Linear Algebra and the C Language/a08g

__NOTOC__ Install and compile this file in your working directory. /* ------------------------------------ */ /* Save as : c00a.c */ /* ------------------------------------ */ #include "v_a.h" /* ------------------------------------ */ #define RA R4 #define CA C6 #define Cb C1 /* ------------------------------------ */ #define CbFREE Cb+C3 /* ------------------------------------ */ int main(void) { double ab[RA*(CA+Cb)]={ +1, 3, -2, 0, +2, +0, 0, +2, 6, -5, -2, +4, -3, 0, 0, 0, +5, 10, 0, 15, 0, 2, 6, 0, 8, 4, 18, 0, }; double **Ab = ca_A_mR(ab, i_Abr_Ac_bc_mR(RA,CA,Cb)); double **A = c_Ab_A_mR(Ab, i_mR(RA,CA)); double **b = c_Ab_b_mR(Ab, i_mR(RA,Cb)); double **Ab_free = i_Abr_Ac_bc_mR(CA,CA,CbFREE); double **b_free = i_mR(CA,CbFREE); double **A_bfree = i_mR(RA,CbFREE); int r; clrscrn(); printf("Find a basis for the orthogonal complement of A :\n\n"); printf(" A :"); p_mR(A,S6,P1,C10); printf(" b :"); p_mR(b,S6,P1,C10); printf(" Ab :"); p_mR(Ab,S6,P1,C10); stop(); clrscrn(); printf(" Ab : gj_PP_mR(Ab,NO) :"); gj_PP_mR(Ab,NO); p_mR(Ab,S7,P3,C10); put_zeroR_mR(Ab,Ab_free); printf(" Ab_free : put_zeroR_mR(Ab,Ab_free);"); p_mR(Ab_free,S7,P3,C10); put_freeV_mR(Ab_free); printf(" Ab_free : put_freeV_mR(Ab_free);"); p_mR(Ab_free,S7,P3,C10); stop(); clrscrn(); r = rsize_R(Ab_free); while(r>R1) zero_above_pivot_gj1Ab_mR(Ab_free,r--); printf(" Ab_free : zero_above_pivot_gj1Ab_mR(Ab_free,r--);"); p_mR(Ab_free,S7,P3,C10); c_Ab_b_mR(Ab_free,b_free); printf(" b_free is a basis for the orthogonal complement of A"); p_mR(b_free,S10,P3,C7); stop(); clrscrn(); printf(" A :"); p_mR(A,S10,P3,C10); printf(" b_free is a basis for the orthogonal complement of A"); p_mR(b_free,S10,P3,C7); printf(" The row vectors of A" " are orthogonal to the column vectors of bfree\n\n"); printf(" A * bfree: "); p_mR(mul_mR(A,b_free,A_bfree),S10,P3,C7); stop(); f_mR(Ab); f_mR(A); f_mR(b); f_mR(Ab_free); f_mR(b_free); f_mR(A_bfree); return 0; } /* ------------------------------------ */ /* ------------------------------------ */ We begin by calculating the free variables. The columns of b_free form a basis for the orthogonal complement of A. A * b_free = 0 This proves that the row vectors of A are orthogonal to the column vectors of b_free. Screen output example: Ab : gj_PP_mR(Ab,NO) : +1.000 +3.000 -2.500 -1.000 +2.000 -1.500 +0.000 +0.000 +0.000 +1.000 +2.000 +0.000 +3.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 Ab_free : put_zeroR_mR(Ab,Ab_free); +1.000 +3.000 -2.500 -1.000 +2.000 -1.500 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +2.000 +0.000 +3.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 Ab_free : put_freeV_mR(Ab_free); +1.000 +3.000 -2.500 -1.000 +2.000 -1.500 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +1.000 +2.000 +0.000 +3.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 Press return to continue. Ab_free : zero_above_pivot_gj1Ab_mR(Ab_free,r--); +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 -3.000 -4.000 -2.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 -2.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 b_free is a basis for the orthogonal complement of A +0.000 -3.000 -4.000 -2.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 -2.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 Press return to continue. A : +1.000 +3.000 -2.000 +0.000 +2.000 +0.000 +2.000 +6.000 -5.000 -2.000 +4.000 -3.000 +0.000 +0.000 +5.000 +10.000 +0.000 +15.000 +2.000 +6.000 +0.000 +8.000 +4.000 +18.000 b_free is a basis for the orthogonal complement of A +0.000 -3.000 -4.000 -2.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 -2.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 The row vectors of A are orthogonal to the column vectors of bfree. A * bfree: +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 Press return to continue.