class 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:
To enumerate the elements:
Properties of vectors:
Vector arithmetic:
-
#*(x) “is matrix or number”
-
#+(v)
-
#-(v)
-
#+@
-
#-@
Vector functions:
-
inner_product(v), dot(v)
-
cross_product(v), cross(v)
Conversion to other data types:
String representations:
Attributes
INSTANCE CREATION
Public Class Methods
Creates a Vector from a list of elements.
Vector[7, 4, ...]
# File lib/matrix.rb, line 1741 def Vector.[](*array) new convert_to_array(array, false) end
Returns a standard basis n
-vector, where k is the index.
Vector.basis(size:, index:) # => Vector[0, 1, 0]
# File lib/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
Creates a vector from an Array. The optional second argument specifies whether the array itself or a copy is used internally.
# File lib/matrix.rb, line 1749 def Vector.elements(array, copy = true) new convert_to_array(array, copy) end
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 lib/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 is private; use Vector[] or ::elements to create.
# File lib/matrix.rb, line 1769 def initialize(array) # No checking is done at this point. @elements = array end
Public Instance Methods
Multiplies the vector by x
, where x
is a number
or another vector.
# File lib/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
Vector addition.
# File lib/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
# File lib/matrix.rb, line 1976 def +@ self end
Vector subtraction.
# File lib/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
# File lib/matrix.rb, line 1980 def -@ collect {|e| -e } end
Vector division.
# File lib/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
Returns true
iff the two vectors have the same elements in the
same order.
# File lib/matrix.rb, line 1880 def ==(other) return false unless Vector === other @elements == other.elements end
Returns element number i
(starting at zero) of the vector.
# File lib/matrix.rb, line 1779 def [](i) @elements[i] end
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 lib/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
Returns a copy of the vector.
# File lib/matrix.rb, line 1893 def clone self.class.elements(@elements) end
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 lib/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
Like Array#collect.
# File lib/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
Collects (as in Enumerable#collect) over the
elements of this vector and v
in conjunction.
# File lib/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
Creates a single-row matrix from this vector.
# File lib/matrix.rb, line 2101 def covector Matrix.row_vector(self) end
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 lib/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
Iterate over the elements of this vector
# File lib/matrix.rb, line 1806 def each(&block) return to_enum(:each) unless block_given? @elements.each(&block) self end
Iterate over the elements of this vector and v
in conjunction.
# File lib/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
# File lib/matrix.rb, line 2112 def elements_to_f warn "#{caller(1)[0]}: warning: Vector#elements_to_f is deprecated" map(&:to_f) end
# File lib/matrix.rb, line 2117 def elements_to_i warn "#{caller(1)[0]}: warning: Vector#elements_to_i is deprecated" map(&:to_i) end
# File lib/matrix.rb, line 2122 def elements_to_r warn "#{caller(1)[0]}: warning: Vector#elements_to_r is deprecated" map(&:to_r) end
# File lib/matrix.rb, line 1885 def eql?(other) return false unless Vector === other @elements.eql? other.elements end
Returns a hash-code for the vector.
# File lib/matrix.rb, line 1900 def hash @elements.hash end
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 lib/matrix.rb, line 1869 def independent?(*vs) self.class.independent?(self, *vs) end
Returns the inner product of this vector with the other.
Vector[4,7].inner_product Vector[10,1] => 47
# File lib/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
Overrides Object#inspect
# File lib/matrix.rb, line 2157 def inspect "Vector" + @elements.inspect end
Returns the modulus (Pythagorean distance) of the vector.
Vector[5,8,2].r => 9.643650761
# File lib/matrix.rb, line 2051 def magnitude Math.sqrt(@elements.inject(0) {|v, e| v + e.abs2}) end
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 lib/matrix.rb, line 2074 def normalize n = magnitude raise ZeroVectorError, "Zero vectors can not be normalized" if n == 0 self / n end
Returns the number of elements in the vector.
# File lib/matrix.rb, line 1795 def size @elements.size end
Returns the elements of the vector in an array.
# File lib/matrix.rb, line 2108 def to_a @elements.dup end
Overrides Object#to_s
# File lib/matrix.rb, line 2150 def to_s "Vector[" + @elements.join(", ") + "]" end
Private Instance Methods
# File lib/matrix.rb, line 1785 def []=(i, v) @elements[i]= v end