Class: Vector

Inherits:
Object
  • Object
show all
Extended by:
Matrix::ConversionHelper
Includes:
Enumerable, ExceptionForMatrix, Matrix::CoercionHelper
Defined in:
lib/matrix.rb

Overview

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:

  • Vector.[](*array)

  • Vector.elements(array, copy = true)

  • Vector.basis(size: n, index: k)

To access elements:

  • #[](i)

To enumerate the elements:

  • #each2(v)

  • #collect2(v)

Properties of vectors:

  • #angle_with(v)

  • Vector.independent?(*vs)

  • #independent?(*vs)

Vector arithmetic:

Vector functions:

  • #inner_product(v), dot(v)

  • #cross_product(v), cross(v)

  • #collect

  • #magnitude

  • #map

  • #map2(v)

  • #norm

  • #normalize

  • #r

  • #round

  • #size

Conversion to other data types:

  • #covector

  • #to_a

  • #coerce(other)

String representations:

  • #to_s

  • #inspect

Defined Under Namespace

Classes: ZeroVectorError

Class Method Summary collapse

Instance Method Summary collapse

Methods included from Matrix::CoercionHelper

coerce_to, coerce_to_int

Constructor Details

#initialize(array) ⇒ Vector

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


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# File 'lib/matrix.rb', line 1774

def initialize(array)
  # No checking is done at this point.
  @elements = array
end

Class Method Details

.[](*array) ⇒ Object

Creates a Vector from a list of elements.

Vector[7, 4, ...]

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# File 'lib/matrix.rb', line 1746

def Vector.[](*array)
  new convert_to_array(array, false)
end

.basis(size:, index:) ⇒ Object

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

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

Raises:

  • (ArgumentError)

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# File 'lib/matrix.rb', line 1763

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) ⇒ Object

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


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# File 'lib/matrix.rb', line 1754

def Vector.elements(array, copy = true)
  new convert_to_array(array, copy)
end

.independent?(*vs) ⇒ Boolean

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

Returns:

  • (Boolean)

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# File 'lib/matrix.rb', line 1863

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

Instance Method Details

#*(x) ⇒ Object

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


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# File 'lib/matrix.rb', line 1923

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) ⇒ Object

Vector addition.


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# File 'lib/matrix.rb', line 1940

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

#[email protected]Object


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# File 'lib/matrix.rb', line 1988

def [email protected]
  self
end

#-(v) ⇒ Object

Vector subtraction.


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# File 'lib/matrix.rb', line 1958

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

#[email protected]Object


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# File 'lib/matrix.rb', line 1992

def [email protected]
  collect {|e| -e }
end

#/(x) ⇒ Object

Vector division.


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# File 'lib/matrix.rb', line 1976

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) ⇒ Object

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


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# File 'lib/matrix.rb', line 1892

def ==(other)
  return false unless Vector === other
  @elements == other.elements
end

#[](i) ⇒ Object Also known as: element, component

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


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# File 'lib/matrix.rb', line 1784

def [](i)
  @elements[i]
end

#[]=(i, v) ⇒ Object Also known as: set_element, set_component


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# File 'lib/matrix.rb', line 1790

def []=(i, v)
  @elements[i]= v
end

#angle_with(v) ⇒ Object

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

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

Raises:

  • (TypeError)

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# File 'lib/matrix.rb', line 2097

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

#cloneObject

Returns a copy of the vector.


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# File 'lib/matrix.rb', line 1905

def clone
  self.class.elements(@elements)
end

#coerce(other) ⇒ Object

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.


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# File 'lib/matrix.rb', line 2146

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(&block) ⇒ Object Also known as: map

Like Array#collect.


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# File 'lib/matrix.rb', line 2052

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

#collect2(v) ⇒ Object

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

Raises:

  • (TypeError)

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# File 'lib/matrix.rb', line 1841

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

#covectorObject

Creates a single-row matrix from this vector.


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# File 'lib/matrix.rb', line 2113

def covector
  Matrix.row_vector(self)
end

#cross_product(*vs) ⇒ Object Also known as: cross

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]

Raises:

  • (ErrOperationNotDefined)

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# File 'lib/matrix.rb', line 2027

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

#each(&block) ⇒ Object

Iterate over the elements of this vector


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# File 'lib/matrix.rb', line 1818

def each(&block)
  return to_enum(:each) unless block_given?
  @elements.each(&block)
  self
end

#each2(v) ⇒ Object

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

Raises:

  • (TypeError)

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# File 'lib/matrix.rb', line 1827

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

#elements_to_fObject


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# File 'lib/matrix.rb', line 2124

def elements_to_f
  warn "#{caller(1)[0]}: warning: Vector#elements_to_f is deprecated"
  map(&:to_f)
end

#elements_to_iObject


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# File 'lib/matrix.rb', line 2129

def elements_to_i
  warn "#{caller(1)[0]}: warning: Vector#elements_to_i is deprecated"
  map(&:to_i)
end

#elements_to_rObject


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# File 'lib/matrix.rb', line 2134

def elements_to_r
  warn "#{caller(1)[0]}: warning: Vector#elements_to_r is deprecated"
  map(&:to_r)
end

#eql?(other) ⇒ Boolean

Returns:

  • (Boolean)

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# File 'lib/matrix.rb', line 1897

def eql?(other)
  return false unless Vector === other
  @elements.eql? other.elements
end

#hashObject

Returns a hash-code for the vector.


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# File 'lib/matrix.rb', line 1912

def hash
  @elements.hash
end

#independent?(*vs) ⇒ Boolean

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

Returns:

  • (Boolean)

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# File 'lib/matrix.rb', line 1881

def independent?(*vs)
  self.class.independent?(self, *vs)
end

#inner_product(v) ⇒ Object Also known as: dot

Returns the inner product of this vector with the other.

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

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# File 'lib/matrix.rb', line 2004

def inner_product(v)
  Vector.Raise ErrDimensionMismatch if size != v.size

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

#inspectObject

Overrides Object#inspect


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# File 'lib/matrix.rb', line 2169

def inspect
  "Vector" + @elements.inspect
end

#magnitudeObject Also known as: r, norm

Returns the modulus (Pythagorean distance) of the vector.

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

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# File 'lib/matrix.rb', line 2063

def magnitude
  Math.sqrt(@elements.inject(0) {|v, e| v + e.abs2})
end

#map2(v, &block) ⇒ Object

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


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# File 'lib/matrix.rb', line 2072

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

#normalizeObject

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

Raises:


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# File 'lib/matrix.rb', line 2086

def normalize
  n = magnitude
  raise ZeroVectorError, "Zero vectors can not be normalized" if n == 0
  self / n
end

#round(ndigits = 0) ⇒ Object

Returns a vector with entries rounded to the given precision (see Float#round)


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# File 'lib/matrix.rb', line 1800

def round(ndigits=0)
  map{|e| e.round(ndigits)}
end

#sizeObject

Returns the number of elements in the vector.


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# File 'lib/matrix.rb', line 1807

def size
  @elements.size
end

#to_aObject

Returns the elements of the vector in an array.


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# File 'lib/matrix.rb', line 2120

def to_a
  @elements.dup
end

#to_sObject

Overrides Object#to_s


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# File 'lib/matrix.rb', line 2162

def to_s
  "Vector[" + @elements.join(", ") + "]"
end