Hermite,對應其正交多項式function OP = Orthogonal_polynomial(type)sym type;if strcmp(type,'Legendre') == 1    disp('目前輸出為勒讓德多項式');    disp('定義區間為:');[-1 1]    N = input('請輸入勒讓德多項式的項數:');    L = Legendre(N);    OP = simplify(L(N));elseif strcmp(type,'Chebyshev') == 1    disp('目前輸出為切比雪夫多項式');    disp('定義區間為:');[-1 1]    disp('在這里,規定第一類切比雪夫多項式是以1/sqrt(1-x^2)作為權函數,第二類切比雪夫多項式以sqrt(1-x^2)作為權函數得到的');    class = input('請輸入需要輸出第幾類切比雪夫多項式(輸入1,2)即可:');    N = input('請輸入切比雪夫多項式的項數:');    Che = Chebyshve(N,class);    OP = simplify(Che(N));elseif strcmp(type,'Laguerre') == 1    disp('目前輸出為拉蓋爾多項式');    disp('定義區間為:');[0 +inf]    N = input('請輸入拉蓋爾多項式的項數:');    La = Laguerre(N);    OP = simplify(La(N));elseif strcmp(type,'Hermite') == 1    disp('目前輸出為埃爾米特多項式');    disp('定義區間為:');[-inf +inf]    N = input('請輸入埃爾米特多項式的項數:');    H = Hermite(N);    OP = simplify(H(N));end%%勒讓德多項式    function L = Legendre(N)        x = sym('x');        for i = 1:N            Leg(i) = diff((x^2-1)^(i-1),i-1)/(factorial(i-1)*2^(i-1));        end        L = Leg;    end%%切比雪夫多項式    function C = Chebyshve(n,class)        x = sym('x');        if class == 1            T = string([1 x]);            T = sym(T);            if n <=2                C = T(1:n);            else                for i = 2:n                    T(i+1) = 2*x*T(i)-T(i-1);                end                C = T(1:n);            end        elseif class ==2            U = string([1]);            U = sym(U);            U = [U 2*x];            if n <=2                C = U(1:n);            else                for i = 2:n                    U(i+1) = 2*x*U(i)-U(i-1);                end                C = U(1:n);            end        end    end%%埃爾米特多項式    function H = Hermite(N)        x = sym('x');        for i = 1:N            He(i) = (-1)^N*exp(x^2)*diff(exp(-x^2),(i-1));        end        H = simplify(He);    end%%拉蓋爾多項式    function La = Laguerre(N)        x = sym('x');        for i = 1:N            Lag(i) = exp(x)*diff(x^(i-1)*exp(-x),(i-1));        end        La = simplify(Lag);    end%%階乘函數    function F = factorial(n)        if n == 0            F = 1;        else            F = factorial(n-1)*n;        end    endend

%%正交多項式%%此函數包括勒讓德正交多項式,切比雪夫正交多項式(兩類),拉蓋爾正交多項式,埃爾米特正交多項式,輸入項數應從1開始%%n是多項式的項數,n>=0,type是類型,分為Legendre、Chebyshev、Laguerre、Hermite,對應其正交多項式function OP = Orthogonal_polynomial(type)sym type;if strcmp(type,'Legendre') == 1 disp('目前輸出為勒讓德多項式'); disp('定義區間為:');[-1 1] N = input('請輸入勒讓德多項式的項數:'); L = Legendre(N); OP = simplify(L(N));elseif strcmp(type,'Chebyshev') == 1 disp('目前輸出為切比雪夫多項式'); disp('定義區間為:');[-1 1] disp('在這里,規定第一類切比雪夫多項式是以1/sqrt(1-x^2)作為權函數,第二類切比雪夫多項式以sqrt(1-x^2)作為權函數得到的'); class = input('請輸入需要輸出第幾類切比雪夫多項式(輸入1,2)即可:'); N = input('請輸入切比雪夫多項式的項數:'); Che = Chebyshve(N,class); OP = simplify(Che(N));elseif strcmp(type,'Laguerre') == 1 disp('目前輸出為拉蓋爾多項式'); disp('定義區間為:');[0 +inf] N = input('請輸入拉蓋爾多項式的項數:'); La = Laguerre(N); OP = simplify(La(N));elseif strcmp(type,'Hermite') == 1 disp('目前輸出為埃爾米特多項式'); disp('定義區間為:');[-inf +inf] N = input('請輸入埃爾米特多項式的項數:'); H = Hermite(N); OP = simplify(H(N));end%%勒讓德多項式 function L = Legendre(N) x = sym('x'); for i = 1:N Leg(i) = diff((x^2-1)^(i-1),i-1)/(factorial(i-1)*2^(i-1)); end L = Leg; end%%切比雪夫多項式 function C = Chebyshve(n,class) x = sym('x'); if class == 1 T = string([1 x]); T = sym(T); if n <=2 C = T(1:n); else for i = 2:n T(i+1) = 2*x*T(i)-T(i-1); end C = T(1:n); end elseif class ==2 U = string([1]); U = sym(U); U = [U 2*x]; if n <=2 C = U(1:n); else for i = 2:n U(i+1) = 2*x*U(i)-U(i-1); end C = U(1:n); end end end%%埃爾米特多項式 function H = Hermite(N) x = sym('x'); for i = 1:N He(i) = (-1)^N*exp(x^2)*diff(exp(-x^2),(i-1)); end H = simplify(He); end%%拉蓋爾多項式 function La = Laguerre(N) x = sym('x'); for i = 1:N Lag(i) = exp(x)*diff(x^(i-1)*exp(-x),(i-1)); end La = simplify(Lag); end%%階乘函數 function F = factorial(n) if n == 0 F = 1; else F = factorial(n-1)*n; end endend