这是一个 MATLAB 脚本,用于进行最小二乘法拟合。脚本首先要求用户输入已知点的 x 和 y 坐标,然后输入拟合的多项式次数 n。脚本使用最小二乘法拟合数据,并绘制了原始数据点和拟合曲线的图表。以下是对代码的主要部分的解释:/ @8 N9 G( b' F& e; {& l8 E3 q
function fp = fitpt()$ A+ j1 S+ J6 S( M1 X
% 最小二乘' J% p6 c; n; s! B! e5 w, |
% 基取 {1, x, ...}5 ]) B$ B2 m: }( P0 b/ {
% fitpt.m- V5 g7 |9 q# u1 Z! W4 a, F
8 P" l4 K7 ?% C; Y |; I- B3 a % 默认算例为课本:P65,例3.27 Q$ e! N) A! j% W C2 B8 G
% x = [0,1,2,3,4,5,6,7] ) d. C! f* i0 g* u* k5 ]5 {2 T % y = [3.95,6.82,9.78,12.91,15.74,19.26,21.73,24.07] 2 q$ Q, y' z6 m4 [1 c % 结果:P(x) = 4.005 + 2.936x 平方误差=0.6162 & t8 @" K0 i2 }( x $ e7 n: X# i5 U2 a. D % MatLab函数:polyfit(x, y, n)+ x2 e4 N3 i& d: W# t' N4 X
+ S* c% K$ q8 N# ]6 |+ I" |: e& C s = input('<最小二乘>\n输入已知点的x坐标:(回车表示[0,1,2,3,4,5,6,7])\n', 's');9 i! Q7 Y8 D. u
if isempty(s): N; x7 n8 J2 a8 ^) B9 r4 d
s = '[0,1,2,3,4,5,6,7]';2 e6 o7 u1 M* D j, V+ I
else5 _% j. U# D& R3 B
if (s(1) ~= '[')" I# M. U( h- w! [
s = strcat('[', s);4 a) T+ K) `" `
s = strcat(s, ']'); ) @5 ~! X: E, y end 5 K8 C( t5 ?4 ~) i end ) ^; M t8 f4 u8 B x = sym(s); & \- K+ m* N: w9 a- \' R6 Q: i+ G w4 a1 I D
s = input('输入已知点的y坐标:(回车表示[3.95,6.82,9.78,12.91,15.74,19.26,21.73,24.07])\n', 's');) ]# X6 U" x; ]* w3 U
if isempty(s) & c3 S/ X( Z: g1 J4 \! H: W& {. f s = '[3.95,6.82,9.78,12.91,15.74,19.26,21.73,24.07]'; ) \$ `, a8 j" e- }( x else - f* n& m# H; ~/ s9 c& k+ [ if (s(1) ~= '[') # T: f7 H. M7 Z% n9 C6 Q9 ] s = strcat('[', s);- ^* ~% x7 J% i( f" H: P u. I
s = strcat(s, ']'); 3 ~- G' m( {: E2 | end 4 }$ H1 [7 l8 f end 0 f) a Z( k- D& z% t y = sym(s);. n. K/ P; P6 d$ p
sz = size(x);' x0 U0 B6 _+ a) _. R( U
sz = sz(2); 1 k% M+ D" M$ j9 `! Z n = input('输入多项式次数n:');" u6 S$ l0 Z8 T5 f, f9 o
if (n + 1 > sz) T# g6 b1 l' i/ b- O' L/ }" p n = input('多项式次数需要小于已知点个数,请重新输入n:'); 5 P' d3 Q3 r' ?8 q: W7 l end" G# ]8 }, x. a6 i" |
if (n + 1 > sz) - o1 |5 d; f: {6 G0 j2 E error('多项式次数不能小于已知点个数!');+ _) v" b3 {) ]5 J' h( ?
end - w/ l% G' Y! o. L7 w# C fp = s_fitpt_p(x, y, n);7 |: }$ I0 x6 y( b
5 v! \: Z) G: f% _! K/ \5 _; } % 绘制原始数据点和拟合曲线6 k% A- {, o- K* d: Y
plot(double(x), double(y), 'r*') ) H" r7 T! @! M8 {: U S8 ^# I9 V hold on : y4 a1 X4 x0 Q+ R S a = double(x(1));" y' P4 m' U& h
b = double(x(sz));: Z Z p# L7 V. P, d/ }
x = a:abs(b - 1)/100:b;3 g/ Q# q! c' r
y = subs(fp, x);5 y. {; ?8 D7 K# l" W
plot(x, y) & L6 C6 R( e; M2 o& vend ! y8 D; Y ?9 h3 \7 t7 v, G! D$ n( s" f; @, D
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%- J& _8 n- Q: q
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function f = s_fitpt_p(x, y, n)( _" j2 T7 x: g1 ^1 A
% 用 n 次多项式实现的最小二乘法1 K4 Y4 E5 e# p0 g U: ^: c1 j
, ?+ R8 q1 b3 g0 B% l/ m, m sz = size(x); 6 ^8 L9 F9 A$ a sz = sz(2);6 v* r- U! W& j8 V; U6 q' \
A = zeros(sz, n + 1); ! Q. R( b; b( O8 M! X, u3 X v = vh(n); , f. }; P# D/ F% X for i = 1:sz 0 u3 o0 y" v2 T* S A(i, = subs(v, double(x(i)));1 g. D2 T" f2 A: }9 ]2 g
end ! g3 R' l% d" b# f) a4 i) s/ w3 H+ I f = linsolve(A' * A, A' * y');5 r S8 i2 `. E
f = vpa(f, 4); 0 S3 W, [8 x. j/ i7 y$ V9 D f = v * f; & u$ [1 r: r4 s8 pend3 ]4 Z' E+ H) f5 z+ m
; n3 K. x, e% J/ t%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1 i7 ~9 F) ]- N+ l k; \# U1 Z3 E
function v = vh(n) . w" ~1 ~0 p3 J! l; C4 | % Create vector in horizontal style, such as & h/ j: a* S- o- z7 R0 m % v = [1, x, x^2, ..., x^n]3 n7 K( a( [ P( v: ?1 ]
9 v* o9 K% C* P. {) J if (n < 0 || n > 9)3 w7 h; T3 R6 M# H9 s( o m! q& K
error('Make sure ''n'' is in range of [0, 9]')1 Q9 A$ [ F: R
end9 o7 j# k6 l3 q5 s+ n
s = ''; ( G0 `. q8 b& Q2 D+ d0 A4 g for i = 0:n& } X9 v( `6 x0 G% g4 L
s = strcat(s, ',x^'); 3 V9 d2 \- D+ u) J- ^' y! G; s s = strcat(s, num2str(i)); # x2 h1 b; E8 ~7 W end % B5 E: C4 [/ i x5 P9 g s(1) = '['; 5 U% s7 L% }' f! x' C3 l4 R sz = size(s); ! ^9 z' r3 j. B; a- w s(sz(2) + 1) = ']';' `: K7 s9 h* {/ B) m
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v = simplify(sym(s));1 Z" o) \ V* `( B( L5 t
end7 t% d: H& O$ u; d# {+ C
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这个脚本首先获取用户输入的已知点的 x 和 y 坐标,然后使用最小二乘法进行拟合。最后,脚本绘制了原始数据点和拟合曲线的图表。) m* ]5 A E' ~+ X2 X2 Z( T4 M