Module: SyMath::Operation::Differential

Includes:
SyMath::Operation
Included in:
Value
Defined in:
lib/symath/operation/differential.rb

Defined Under Namespace

Classes: DifferentialError

Class Method Summary collapse

Instance Method Summary collapse

Methods included from SyMath::Operation

#iterate, #recurse

Class Method Details

.initialize ⇒ Object

Module initialization



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# File 'lib/symath/operation/differential.rb', line 20

def self.initialize()
  # Map of single argument functions to their derivative.
  # FIXME: Check whether this still works if the symbol a is defined?
  @@functions = {
    # Exponential and trigonometric functions
    :exp => definition(:exp),
    :ln  => lmd(1.to_m/:a.to_m, :a),
    # Trigonometric functions
    :sin => definition(:cos),
    :cos => lmd(- fn(:sin, :a), :a),
    :tan => lmd(1.to_m + fn(:tan, :a)**2, :a),
    :cot => lmd(- (1.to_m + fn(:cot, :a)**2), :a),
    :sec => lmd(fn(:sec, :a)*fn(:tan, :a), :a),
    :csc => lmd(- fn(:cot, :a.to_m)*fn(:csc, :a.to_m), :a),
    # Inverse trigonometric functions
    :arcsin => lmd(1.to_m/fn(:sqrt, 1.to_m - :a.to_m**2), :a),
    :arccos => lmd(- 1.to_m/fn(:sqrt, 1.to_m - :a.to_m**2), :a),
    :arctan => lmd(1.to_m/fn(:sqrt, 1.to_m + :a.to_m**2), :a),
    :arcsec => lmd(1.to_m/(fn(:abs, :a)*fn(:sqrt, :a.to_m**2 - 1)), :a),
    :arccsc => lmd(- 1.to_m/(fn(:abs, :a)*fn(:sqrt, :a.to_m**2 - 1)), :a),
    :arccot => lmd(- 1.to_m/(1.to_m + :a.to_m**2), :a),
    # Hyperbolic functions
    :sinh => definition(:cosh),
    :cosh => definition(:sinh),
    :tanh => lmd(fn(:sech, :a)**2, :a),
    :sech => lmd(- fn(:tanh, :a)*fn(:sech, :a), :a),
    :csch => lmd(- fn(:coth, :a)*fn(:csch, :a), :a),
    :coth => lmd(- fn(:csch, :a)**2, :a),
    # Inverse hyperbolic functions
    :arsinh => lmd(1.to_m/fn(:sqrt, :a.to_m**2 + 1), :a),
    :arcosh => lmd(1.to_m/fn(:sqrt, :a.to_m**2 - 1), :a),
    :artanh => lmd(1.to_m/(1.to_m - :a.to_m**2), :a),
    :arsech => lmd(- 1.to_m/(:a.to_m*fn(:sqrt, 1.to_m - :a.to_m**2)), :a),
    :arcsch => lmd(- 1.to_m/(fn(:abs, :a.to_m)*fn(:sqrt, :a.to_m**2 + 1)), :a),
    :arcoth => lmd(1.to_m/(1.to_m - :a.to_m**2), :a),
  }
end

Instance Method Details

#_d_wedge(exp1, exp2) ⇒ Object

Apply wedge product or ordinary product between two expressions, depending on whether or not they have vector parts.



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# File 'lib/symath/operation/differential.rb', line 162

def _d_wedge(exp1, exp2)
  # The product operator will determine whether this is a scalar
  # or a wedge product.
  return (exp1.factors.to_a + exp2.factors.to_a).inject(:*)
end

#d(vars) ⇒ Object 



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# File 'lib/symath/operation/differential.rb', line 58

def d(vars)
  if self.is_a?(SyMath::Definition::Function)
    return d_function_def(vars)
  end

  # d(c) = 0 for constant c
  if is_constant?(vars)
    return 0.to_m
  end

  # d(v) = dv for variable v
  if vars.member?(self)
    return to_d
  end

  # d(a + b + ...) = d(a) + d(b) + ...
  if is_a?(SyMath::Sum)
    return term1.d(vars) + term2.d(vars)
  end

  # d(-a) = -d(a)
  if is_a?(SyMath::Minus)
    return -argument.d(vars)
  end

  # Product rule
  if is_a?(SyMath::Product)
    return d_product(vars)
  end

  # Fraction rule
  if is_a?(SyMath::Fraction)
    return d_fraction(vars)
  end

  # Power rule
  if is_a?(SyMath::Power)
    return d_power(vars)
  end

  # Derivative of function
  return d_function(vars)
end

#d_failure ⇒ Object

Raises:



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# File 'lib/symath/operation/differential.rb', line 102

def d_failure()
  raise DifferentialError, 'Cannot calculate differential of expression ' + to_s
end

#d_fraction(vars) ⇒ Object 



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# File 'lib/symath/operation/differential.rb', line 113

def d_fraction(vars)
  return (_d_wedge(dividend.d(vars), divisor) -
          _d_wedge(dividend, divisor.d(vars))) /
         (divisor**2)
end

#d_function(vars) ⇒ Object 



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# File 'lib/symath/operation/differential.rb', line 142

def d_function(vars)
  if !self.is_a?SyMath::Operator
    d_failure
  end

  if name != '' and @@functions.key?(name.to_sym)
    df = @@functions[name.to_sym]
    dfcall = df.(args[0]).evaluate
    return _d_wedge(dfcall, args[0].d(vars))
  end

  if !definition.exp.nil?
    return definition.(*args).evaluate.d(vars)
  end

  d_failure
end

#d_function_def(vars) ⇒ Object 



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# File 'lib/symath/operation/differential.rb', line 128

def d_function_def(vars)
  if name != '' and @@functions.key?(name.to_sym)
    df = @@functions[name.to_sym]
    dfcall = df.(args[0]).evaluate
    return _d_wedge(dfcall, args[0].d(vars))
  end

  if !exp.nil?
    return self.(*args).evaluate.d(vars)
  end

  d_failure
end

#d_power(vars) ⇒ Object 



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# File 'lib/symath/operation/differential.rb', line 119

def d_power(vars)
  if (exponent.is_constant?(vars))
    return _d_wedge(_d_wedge(exponent, base**(exponent - 1)), base.d(vars))
  else
    return _d_wedge(_d_wedge(self, fn(:ln, base)), exponent.d(vars)) +
      _d_wedge(_d_wedge(exponent, base**(exponent - 1)), base.d(vars))
  end
end

#d_product(vars) ⇒ Object

For simplicity, just use wedge products all the time. They will be normalized to scalar products afterwards.



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# File 'lib/symath/operation/differential.rb', line 108

def d_product(vars)
  return (_d_wedge(factor1.d(vars), factor2) +
          _d_wedge(factor1, factor2.d(vars)))
end