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require â€˜bigdecimal/jacobianâ€™
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:
returns the values of all functions at x
returns 0.0
returns 1.0
returns 2.0
returns 10.0
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).
Computes the derivative of f at x. fx is the value of f at x.
# File bigdecimal/lib/bigdecimal/jacobian.rb, line 44 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 = [] nRetry += 1 if nRetry > 100 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
Determines the equality of two numbers by comparing to zero, or using the epsilon value
# File bigdecimal/lib/bigdecimal/jacobian.rb, line 27 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 Jacobian of f at x. fx is the value of f at x.
# File bigdecimal/lib/bigdecimal/jacobian.rb, line 77 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
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