/* Square root, sine, cosine, and arctangent of polynomial. * See polyn.c for data structures and discussion. */ #include #include #ifdef ANSIPROT extern double atan2 ( double, double ); extern double sqrt ( double ); extern double fabs ( double ); extern double sin ( double ); extern double cos ( double ); extern void polclr ( double *a, int n ); extern void polmov ( double *a, int na, double *b ); extern void polmul ( double a[], int na, double b[], int nb, double c[] ); extern void poladd ( double a[], int na, double b[], int nb, double c[] ); extern void polsub ( double a[], int na, double b[], int nb, double c[] ); extern int poldiv ( double a[], int na, double b[], int nb, double c[] ); extern void polsbt ( double a[], int na, double b[], int nb, double c[] ); extern void * malloc ( long ); extern void free ( void * ); #else double atan2(), sqrt(), fabs(), sin(), cos(); void polclr(), polmov(), polsbt(), poladd(), polsub(), polmul(); int poldiv(); void * malloc(); void free (); #endif /* Highest degree of polynomial to be handled by the polyn.c subroutine package. */ #define N 16 /* Highest degree actually initialized at runtime. */ extern int MAXPOL; /* Taylor series coefficients for various functions */ double patan[N+1] = { 0.0, 1.0, 0.0, -1.0/3.0, 0.0, 1.0/5.0, 0.0, -1.0/7.0, 0.0, 1.0/9.0, 0.0, -1.0/11.0, 0.0, 1.0/13.0, 0.0, -1.0/15.0, 0.0 }; double psin[N+1] = { 0.0, 1.0, 0.0, -1.0/6.0, 0.0, 1.0/120.0, 0.0, -1.0/5040.0, 0.0, 1.0/362880.0, 0.0, -1.0/39916800.0, 0.0, 1.0/6227020800.0, 0.0, -1.0/1.307674368e12, 0.0}; double pcos[N+1] = { 1.0, 0.0, -1.0/2.0, 0.0, 1.0/24.0, 0.0, -1.0/720.0, 0.0, 1.0/40320.0, 0.0, -1.0/3628800.0, 0.0, 1.0/479001600.0, 0.0, -1.0/8.7179291e10, 0.0, 1.0/2.0922789888e13}; double pasin[N+1] = { 0.0, 1.0, 0.0, 1.0/6.0, 0.0, 3.0/40.0, 0.0, 15.0/336.0, 0.0, 105.0/3456.0, 0.0, 945.0/42240.0, 0.0, 10395.0/599040.0 , 0.0, 135135.0/9676800.0 , 0.0 }; /* Square root of 1 + x. */ double psqrt[N+1] = { 1.0, 1./2., -1./8., 1./16., -5./128., 7./256., -21./1024., 33./2048., -429./32768., 715./65536., -2431./262144., 4199./524288., -29393./4194304., 52003./8388608., -185725./33554432., 334305./67108864., -9694845./2147483648.}; /* Arctangent of the ratio num/den of two polynomials. */ void polatn( num, den, ans, nn ) double num[], den[], ans[]; int nn; { double a, t; double *polq, *polu, *polt; int i; if (nn > N) { mtherr ("polatn", OVERFLOW); return; } /* arctan( a + b ) = arctan(a) + arctan( b/(1 + ab + a**2) ) */ t = num[0]; a = den[0]; if( (t == 0.0) && (a == 0.0 ) ) { t = num[1]; a = den[1]; } t = atan2( t, a ); /* arctan(num/den), the ANSI argument order */ polq = (double * )malloc( (MAXPOL+1) * sizeof (double) ); polu = (double * )malloc( (MAXPOL+1) * sizeof (double) ); polt = (double * )malloc( (MAXPOL+1) * sizeof (double) ); polclr( polq, MAXPOL ); i = poldiv( den, nn, num, nn, polq ); a = polq[0]; /* a */ polq[0] = 0.0; /* b */ polmov( polq, nn, polu ); /* b */ /* Form the polynomial 1 + ab + a**2 where a is a scalar. */ for( i=0; i<=nn; i++ ) polu[i] *= a; polu[0] += 1.0 + a * a; poldiv( polu, nn, polq, nn, polt ); /* divide into b */ polsbt( polt, nn, patan, nn, polu ); /* arctan(b) */ polu[0] += t; /* plus arctan(a) */ polmov( polu, nn, ans ); free( polt ); free( polu ); free( polq ); } /* Square root of a polynomial. * Assumes the lowest degree nonzero term is dominant * and of even degree. An error message is given * if the Newton iteration does not converge. */ void polsqt( pol, ans, nn ) double pol[], ans[]; int nn; { double t; double *x, *y; int i, n; #if 0 double z[N+1]; double u; #endif if (nn > N) { mtherr ("polatn", OVERFLOW); return; } x = (double * )malloc( (MAXPOL+1) * sizeof (double) ); y = (double * )malloc( (MAXPOL+1) * sizeof (double) ); polmov( pol, nn, x ); polclr( y, MAXPOL ); /* Find lowest degree nonzero term. */ t = 0.0; for( n=0; n 0 ) { if (n & 1) { printf("error, sqrt of odd polynomial\n"); return; } /* Divide by x^n. */ y[n] = x[n]; poldiv (y, nn, pol, N, x); } t = x[0]; for( i=1; i<=nn; i++ ) x[i] /= t; x[0] = 0.0; /* series development sqrt(1+x) = 1 + x / 2 - x**2 / 8 + x**3 / 16 hopes that first (constant) term is greater than what follows */ polsbt( x, nn, psqrt, nn, y); t = sqrt( t ); for( i=0; i<=nn; i++ ) y[i] *= t; /* If first nonzero coefficient was at degree n > 0, multiply by x^(n/2). */ if (n > 0) { polclr (x, MAXPOL); x[n/2] = 1.0; polmul (x, nn, y, nn, y); } #if 0 /* Newton iterations */ for( n=0; n<10; n++ ) { poldiv( y, nn, pol, nn, z ); poladd( y, nn, z, nn, y ); for( i=0; i<=nn; i++ ) y[i] *= 0.5; for( i=0; i<=nn; i++ ) { u = fabs( y[i] - z[i] ); if( u > 1.0e-15 ) goto more; } goto done; more: ; } printf( "square root did not converge\n" ); done: #endif /* 0 */ polmov( y, nn, ans ); free( y ); free( x ); } /* Sine of a polynomial. * The computation uses * sin(a+b) = sin(a) cos(b) + cos(a) sin(b) * where a is the constant term of the polynomial and * b is the sum of the rest of the terms. * Since sin(b) and cos(b) are computed by series expansions, * the value of b should be small. */ void polsin( x, y, nn ) double x[], y[]; int nn; { double a, sc; double *w, *c; int i; if (nn > N) { mtherr ("polatn", OVERFLOW); return; } w = (double * )malloc( (MAXPOL+1) * sizeof (double) ); c = (double * )malloc( (MAXPOL+1) * sizeof (double) ); polmov( x, nn, w ); polclr( c, MAXPOL ); polclr( y, nn ); /* a, in the description, is x[0]. b is the polynomial x - x[0]. */ a = w[0]; /* c = cos (b) */ w[0] = 0.0; polsbt( w, nn, pcos, nn, c ); sc = sin(a); /* sin(a) cos (b) */ for( i=0; i<=nn; i++ ) c[i] *= sc; /* y = sin (b) */ polsbt( w, nn, psin, nn, y ); sc = cos(a); /* cos(a) sin(b) */ for( i=0; i<=nn; i++ ) y[i] *= sc; poladd( c, nn, y, nn, y ); free( c ); free( w ); } /* Cosine of a polynomial. * The computation uses * cos(a+b) = cos(a) cos(b) - sin(a) sin(b) * where a is the constant term of the polynomial and * b is the sum of the rest of the terms. * Since sin(b) and cos(b) are computed by series expansions, * the value of b should be small. */ void polcos( x, y, nn ) double x[], y[]; int nn; { double a, sc; double *w, *c; int i; double sin(), cos(); if (nn > N) { mtherr ("polatn", OVERFLOW); return; } w = (double * )malloc( (MAXPOL+1) * sizeof (double) ); c = (double * )malloc( (MAXPOL+1) * sizeof (double) ); polmov( x, nn, w ); polclr( c, MAXPOL ); polclr( y, nn ); a = w[0]; w[0] = 0.0; /* c = cos(b) */ polsbt( w, nn, pcos, nn, c ); sc = cos(a); /* cos(a) cos(b) */ for( i=0; i<=nn; i++ ) c[i] *= sc; /* y = sin(b) */ polsbt( w, nn, psin, nn, y ); sc = sin(a); /* sin(a) sin(b) */ for( i=0; i<=nn; i++ ) y[i] *= sc; polsub( y, nn, c, nn, y ); free( c ); free( w ); }