# Module: Jacobian

Included in:
Newton
Defined in:
lib/bigdecimal/jacobian.rb

## Overview

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 2.0

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).

## Class Method Summary collapse

• Computes the derivative of f at x.

• Determines the equality of two numbers by comparing to zero, or using the epsilon value.

• Computes the Jacobian of f at x.

## Class Method Details

### .dfdxi(f, fx, x, i) ⇒ Object

Computes the derivative of f at x. fx is the value of f at x.

 ``` 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``` ```# File '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```

### .isEqual(a, b, zero = 0.0, e = 1.0e-8) ⇒ Object

Determines the equality of two numbers by comparing to zero, or using the epsilon value

 ``` 27 28 29 30 31 32 33 34 35 36 37 38 39``` ```# File '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```

### .jacobian(f, fx, x) ⇒ Object

Computes the Jacobian of f at x. fx is the value of f at x.

 ``` 77 78 79 80 81 82 83 84 85 86 87``` ```# File '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```