10.3 One-Dimensional Search with First Derivatives 405 etemp=e; e=d; if (fabs(p)>=fabs(0.5*q*etemp)Il p <=q*(a-x)II p>=q*(b-x)) d=CGOLD*(e=(x >xm a-x b-x)); The above conditions determine the acceptability of the parabolic fit.Here we take the golden section step into the larger of the two segments. else d=p/q; Take the parabolic step. u=x+d; if (u-a tol2 II b-u tol2) d=SIGN(tol1,xm-x); 2 else d=CGOLD(e=(x >xm a-x b-x)); 83g u=(fabs(d)>=tol1 x+d x+SIGN(tol1,d)); granted for 19881992 fu=(*f)(u); This is the one function evaluation per iteration 1.800 if (fu <fx){ Now decide what to do with our func- if (u >x)a=x;else b=x; tion evaluation. SHFT(v,w,x,u) Housekeeping follows: from NUMERICAL RECIPESI SHFT(fv,fw,fx,fu) else if (u x)a=u;else b=u; if (fu <fw l v==x){ v=W; w=u; (North America 州bMe se to make one paper University Press. THE fvefw; ART fw=fu; else if (fu <fv l v =x Il v==){ v=u; fv=fu; ictly proh Programs 2 2 OF SCIENTIFIC Done with housekeeping.Back for another iteration. nrerror("Too many iterations in brent"); 来xmin=x: Never get here. return fx; 1920 COMPUTING(ISBN Numerical Recipes 10-621 CITED REFERENCES AND FURTHER READING: Brent.R.P.1973.Algorithms for Minimization without Derivatives(Englewood Cliffs.NJ:Prentice- 43108 Hall),Chapter 5.[1] Forsythe,G.E.,Malcolm,M.A.,and Moler,C.B.1977,Computer Methods for Mathematical Computations (Englewood Cliffs,NJ:Prentice-Hall),$8.2. (outside North Software. 10.3 One-Dimensional Search with First visit website machine Derivatives Here we want to accomplish precisely the same goal as in the previous section,namely to isolate a functional minimum that is bracketed by the triplet of abscissas (a,6,c),but utilizing an additional capability to compute the function's first derivative as well as its value.10.3 One-Dimensional Search with First Derivatives 405 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). etemp=e; e=d; if (fabs(p) >= fabs(0.5*q*etemp) || p <= q*(a-x) || p >= q*(b-x)) d=CGOLD*(e=(x >= xm ? a-x : b-x)); The above conditions determine the acceptability of the parabolic fit. Here we take the golden section step into the larger of the two segments. else { d=p/q; Take the parabolic step. u=x+d; if (u-a < tol2 || b-u < tol2) d=SIGN(tol1,xm-x); } } else { d=CGOLD*(e=(x >= xm ? a-x : b-x)); } u=(fabs(d) >= tol1 ? x+d : x+SIGN(tol1,d)); fu=(*f)(u); This is the one function evaluation per iteration. if (fu <= fx) { Now decide what to do with our func￾if (u >= x) a=x; else b=x; tion evaluation. SHFT(v,w,x,u) Housekeeping follows: SHFT(fv,fw,fx,fu) } else { if (u < x) a=u; else b=u; if (fu <= fw || w == x) { v=w; w=u; fv=fw; fw=fu; } else if (fu <= fv || v == x || v == w) { v=u; fv=fu; } } Done with housekeeping. Back for } another iteration. nrerror("Too many iterations in brent"); *xmin=x; Never get here. return fx; } CITED REFERENCES AND FURTHER READING: Brent, R.P. 1973, Algorithms for Minimization without Derivatives (Englewood Cliffs, NJ: Prentice￾Hall), Chapter 5. [1] Forsythe, G.E., Malcolm, M.A., and Moler, C.B. 1977, Computer Methods for Mathematical Computations (Englewood Cliffs, NJ: Prentice-Hall), §8.2. 10.3 One-Dimensional Search with First Derivatives Here we want to accomplish precisely the same goal as in the previous section, namely to isolate a functional minimum that is bracketed by the triplet of abscissas (a, b, c), but utilizing an additional capability to compute the function’s first derivative as well as its value