Linear Algebra and the C Language/a07m

__NOTOC__ Install and compile this file in your working directory. /* ------------------------------------ */ /* Save as : c00f.c */ /* ------------------------------------ */ #include "v_a.h" #include "dpoly.h" /* ------------------------------------ */ int main(void) { double xy[10] ={ 1, -2, 2, -2, 3, 3, 4, -9, 5, 4, }; double **XY = ca_A_mR(xy,i_mR(R5,C2)); double **A = i_mR(R5,C5); double **b = i_mR(R5,C1); double **Ab = i_Abr_Ac_bc_mR(R5,C5,C1); clrscrn(); printf(" Find the coefficients a, b, c of the curve \n\n"); printf(" y = ax**4 + bx**3 + cx**2 + dx + e \n\n"); printf(" that passes through the points. \n\n"); printf(" x y"); p_mR(XY,S5,P0,C6); printf(" Using the given points, we obtain this matrix\n\n"); printf(" x**4 x**3 x**2 x**1 x**0 y"); i_A_b_with_XY_mR(XY,A,b); c_A_b_Ab_mR(A,b,Ab); p_mR(Ab,S7,P2,C6); stop(); clrscrn(); printf(" The Gauss Jordan process will reduce this matrix to : \n"); gj_TP_mR(Ab); p_mR(Ab,S7,P2,C6); printf("\n The coefficients a, b, c of the curve are : \n\n"); p_eq_poly_mR(Ab); stop(); clrscrn(); printf(" x y \n"); p_mR(XY,S5,P0,C6); printf("\n"); printf(" Verify the result : \n\n"); verify_X_mR(Ab,XY[R1][C1]); verify_X_mR(Ab,XY[R2][C1]); verify_X_mR(Ab,XY[R3][C1]); verify_X_mR(Ab,XY[R4][C1]); verify_X_mR(Ab,XY[R5][C1]); printf("\n\n\n"); stop(); f_mR(XY); f_mR(A); f_mR(b); f_mR(Ab); 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 gj_TP_mR() function to solve the system that will give us the coefficients a, b, c, d, e Screen output example: Find the coefficients a, b, c of the curve y = ax**4 + bx**3 + cx**2 + dx + e that passes through the points. x y +1 -2 +2 -2 +3 +3 +4 -9 +5 +4 Using the given points, we obtain this matrix x**4 x**3 x**2 x**1 x**0 y +1.00 +1.00 +1.00 +1.00 +1.00 -2.00 +16.00 +8.00 +4.00 +2.00 +1.00 -2.00 +81.00 +27.00 +9.00 +3.00 +1.00 +3.00 +256.00 +64.00 +16.00 +4.00 +1.00 -9.00 +625.00 +125.00 +25.00 +5.00 +1.00 +4.00 Press return to continue. The Gauss Jordan process will reduce this matrix to : +1.00 +0.00 +0.00 +0.00 +0.00 +2.67 +0.00 +1.00 +0.00 +0.00 +0.00 -30.33 +0.00 +0.00 +1.00 +0.00 +0.00 +117.83 -0.00 -0.00 -0.00 +1.00 -0.00 -181.17 +0.00 +0.00 +0.00 +0.00 +1.00 +89.00 The coefficients a, b, c of the curve are : y = +2.667x**4 -30.333x**3 +117.833x**2 -181.167x +89.000 Press return to continue. x y +1 -2 +2 -2 +3 +3 +4 -9 +5 +4 Verify the result : With x = +1.000, y = -2.000 With x = +2.000, y = -2.000 With x = +3.000, y = +3.000 With x = +4.000, y = -9.000 With x = +5.000, y = +4.000 Press return to continue. Copy and paste in Octave: function y = f (x) y = +2.666665810*x^4 -30.333323719*x^3 +117.833298608*x^2 -181.166625268*x +88.999995106; endfunction f (+1) f (+2) f (+3) f (+4) f (+5)