added generated files

Author: tux-mind

.ext/common/bigdecimal/jacobian.rb

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+#
+# 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

.ext/common/bigdecimal/ludcmp.rb

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+require 'bigdecimal'
+
+#
+# Solves a*x = b for x, using LU decomposition.
+#
+module LUSolve
+  module_function
+
+  # Performs LU decomposition of the n by n matrix a.
+  def ludecomp(a,n,zero=0,one=1)
+    prec = BigDecimal.limit(nil)
+    ps     = []
+    scales = []
+    for i in 0...n do  # pick up largest(abs. val.) element in each row.
+      ps <<= i
+      nrmrow  = zero
+      ixn = i*n
+      for j in 0...n do
+        biggst = a[ixn+j].abs
+        nrmrow = biggst if biggst>nrmrow
+      end
+      if nrmrow>zero then
+        scales <<= one.div(nrmrow,prec)
+      else
+        raise "Singular matrix"
+      end
+    end
+    n1          = n - 1
+    for k in 0...n1 do # Gaussian elimination with partial pivoting.
+      biggst  = zero;
+      for i in k...n do
+        size = a[ps[i]*n+k].abs*scales[ps[i]]
+        if size>biggst then
+          biggst = size
+          pividx  = i
+        end
+      end
+      raise "Singular matrix" if biggst<=zero
+      if pividx!=k then
+        j = ps[k]
+        ps[k] = ps[pividx]
+        ps[pividx] = j
+      end
+      pivot   = a[ps[k]*n+k]
+      for i in (k+1)...n do
+        psin = ps[i]*n
+        a[psin+k] = mult = a[psin+k].div(pivot,prec)
+        if mult!=zero then
+          pskn = ps[k]*n
+          for j in (k+1)...n do
+            a[psin+j] -= mult.mult(a[pskn+j],prec)
+          end
+        end
+      end
+    end
+    raise "Singular matrix" if a[ps[n1]*n+n1] == zero
+    ps
+  end
+
+  # Solves a*x = b for x, using LU decomposition.
+  #
+  # a is a matrix, b is a constant vector, x is the solution vector.
+  #
+  # ps is the pivot, a vector which indicates the permutation of rows performed
+  # during LU decomposition.
+  def lusolve(a,b,ps,zero=0.0)
+    prec = BigDecimal.limit(nil)
+    n = ps.size
+    x = []
+    for i in 0...n do
+      dot = zero
+      psin = ps[i]*n
+      for j in 0...i do
+        dot = a[psin+j].mult(x[j],prec) + dot
+      end
+      x <<= b[ps[i]] - dot
+    end
+    (n-1).downto(0) do |i|
+      dot = zero
+      psin = ps[i]*n
+      for j in (i+1)...n do
+        dot = a[psin+j].mult(x[j],prec) + dot
+      end
+      x[i]  = (x[i]-dot).div(a[psin+i],prec)
+    end
+    x
+  end
+end

.ext/common/bigdecimal/math.rb

@@ -0,0 +1,206 @@
+require 'bigdecimal'
+
+#
+#--
+# Contents:
+#   sqrt(x, prec)
+#   sin (x, prec)
+#   cos (x, prec)
+#   atan(x, prec)  Note: |x|<1, x=0.9999 may not converge.
+#   log (x, prec)
+#   PI  (prec)
+#   E   (prec) == exp(1.0,prec)
+#
+# where:
+#   x    ... BigDecimal number to be computed.
+#            |x| must be small enough to get convergence.
+#   prec ... Number of digits to be obtained.
+#++
+#
+# Provides mathematical functions.
+#
+# Example:
+#
+#   require "bigdecimal"
+#   require "bigdecimal/math"
+#
+#   include BigMath
+#
+#   a = BigDecimal((PI(100)/2).to_s)
+#   puts sin(a,100) # -> 0.10000000000000000000......E1
+#
+module BigMath
+  module_function
+
+  # Computes the square root of x to the specified number of digits of
+  # precision.
+  #
+  # BigDecimal.new('2').sqrt(16).to_s
+  #  -> "0.14142135623730950488016887242096975E1"
+  #
+  def sqrt(x,prec)
+    x.sqrt(prec)
+  end
+
+  # Computes the sine of x to the specified number of digits of precision.
+  #
+  # If x is infinite or NaN, returns NaN.
+  def sin(x, prec)
+    raise ArgumentError, "Zero or negative precision for sin" if prec <= 0
+    return BigDecimal("NaN") if x.infinite? || x.nan?
+    n    = prec + BigDecimal.double_fig
+    one  = BigDecimal("1")
+    two  = BigDecimal("2")
+    x = -x if neg = x < 0
+    if x > (twopi = two * BigMath.PI(prec))
+      if x > 30
+        x %= twopi
+      else
+        x -= twopi while x > twopi
+      end
+    end
+    x1   = x
+    x2   = x.mult(x,n)
+    sign = 1
+    y    = x
+    d    = y
+    i    = one
+    z    = one
+    while d.nonzero? && ((m = n - (y.exponent - d.exponent).abs) > 0)
+      m = BigDecimal.double_fig if m < BigDecimal.double_fig
+      sign = -sign
+      x1  = x2.mult(x1,n)
+      i  += two
+      z  *= (i-one) * i
+      d   = sign * x1.div(z,m)
+      y  += d
+    end
+    neg ? -y : y
+  end
+
+  # Computes the cosine of x to the specified number of digits of precision.
+  #
+  # If x is infinite or NaN, returns NaN.
+  def cos(x, prec)
+    raise ArgumentError, "Zero or negative precision for cos" if prec <= 0
+    return BigDecimal("NaN") if x.infinite? || x.nan?
+    n    = prec + BigDecimal.double_fig
+    one  = BigDecimal("1")
+    two  = BigDecimal("2")
+    x = -x if x < 0
+    if x > (twopi = two * BigMath.PI(prec))
+      if x > 30
+        x %= twopi
+      else
+        x -= twopi while x > twopi
+      end
+    end
+    x1 = one
+    x2 = x.mult(x,n)
+    sign = 1
+    y = one
+    d = y
+    i = BigDecimal("0")
+    z = one
+    while d.nonzero? && ((m = n - (y.exponent - d.exponent).abs) > 0)
+      m = BigDecimal.double_fig if m < BigDecimal.double_fig
+      sign = -sign
+      x1  = x2.mult(x1,n)
+      i  += two
+      z  *= (i-one) * i
+      d   = sign * x1.div(z,m)
+      y  += d
+    end
+    y
+  end
+
+  # Computes the arctangent of x to the specified number of digits of precision.
+  #
+  # If x is NaN, returns NaN.
+  def atan(x, prec)
+    raise ArgumentError, "Zero or negative precision for atan" if prec <= 0
+    return BigDecimal("NaN") if x.nan?
+    pi = PI(prec)
+    x = -x if neg = x < 0
+    return pi.div(neg ? -2 : 2, prec) if x.infinite?
+    return pi / (neg ? -4 : 4) if x.round(prec) == 1
+    x = BigDecimal("1").div(x, prec) if inv = x > 1
+    x = (-1 + sqrt(1 + x**2, prec))/x if dbl = x > 0.5
+    n    = prec + BigDecimal.double_fig
+    y = x
+    d = y
+    t = x
+    r = BigDecimal("3")
+    x2 = x.mult(x,n)
+    while d.nonzero? && ((m = n - (y.exponent - d.exponent).abs) > 0)
+      m = BigDecimal.double_fig if m < BigDecimal.double_fig
+      t = -t.mult(x2,n)
+      d = t.div(r,m)
+      y += d
+      r += 2
+    end
+    y *= 2 if dbl
+    y = pi / 2 - y if inv
+    y = -y if neg
+    y
+  end
+
+  # Computes the value of pi to the specified number of digits of precision.
+  def PI(prec)
+    raise ArgumentError, "Zero or negative argument for PI" if prec <= 0
+    n      = prec + BigDecimal.double_fig
+    zero   = BigDecimal("0")
+    one    = BigDecimal("1")
+    two    = BigDecimal("2")
+
+    m25    = BigDecimal("-0.04")
+    m57121 = BigDecimal("-57121")
+
+    pi     = zero
+
+    d = one
+    k = one
+    w = one
+    t = BigDecimal("-80")
+    while d.nonzero? && ((m = n - (pi.exponent - d.exponent).abs) > 0)
+      m = BigDecimal.double_fig if m < BigDecimal.double_fig
+      t   = t*m25
+      d   = t.div(k,m)
+      k   = k+two
+      pi  = pi + d
+    end
+
+    d = one
+    k = one
+    w = one
+    t = BigDecimal("956")
+    while d.nonzero? && ((m = n - (pi.exponent - d.exponent).abs) > 0)
+      m = BigDecimal.double_fig if m < BigDecimal.double_fig
+      t   = t.div(m57121,n)
+      d   = t.div(k,m)
+      pi  = pi + d
+      k   = k+two
+    end
+    pi
+  end
+
+  # Computes e (the base of natural logarithms) to the specified number of
+  # digits of precision.
+  def E(prec)
+    raise ArgumentError, "Zero or negative precision for E" if prec <= 0
+    n    = prec + BigDecimal.double_fig
+    one  = BigDecimal("1")
+    y  = one
+    d  = y
+    z  = one
+    i  = 0
+    while d.nonzero? && ((m = n - (y.exponent - d.exponent).abs) > 0)
+      m = BigDecimal.double_fig if m < BigDecimal.double_fig
+      i += 1
+      z *= i
+      d  = one.div(z,m)
+      y += d
+    end
+    y
+  end
+end