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 ```1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 ``` ``````# # require 'bigdecimal/jacobian' # # Provides methods to compute the Jacobian matrix of a set of equations at a # point x. In the methods below: # # f is an Object which is used to compute the Jacobian matrix of the equations. # It must provide the following methods: # # f.values(x):: returns the values of all functions at x # # f.zero:: returns 0.0 # f.one:: returns 1.0 # f.two:: returns 1.0 # f.ten:: returns 10.0 # # f.eps:: returns the convergence criterion (epsilon value) used to determine whether two values are considered equal. If |a-b| < epsilon, the two values are considered equal. # # x is the point at which to compute the Jacobian. # # fx is f.values(x). # module Jacobian module_function # Determines the equality of two numbers by comparing to zero, or using the epsilon value def isEqual(a,b,zero=0.0,e=1.0e-8) aa = a.abs bb = b.abs if aa == zero && bb == zero then true else if ((a-b)/(aa+bb)).abs < e then true else false end end end # Computes the derivative of f[i] at x[i]. # fx is the value of f at x. def dfdxi(f,fx,x,i) nRetry = 0 n = x.size xSave = x[i] ok = 0 ratio = f.ten*f.ten*f.ten dx = x[i].abs/ratio dx = fx[i].abs/ratio if isEqual(dx,f.zero,f.zero,f.eps) dx = f.one/f.ten if isEqual(dx,f.zero,f.zero,f.eps) until ok>0 do s = f.zero deriv = [] if(nRetry>100) then raise "Singular Jacobian matrix. No change at x[" + i.to_s + "]" end dx = dx*f.two x[i] += dx fxNew = f.values(x) for j in 0...n do if !isEqual(fxNew[j],fx[j],f.zero,f.eps) then ok += 1 deriv <<= (fxNew[j]-fx[j])/dx else deriv <<= f.zero end end x[i] = xSave end deriv end # Computes the Jacobian of f at x. fx is the value of f at x. def jacobian(f,fx,x) n = x.size dfdx = Array::new(n*n) for i in 0...n do df = dfdxi(f,fx,x,i) for j in 0...n do dfdx[j*n+i] = df[j] end end dfdx end end ``````