% matlab语言把多项式表达成一个行向量，该向量中的元素是按多项式   %降幂排列的。

% f(x)=an^n+an-1^n-1+……+a0

% 可用行向量 p=[an an-1 …… a1 a0]表示

clear all ;

clc;

close all;

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% 1. poly —— 产生特征多项式系数向量

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% 特征多项式一定是n+1维的

% 特征多项式第一个元素一定是1

a=[1 2 3;4 5 6;7 8 0];

p=poly(a)

% p =1.00 -6.00 -72.00 -27.00

% p是多项式p(x)=x^3-6x^2-72x-27的matlab描述方法，我们可用：

p1=poly2str(p,'x') %— 函数文件，显示

% 数学多项式的形式

% p1 =x^3 - 6 x^2 - 72 x - 27

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% 2.roots —— 求多项式的根

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r=roots(p)

% r = 12.12

% -5.73 ——显然 r是矩阵a的特征值

% -0.39

% 当然我们可用poly令其返回多项式形式

p2 = poly(r)

% p2 =

% 1.00 -6.00 -72.00 -27.00

% matlab规定多项式系数向量用行向量表示，一组根用列向量表示

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% 3.conv,convs多项式乘运算

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% 例:a(x)=x^2+2x+3; b(x)=4x^2+5x+6;

% c = (x^2+2x+3)(4x^2+5x+6)

a=[1 2 3];b=[4 5 6];

c=conv(a,b) %=conv([1 2 3],[4 5 6])

% c = 4.00 13.00 28.00 27.00 18.00

p=poly2str(c,'x')

% p = 4 x^4 + 13 x^3 + 28 x^2 + 27 x + 18

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% 4.deconv多项式除运算

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a=[1 2 3];

c = [4.00 13.00 28.00 27.00 18.00]

d=deconv(c,a)

% d =4.00 5.00 6.00

% [d,r]=deconv(c,a)

% r为余数

% d为c除a后的整数

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% 5.多项式微分

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% matlab提供了polyder函数多项式的微分。

% 命令格式：

% polyder(p): 求p的微分

% polyder(a,b): 求多项式a,b乘积的微分

% [p,q]=polyder(a,b): 求多项式a,b商的微分

% 例：

a=[1 2 3 4 5];

poly2str(a,'x')

% ans = x^4 + 2 x^3 + 3 x^2 + 4 x + 5

b=polyder(a)

% b = 4 6 6 4

poly2str(b,'x')

% ans =4 x^3 + 6 x^2 + 6 x + 4

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%6.多项式拟合

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% P = POLYFIT(X,Y,N) finds the coefficients of a polynomial P(X) of

% degree N that fits the data Y best in a least-squares sense. P is a

% row vector of length N+1 containing the polynomial coefficients in

% descending powers, P(1)*X^N + P(2)*X^(N-1) +...+ P(N)*X + P(N+1).

x0=0:0.1:1;

y0=[-.447 1.978 3.11 5.25 5.02 4.66 4.01 4.58 3.45 5.35 9.22];

p=polyfit(x0,y0,3)

% p = 56.6915 -87.1174 40.0070 -0.9043

% Y = POLYVAL(P,X) returns the value of a polynomial P evaluated at X. P

% is a vector of length N+1 whose elements are the coefficients of the

% polynomial in descending powers.

% Y = P(1)*X^N + P(2)*X^(N-1) + ... + P(N)*X + P(N+1)

xx=0:0.01:1;yy=polyval(p,xx);

plot(xx,yy,'-b',x0,y0,'or')