411 REFERENCE · Wikibooks
Linear Algebra and the C Language/a0jy
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__NOTOC__
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/* ------------------------------------ */
/* Save as : c00e.c */
/* ------------------------------------ */
#include "v_a.h"
/* ------------------------------------ */
/* ------------------------------------ */
#define RA R5
#define CA C5
#define CXY C2
/* ------------------------------------ */
int main(void)
{
double tA[RA*CA]={
/* x**4 x**3 x**2 x**1 x**0 */
1, 1., 1., 1., 1.,
16., 8., 4., 2., 1,
81., 27., 9., 3., 1,
256., 64., 16., 4., 1,
625., 125., 25., 5., 1,
};
double tb[RA*C1]={
/* y */
-5.,
8.,
-7.,
1.,
-4.
};
double xy[RA*CXY] ={
1, -5,
2, 8,
3, -7,
4, 1,
5, -4 };
double **XY = ca_A_mR(xy,i_mR(RA,CXY));
double **A = ca_A_mR(tA,i_mR(RA,CA));
double **b = ca_A_mR(tb,i_mR(RA,C1));
double **Inv = i_mR(CA,RA);
double **Invb = i_mR(CA,C1);
clrscrn();
printf(" Fitting a linear Curve to Data :\n\n");
printf(" x y");
p_mR(XY,S5,P0,C6);
printf(" A :\n x**4 x**3 x**2 x**1 x**0");
p_mR(A,S7,P2,C7);
printf(" b :\n y ");
p_mR(b,S7,P2,C7);
stop();
clrscrn();
printf(" Inv : ");
invgj_mR(A,Inv);
pE_mR(Inv,S12,P4,C10);
printf(" x = Inv * b ");
mul_mR(Inv,b,Invb);
p_mR(Invb,S10,P4,C10);
printf(" The Quartic equation Curve to Data : \n\n"
" s = %+.3ft**4 %+.3ft**3 %+.3f*t**2 %+.3ft %+.3f\n\n"
,Invb[R1][C1],Invb[R2][C1],Invb[R3][C1],
Invb[R4][C1],Invb[R5][C1]);
stop();
f_mR(XY);
f_mR(b);
f_mR(A);
f_mR(Inv);
f_mR(Invb);
return 0;
}
/* ------------------------------------ */
/* ------------------------------------ */
Presentation :
Let's calculate the coefficients of a polynomial.
y = ax**4 + bx**3 + cx**2 + dx + e
Which passes through these five points.
x[1], y[1]
x[2], y[2]
x[3], y[3]
x[4], y[4]
x[5], y[5]
Using the points we obtain the matrix:
x**4 x**3 x**2 x**1 x**0 y
x[1]**4 x[1]**3 x[1]**2 x[1]**1 x[1]**0 y[1]
x[2]**4 x[2]**3 x[2]**2 x[2]**1 x[2]**0 y[2]
x[3]**4 x[3]**3 x[3]**2 x[3]**1 x[3]**0 y[3]
x[4]**4 x[4]**3 x[4]**2 x[4]**1 x[4]**0 y[4]
x[5]**4 x[5]**3 x[5]**2 x[5]**1 x[5]**0 y[5]
That we can write:
x**4 x**3 x**2 x 1 y
x[1]**4 x[1]**3 x[1]**2 x[1] 1 y[1]
x[2]**4 x[2]**3 x[2]**2 x[2] 1 y[2]
x[3]**4 x[3]**3 x[3]**2 x[3] 1 y[3]
x[4]**4 x[4]**3 x[4]**2 x[4] 1 y[4]
x[5]**4 x[5]**3 x[5]**2 x[5] 1 y[5]
Let's use the invgj_mR() function to solve
the system that will give us the coefficients a, b, c, d, e
Screen output example:
Fitting a linear Curve to Data :
x y
+1 -5
+2 +8
+3 -7
+4 +1
+5 -4
A :
x**4 x**3 x**2 x**1 x**0
+1.00 +1.00 +1.00 +1.00 +1.00
+16.00 +8.00 +4.00 +2.00 +1.00
+81.00 +27.00 +9.00 +3.00 +1.00
+256.00 +64.00 +16.00 +4.00 +1.00
+625.00 +125.00 +25.00 +5.00 +1.00
b :
y
-5.00
+8.00
-7.00
+1.00
-4.00
Press return to continue.
Inv :
+4.1667e-02 -1.6667e-01 +2.5000e-01 -1.6667e-01 +4.1667e-02
-5.8333e-01 +2.1667e+00 -3.0000e+00 +1.8333e+00 -4.1667e-01
+2.9583e+00 -9.8333e+00 +1.2250e+01 -6.8333e+00 +1.4583e+00
-6.4167e+00 +1.7833e+01 -1.9500e+01 +1.0167e+01 -2.0833e+00
+5.0000e+00 -1.0000e+01 +1.0000e+01 -5.0000e+00 +1.0000e+00
x = Inv * b
-3.6250
+44.7500
-191.8750
+329.7500
-184.0000
The Quartic equation Curve to Data :
s = -3.625t**4 +44.750t**3 -191.875*t**2 +329.750t -184.000
Press return to continue.
Copy and paste in Octave:
function y = f (x)
y = -3.625*x^4 +44.750*x^3 -191.875*x^2 +329.750*x -184.000;
endfunction
f (+1)
f (+2)
f (+3)
f (+4)
f (+5)