Class: AppMath::Vec

Inherits:
Object
  • Object
show all
Includes:
Comparable, Enumerable
Defined in:
lib/linalg.rb

Overview

Vector space of arbitrary dimension. The intended usage is that the components of a vector are all either real or complex. Since 

x = Vec.new(anyArray); x[1] = anyObject

works, there is no guaranty for type-uniformity of the components of a vector.

Instance Attribute Summary collapse

Class Method Summary collapse

Instance Method Summary collapse

Constructor Details

#initialize(*arg) ⇒ Vec

These are the 3 mehods to generate a vector via ‘new’

a = Vec.new(anArray)
b = Vec.new(aVec)
c = Vec.new(aPositiveInteger, aRealOrComplex)



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# File 'lib/linalg.rb', line 59

def initialize(*arg)
  case arg.size
  when 1
    a0 = arg[0]
    if a0.is_a?(Array)
      @x = Array.new(a0)
     # @x = a0 # seems to work but can't be safe
    elsif a0.is_a?(Vec)
      @x = Array.new(a0.x)
    else
      fail "object can't be used to build a vector"
    end
  when 2
    n = arg[0]
    fail "first argument has to be an integer" unless n.integer?
    fail "first argument must be non-negative" unless n >= 0
    @x = Array.new(n,arg[1])
  end
end

Instance Attribute Details

#x ⇒ Object

Returns the value of attribute x.



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# File 'lib/linalg.rb', line 49

def x
  @x
end

Class Method Details

.test(n0, verbose = true, complex = false) ⇒ Object

Consistency test of class Vec.



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# File 'lib/linalg.rb', line 307

def Vec.test(n0, verbose = true , complex = false)
  puts "Doing Vec.test( n = #{n0}, verbose = #{verbose}, " +
  "complex = #{complex}) for R.prec = #{R.prec}:"
  puts "*************************************************"
  
  t1 = Time.now
  s = R.c0
  puts "class of s is " + s.class.to_s
  i = n0
  a = Vec.tob(n0, i, complex) 
  i += 1
  b = Vec.tob(n0, i, complex) 
  i += 1
  c = Vec.tob(n0, i, complex) 
  i += 1
  s1 = complex ? C.ran(i) : R.ran(i)
  i += 1
  s2 = complex ? C.ran(i) : R.ran(i)

  r = (a + b) + c
  l = a + (b + c)
  ds = r.dis(l)
  puts "associativity: ds = " + ds.to_s if verbose
  s += ds

  r = (a - b) + c
  l = a - (b - c)
  ds = r.dis(l)
  puts "associativity 2: ds = " + ds.to_s if verbose
  s += ds

  r = (a + b) * s1
  l = a * s1 + b * s1
  ds = r.dis(l)
  puts "distributivity: ds = " + ds.to_s if verbose
  s += ds

  r = c * (s1*s2)
  l = (c * s1) * s2
  ds = r.dis(l)
  puts "distributivity of multiplication by scalars: ds = " + ds.to_s if verbose
  s += ds

  r = a
  l = -(-a)
  ds = r.dis(l)
  puts "idempotency of unary minus: ds = " + ds.to_s if verbose
  s += ds

  r = (a + b).spr(c)
  l = a.spr(c) + b.spr(c)
  ds = r.dis(l)
  puts "distributivity of spr: ds = " + ds.to_s if verbose
  s += ds
  
  t2 = Time.now

  if verbose
    puts
    a.prn("a")
    puts
    b.prn("b")
    puts
    c.prn("c")
    puts
    s1.prn("s1")
    puts
    s2.prn("s2")
  end

  puts "class of s is " + s.class.to_s + " ."
  puts "The error sum s is " + s.to_s + " ."
  puts "It should be close to 0."
  puts "Computation time was " + (t2-t1).to_s
  s
end

.tob(n, i, complex = false) ⇒ Object

Test object.

Returns a Vec res such that res.dim == n. Vector res depends rather chaotically on the integer argument i. If the last argument is ‘false’ res will have R-typed components, and C-typed components else.



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# File 'lib/linalg.rb', line 89

def Vec.tob(n,i,complex=false)
  vi = complex ? C.tob(i) : R.tob(i)
  res=Vec.new(n, vi)
  if complex 
    rg1 = Ran.new(-vi.re,vi.re)
    rg2 = Ran.new(-vi.im,vi.im)
    for j in 1..res.dim
      res[j] = C.new(rg1.ran,rg2.ran)
    end
  else
    rg = Ran.new(-vi,vi)
    for j in 1..res.dim
      res[j] = rg.ran
    end
  end
  res
end

Instance Method Details

#*(s) ⇒ Object

Returns self * s, where s has the same type as the components of self.



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# File 'lib/linalg.rb', line 178

def *(s)
  res = clone
  for i in 1..dim
    res[i] *= s
  end
  res
end

#+(v) ⇒ Object

Returns self + v, where v is a Vec



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# File 'lib/linalg.rb', line 156

def +(v)
  fail "object can't be added to a Vec" unless v.is_a?(Vec)
  fail "dimension mismatch" unless dim == v.dim
  res = clone
  for i in 1..dim
    res[i] += v[i]
  end
  res
end

#-(v) ⇒ Object

Returns self - v , where v is a Vec



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# File 'lib/linalg.rb', line 167

def -(v)
  fail "object can't be subtracted from a Vec" unless v.is_a?(Vec)
  fail "dimension mismatch" unless dim == v.dim
  res = clone
  for i in 1..dim
    res[i] -= v[i]
  end
  res
end

#-@ ⇒ Object

Returns -self.



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# File 'lib/linalg.rb', line 187

def -@
  res = clone
  for i in 1..dim
    res[i] = -res[i]
  end
  res
end

#<=>(v) ⇒ Object

The order relation is here lexicographic ordering of lists. Needed only for book-keeping purposes. Defines the functionality of self as a Comparable.



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# File 'lib/linalg.rb', line 215

def <=> (v)
  d1 = dim; d2 = v.dim
  if d1 < d2
    return -1
  elsif d1 > d2 
    return 1
  else
    for i in 0...d1
      ci = x[i] <=> v.x[i]
      return ci unless ci == 0
    end
  end
  return 0
end

#[](i) ⇒ Object

Valid indexes start with 1 not with 0. Read access to the components also works via indexes such as

y = x[3]



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# File 'lib/linalg.rb', line 144

def [](i)
  @x[i-1]
end

#[]=(i, a) ⇒ Object

Valid indexes start with 1 not with 0. Write access to the components also works via indexes such as

x[1] = 3.14



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# File 'lib/linalg.rb', line 151

def []=(i,a)
  @x[i-1] = a
end

#abs ⇒ Object

Returns the absolute value of self. This is also known as the L2-norm.



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# File 'lib/linalg.rb', line 267

def abs
  if complex?
    abs2.re.sqrt
  else
    abs2.sqrt
  end
end

#abs2 ⇒ Object

Returns the square of absolute value of self.



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# File 'lib/linalg.rb', line 261

def abs2
  spr(self)
end

#clone ⇒ Object

Returns an independent copy of self.



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# File 'lib/linalg.rb', line 80

def clone
  Vec.new(self)
end

#complex? ⇒ Boolean

Returns ‘true’ if the first component is complex. Notice that the normal usage of Vec is to have all components of the same type.

Returns:

  • (Boolean) 


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# File 'lib/linalg.rb', line 301

def complex? 
  return nil if dim.zero?
  @x[0].complex?
end

#convolution(v) ⇒ Object

Returns a ‘modified scalar product’ in which no complex conjugation is involved.



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# File 'lib/linalg.rb', line 250

def convolution(v)
  fail "dimension mismatch" unless dim == v.dim
  return nil if dim.zero?
  s = self[1] * v[1] 
  for i in 2..dim
    s += self[i] * v[i] 
  end
  s
end

#dim ⇒ Object

Returns the ‘dimension’ of the vector, i.e. the number of its components.



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# File 'lib/linalg.rb', line 53

def dim; @x.size; end

#dis(v) ⇒ Object

Returns a relative distance between self and v.



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# File 'lib/linalg.rb', line 288

def dis(v)
  a = abs
  b = v.abs
  d = (self - v).abs
  s = a + b
  return R.c0 if s.zero?
  d1 = d/s
  Basics.inf(d,d1)
end

#each ⇒ Object

Defines the functionality of self as an Enumerable.



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# File 'lib/linalg.rb', line 231

def each
  @x.each{ |c| yield c}
end

#prn(name) ⇒ Object

Prints the content of self and naming the output.



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# File 'lib/linalg.rb', line 206

def prn(name)
  for i in 1..dim 
    puts " #{name}[#{i}] = " + self[i].to_s
  end
end

#pseudo_inv(acc = 0) ⇒ Object

Gives the pseudoinverse of the vector self. This means that all components get inverted except those that are close to zero in comparison to the component with the largest absolute value. For small components c ( |c| ~ acc * sup |self| ) a continuous transition between the inverse and zero becomes operational.



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# File 'lib/linalg.rb', line 114

def pseudo_inv(acc=0)
  n = dim
  fail "dim = 0" if n.zero?
  res = clone
  if acc.zero? # most common case, thus first and without ordering 
   # overhead
    for i in 1..n 
      si = self[i]
      res[i] = si.zero? ? si.to_0 : si.inv
    end
  else
    arr = @x.clone
    arr.each{ |v| v = v.abs }
    arr.sort!
    a_max = arr.last
    eta = a_max * acc
    eta *= 0.5
    eta *= eta
    for i in 1..n 
      si = self[i]
      ni = si * si + eta
      res[i] = si / ni
    end
  end
  res
end

#spr(v) ⇒ Object

Returns the scalar product (self|v). The complex conjugation (which acts trivially on R) affects here the first factor. This is the convention preferred in physics.



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# File 'lib/linalg.rb', line 238

def spr(v)
  fail "dimension mismatch" unless dim == v.dim
  return nil if dim.zero?
  s = self[1].conj * v[1] 
  for i in 2..dim
    s += self[i].conj * v[i] 
  end
  s
end

#to_s ⇒ Object

Returns a string which consists of a list of the strings which represent the components.



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# File 'lib/linalg.rb', line 197

def to_s
  res = "\n Vec"
  for i in 0...dim
    res += "\n    " + x[i].to_s
  end
  res + "\n end Vec"
end

#uv ⇒ Object

Returns a unit vector which has the same direction as self, (or self if this is the zero-vector).



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# File 'lib/linalg.rb', line 277

def uv
  r = abs
  if r.zero?
    clone
  else 
    ri = r.inv
    self * ri
  end
end