Vector

The Vector class represents a mathematical vector, which is useful in its own right, and also constitutes a row or column of a Matrix.

Method Catalogue¶ ↑

To create a Vector:

::[]
::elements(array, copy = true)
::basis(size: n, index: k)

To access elements:

[](i)

To enumerate the elements:

Properties of vectors:

Vector arithmetic:

#*(x) “is matrix or number”
#+(v)
#-(v)
#+@
#-@

Vector functions:

Conversion to other data types:

String representations:

to_s
inspect

Attributes

elements[R]

INSTANCE CREATION

Public Class Methods

[](*array) click to toggle source

Creates a Vector from a list of elements.

Vector[7, 4, ...]

 
               # File matrix.rb, line 1741
def Vector.[](*array)
  new convert_to_array(array, false)
end

basis(size:, index:) click to toggle source

Returns a standard basis n-vector, where k is the index.

Vector.basis(size:, index:) # => Vector[0, 1, 0]

 
               # File matrix.rb, line 1758
def Vector.basis(size,, index))
  raise ArgumentError, "invalid size (#{size} for 1..)" if size < 1
  raise ArgumentError, "invalid index (#{index} for 0...#{size})" unless 0 <= index && index < size
  array = Array.new(size, 0)
  array[index] = 1
  new convert_to_array(array, false)
end

elements(array, copy = true) click to toggle source

Creates a vector from an Array. The optional second argument specifies whether the array itself or a copy is used internally.

 
               # File matrix.rb, line 1749
def Vector.elements(array, copy = true)
  new convert_to_array(array, copy)
end

independent?(*vs) click to toggle source

Returns true iff all of vectors are linearly independent.

Vector.independent?(Vector[1,0], Vector[0,1])
  => true

Vector.independent?(Vector[1,2], Vector[2,4])
  => false

 
               # File matrix.rb, line 1851
def Vector.independent?(*vs)
  vs.each do |v|
    raise TypeError, "expected Vector, got #{v.class}" unless v.is_a?(Vector)
    Vector.Raise ErrDimensionMismatch unless v.size == vs.first.size
  end
  return false if vs.count > vs.first.size
  Matrix[*vs].rank.eql?(vs.count)
end

new(array) click to toggle source

::new is private; use Vector[] or ::elements to create.

 
               # File matrix.rb, line 1769
def initialize(array)
  # No checking is done at this point.
  @elements = array
end

Public Instance Methods

*(x) click to toggle source

Multiplies the vector by x, where x is a number or another vector.

 
               # File matrix.rb, line 1911
def *(x)
  case x
  when Numeric
    els = @elements.collect{|e| e * x}
    self.class.elements(els, false)
  when Matrix
    Matrix.column_vector(self) * x
  when Vector
    Vector.Raise ErrOperationNotDefined, "*", self.class, x.class
  else
    apply_through_coercion(x, __method__)
  end
end

+(v) click to toggle source

Vector addition.

 
               # File matrix.rb, line 1928
def +(v)
  case v
  when Vector
    Vector.Raise ErrDimensionMismatch if size != v.size
    els = collect2(v) {|v1, v2|
      v1 + v2
    }
    self.class.elements(els, false)
  when Matrix
    Matrix.column_vector(self) + v
  else
    apply_through_coercion(v, __method__)
  end
end

+@() click to toggle source

 
               # File matrix.rb, line 1976
def +@
  self
end

-(v) click to toggle source

Vector subtraction.

 
               # File matrix.rb, line 1946
def -(v)
  case v
  when Vector
    Vector.Raise ErrDimensionMismatch if size != v.size
    els = collect2(v) {|v1, v2|
      v1 - v2
    }
    self.class.elements(els, false)
  when Matrix
    Matrix.column_vector(self) - v
  else
    apply_through_coercion(v, __method__)
  end
end

-@() click to toggle source

 
               # File matrix.rb, line 1980
def -@
  collect {|e| -e }
end

/(x) click to toggle source

Vector division.

 
               # File matrix.rb, line 1964
def /(x)
  case x
  when Numeric
    els = @elements.collect{|e| e / x}
    self.class.elements(els, false)
  when Matrix, Vector
    Vector.Raise ErrOperationNotDefined, "/", self.class, x.class
  else
    apply_through_coercion(x, __method__)
  end
end

==(other) click to toggle source

Returns true iff the two vectors have the same elements in the same order.

 
               # File matrix.rb, line 1880
def ==(other)
  return false unless Vector === other
  @elements == other.elements
end

[](i) click to toggle source

Returns element number i (starting at zero) of the vector.

 
               # File matrix.rb, line 1779
def [](i)
  @elements[i]
end

Also aliased as: element, component

angle_with(v) click to toggle source

Returns an angle with another vector. Result is within the [0…Math::PI].

Vector[1,0].angle_with(Vector[0,1])
# => Math::PI / 2

 
               # File matrix.rb, line 2085
def angle_with(v)
  raise TypeError, "Expected a Vector, got a #{v.class}" unless v.is_a?(Vector)
  Vector.Raise ErrDimensionMismatch if size != v.size
  prod = magnitude * v.magnitude
  raise ZeroVectorError, "Can't get angle of zero vector" if prod == 0

  Math.acos( inner_product(v) / prod )
end

clone() click to toggle source

Returns a copy of the vector.

 
               # File matrix.rb, line 1893
def clone
  self.class.elements(@elements)
end

coerce(other) click to toggle source

The coerce method provides support for Ruby type coercion. This coercion mechanism is used by Ruby to handle mixed-type numeric operations: it is intended to find a compatible common type between the two operands of the operator. See also Numeric#coerce.

 
               # File matrix.rb, line 2134
def coerce(other)
  case other
  when Numeric
    return Matrix::Scalar.new(other), self
  else
    raise TypeError, "#{self.class} can't be coerced into #{other.class}"
  end
end

collect() click to toggle source

Like Array#collect.

 
               # File matrix.rb, line 2040
def collect(&block) # :yield: e
  return to_enum(:collect) unless block_given?
  els = @elements.collect(&block)
  self.class.elements(els, false)
end

Also aliased as: map

collect2(v) click to toggle source

Collects (as in Enumerable#collect) over the elements of this vector and v in conjunction.

 
               # File matrix.rb, line 1829
def collect2(v) # :yield: e1, e2
  raise TypeError, "Integer is not like Vector" if v.kind_of?(Integer)
  Vector.Raise ErrDimensionMismatch if size != v.size
  return to_enum(:collect2, v) unless block_given?
  Array.new(size) do |i|
    yield @elements[i], v[i]
  end
end

component(i) click to toggle source

Alias for: []

covector() click to toggle source

Creates a single-row matrix from this vector.

 
               # File matrix.rb, line 2101
def covector
  Matrix.row_vector(self)
end

cross(*vs) click to toggle source

Alias for: cross_product

cross_product(*vs) click to toggle source

Returns the cross product of this vector with the others.

Vector[1, 0, 0].cross_product Vector[0, 1, 0]   => Vector[0, 0, 1]

It is generalized to other dimensions to return a vector perpendicular to the arguments.

Vector[1, 2].cross_product # => Vector[-2, 1]
Vector[1, 0, 0, 0].cross_product(
   Vector[0, 1, 0, 0],
   Vector[0, 0, 1, 0]
)  #=> Vector[0, 0, 0, 1]

 
               # File matrix.rb, line 2015
def cross_product(*vs)
  raise ErrOperationNotDefined, "cross product is not defined on vectors of dimension #{size}" unless size >= 2
  raise ArgumentError, "wrong number of arguments (#{vs.size} for #{size - 2})" unless vs.size == size - 2
  vs.each do |v|
    raise TypeError, "expected Vector, got #{v.class}" unless v.is_a? Vector
    Vector.Raise ErrDimensionMismatch unless v.size == size
  end
  case size
  when 2
    Vector[-@elements[1], @elements[0]]
  when 3
    v = vs[0]
    Vector[ v[2]*@elements[1] - v[1]*@elements[2],
      v[0]*@elements[2] - v[2]*@elements[0],
      v[1]*@elements[0] - v[0]*@elements[1] ]
  else
    rows = self, *vs, Array.new(size) {|i| Vector.basis(size: size, index: i) }
    Matrix.rows(rows).laplace_expansion(row: size - 1)
  end
end

Also aliased as: cross

dot(v) click to toggle source

Alias for: inner_product

each(&block) click to toggle source

Iterate over the elements of this vector

 
               # File matrix.rb, line 1806
def each(&block)
  return to_enum(:each) unless block_given?
  @elements.each(&block)
  self
end

each2(v) click to toggle source

Iterate over the elements of this vector and v in conjunction.

 
               # File matrix.rb, line 1815
def each2(v) # :yield: e1, e2
  raise TypeError, "Integer is not like Vector" if v.kind_of?(Integer)
  Vector.Raise ErrDimensionMismatch if size != v.size
  return to_enum(:each2, v) unless block_given?
  size.times do |i|
    yield @elements[i], v[i]
  end
  self
end

element(i) click to toggle source

Alias for: []

elements_to_f() click to toggle source

 
               # File matrix.rb, line 2112
def elements_to_f
  warn "#{caller(1)[0]}: warning: Vector#elements_to_f is deprecated"
  map(&:to_f)
end

elements_to_i() click to toggle source

 
               # File matrix.rb, line 2117
def elements_to_i
  warn "#{caller(1)[0]}: warning: Vector#elements_to_i is deprecated"
  map(&:to_i)
end

elements_to_r() click to toggle source

 
               # File matrix.rb, line 2122
def elements_to_r
  warn "#{caller(1)[0]}: warning: Vector#elements_to_r is deprecated"
  map(&:to_r)
end

eql?(other) click to toggle source

 
               # File matrix.rb, line 1885
def eql?(other)
  return false unless Vector === other
  @elements.eql? other.elements
end

hash() click to toggle source

Returns a hash-code for the vector.

 
               # File matrix.rb, line 1900
def hash
  @elements.hash
end

independent?(*vs) click to toggle source

Returns true iff all of vectors are linearly independent.

Vector[1,0].independent?(Vector[0,1])
  => true

Vector[1,2].independent?(Vector[2,4])
  => false

 
               # File matrix.rb, line 1869
def independent?(*vs)
  self.class.independent?(self, *vs)
end

inner_product(v) click to toggle source

Returns the inner product of this vector with the other.

Vector[4,7].inner_product Vector[10,1]  => 47

 
               # File matrix.rb, line 1992
def inner_product(v)
  Vector.Raise ErrDimensionMismatch if size != v.size

  p = 0
  each2(v) {|v1, v2|
    p += v1 * v2.conj
  }
  p
end

Also aliased as: dot

inspect() click to toggle source

Overrides Object#inspect

 
               # File matrix.rb, line 2157
def inspect
  "Vector" + @elements.inspect
end

magnitude() click to toggle source

Returns the modulus (Pythagorean distance) of the vector.

Vector[5,8,2].r => 9.643650761

 
               # File matrix.rb, line 2051
def magnitude
  Math.sqrt(@elements.inject(0) {|v, e| v + e.abs2})
end

Also aliased as: r, norm

map() click to toggle source

Alias for: collect

map2(v) click to toggle source

Like #collect2, but returns a Vector instead of an Array.

 
               # File matrix.rb, line 2060
def map2(v, &block) # :yield: e1, e2
  return to_enum(:map2, v) unless block_given?
  els = collect2(v, &block)
  self.class.elements(els, false)
end

norm() click to toggle source

Alias for: magnitude

normalize() click to toggle source

Returns a new vector with the same direction but with norm 1.

v = Vector[5,8,2].normalize
# => Vector[0.5184758473652127, 0.8295613557843402, 0.20739033894608505]
v.norm => 1.0

 
               # File matrix.rb, line 2074
def normalize
  n = magnitude
  raise ZeroVectorError, "Zero vectors can not be normalized" if n == 0
  self / n
end

r() click to toggle source

Alias for: magnitude

size() click to toggle source

Returns the number of elements in the vector.

 
               # File matrix.rb, line 1795
def size
  @elements.size
end

to_a() click to toggle source

Returns the elements of the vector in an array.

 
               # File matrix.rb, line 2108
def to_a
  @elements.dup
end

to_s() click to toggle source

Overrides Object#to_s

 
               # File matrix.rb, line 2150
def to_s
  "Vector[" + @elements.join(", ") + "]"
end

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