科学计算方法：实战技巧1(非线性方程求根)

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),实战技巧,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;,牛顿法,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, 多项式x3 6x2 72x 27在Matlab中表示为c = 1 -6 -72 -27, 其对应的根通过v = roots(c)计算。,多项式p(x)=c1xn + c2xn-1 + . + cnx +cn+1 在Matlab中表示为c = c1 c2 . cn+1。则给定根v(以向量存储), Matlab通过c = poly(v)生成对应多项式 ; 反之, 给定多项式 c, Matlab通过v = roots(c)计算其对应的根v 。,代数方程求根(roots),Matlab中fzero函数实现的是 Brent方法, 仅需要初始值作为输入。,Example: z = fzero(x)x.3-2*x-5,2),Brent方法是一种复杂但流行的求根方法, 该方法结合了二分法, 割线法和反二次插值方法。当函数的导数难于计算