科學(xué)計(jì)算方法：實(shí)戰(zhàn)技巧1(非線性方程求根)

1、Examine the function graphically (to locate roughly where the roots are and how many there may be) curve sketching is one way or. best: use a Matlab plot to get the lay of the land set the interval or the starting point (find a range of x- values over which the function changes sign) iteratively ref

2、ine the initial guess with a root-finding algorithm (bisection is dependable but slow; Newton is fast if the initial value is good),實(shí)戰(zhàn)技巧,x=-1:0.1:3; y=exp(-x)-sin(pi*x/2); plot(x,y),算例：,二分法,function root = bisect(func,x1,x2,tol) n = ceil(log(abs(x2 - x1)/tol)/log(2.0); for i = 1:n x3 = 0.5*(x1 + x2)

3、; f3 = feval(func,x3); if f3 = 0.0 root = x3; return end if f2*f3 0.0 x1 = x3; f1 = f3; else x2 = x3; f2 = f3; end end root = (x1 + x2)/2;,牛頓法,function root,numIter = newtonsimple(func,dfunc,x,tol,n) % Simple version of Newton method tol = 1.0e(-6); for i = 1:n dx = -feval(func,x)/feval(dfunc,x); x

4、= x + dx; if abs(dx) tol root = x; numIter = i; return end end,例如, 多項(xiàng)式x3 6x2 72x 27在Matlab中表示為c = 1 -6 -72 -27, 其對應(yīng)的根通過v = roots(c)計(jì)算。,多項(xiàng)式p(x)=c1xn + c2xn-1 + . + cnx +cn+1 在Matlab中表示為c = c1 c2 . cn+1。則給定根v(以向量存儲(chǔ)), Matlab通過c = poly(v)生成對應(yīng)多項(xiàng)式 ; 反之, 給定多項(xiàng)式 c, Matlab通過v = roots(c)計(jì)算其對應(yīng)的根v 。,代數(shù)方程求根(roots),Matlab中fzero函數(shù)實(shí)現(xiàn)的是 Brent方法, 僅需要初始值作為輸入。,Example: z = fzero(x)x.3-2*x-5,2),Brent方法是一種復(fù)雜但流行的求根方法, 該方法結(jié)合了二分法, 割線法和反二次插值方法。當(dāng)函數(shù)的導(dǎo)數(shù)難于計(jì)算