Question

matlab编程高手请进!!

求不定方程a^6+b^6+c^6+d^6+e^6+f^6=6*t^6的互不相等的整数解，且a<b<c<d<e<f，5<=t<=55
程序如下

function k
syms int64 t a b c d e f
for t=5:55;
for a=1:t;
for b=a+1:round(1.04*t);
for c=b+1:round(1.07*t);
for d=c+1:round(1.13*t);
for e=d+1:round(1.21*t);
f=round((6*t^6-a^6-b^6-c^6-d^6-e^6)^(1/6));
if f<e
break
elseif int64(f^6)==int64(6*t^6-a^6-b^6-c^6-d^6-e^6)
disp(t)
disp(a)
disp(b)
disp(c)
disp(d)
disp(e)
disp(f)
end
end
end
end
end
end
//
程序中for循环太多影响运算速度，怎样去掉for循环向量化?能否给出最优算法？不胜感激！

Answer 1

用cftool工具。
>>cftool
得到
g=2
Linear model Poly2:
f(x) = p1*x^2 + p2*x + p3
Coefficients (with 95% confidence bounds):
p1 = 8.959e-005 (-6.018e-007, 0.0001798)
p2 = 0.003271 (-0.004365, 0.01091)
p3 = 1.024 (0.9125, 1.136)

Goodness of fit:
SSE: 0.005153
R-square: 0.9922
Adjusted R-square: 0.9871
RMSE: 0.04145
g=4
Linear model Poly4:
f(x) = p1*x^4 + p2*x^3 + p3*x^2 + p4*x + p5
Coefficients (with 95% confidence bounds):
p1 = 3.78e-008 (-3.783e-008, 1.134e-007)
p2 = -3.332e-006 (-1.496e-005, 8.293e-006)
p3 = 6.594e-005 (-0.0004892, 0.0006211)
p4 = 0.007842 (-0.0009549, 0.01664)
p5 = 0.9995 (0.9621, 1.037)

Goodness of fit:
SSE: 8.94e-006
R-square: 1
Adjusted R-square: 0.9999
RMSE: 0.00299

General model Exp1:
f(x) = a*exp(b*x)
Coefficients (with 95% confidence bounds):
a = 0.9802 (0.8973, 1.063)
b = 0.007864 (0.006483, 0.009244)

Goodness of fit:
SSE: 0.009766
R-square: 0.9853
Adjusted R-square: 0.9816
RMSE: 0.04941

Answer 2

用cftool工具。

Answer 3

Linear model Poly2:
f(x) = p1*x^2 + p2*x + p3
Coefficients (with 95% confidence bounds):
p1 = 8.959e-005 (-6.018e-007, 0.0001798)
p2 = 0.003271 (-0.004365, 0.01091)
p3 = 1.024 (0.9125, 1.136)

Goodness of fit:
SSE: 0.005153
R-square: 0.9922
Adjusted R-square: 0.9871
RMSE: 0.04145
g=4
Linear model Poly4:
f(x) = p1*x^4 + p2*x^3 + p3*x^2 + p4*x + p5
Coefficients (with 95% confidence bounds):
p1 = 3.78e-008 (-3.783e-008, 1.134e-007)
p2 = -3.332e-006 (-1.496e-005, 8.293e-006)
p3 = 6.594e-005 (-0.0004892, 0.0006211)
p4 = 0.007842 (-0.0009549, 0.01664)
p5 = 0.9995 (0.9621, 1.037)

Goodness of fit:
SSE: 8.94e-006
R-square: 1
Adjusted R-square: 0.9999
RMSE: 0.00299

General model Exp1:
f(x) = a*exp(b*x)
Coefficients (with 95% confidence bounds):
a = 0.9802 (0.8973, 1.063)
b = 0.007864 (0.006483, 0.009244)

Goodness of fit:
SSE: 0.009766
R-square: 0.9853
Adjusted R-square: 0.9816
RMSE: 0.04941