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.elements(array, copy = true) -
Vector.basis(size: n, index: k)
To access elements:
To enumerate the elements:
Properties of vectors:
Vector arithmetic:
Vector functions:
-
inner_product(v), dot(v) -
cross_product(v), cross(v)
Conversion to other data types:
String representations:
INSTANCE CREATION
# File tmp/rubies/ruby-2.3.8/lib/matrix.rb, line 1746
def Vector.[](*array)
new convert_to_array(array, false)
end
Creates a Vector from a list of elements.
Vector[7, 4, ...]
# File tmp/rubies/ruby-2.3.8/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
Returns a standard basis n-vector, where k is the index.
Vector.basis(size:, index:) # => Vector[0, 1, 0]
# File tmp/rubies/ruby-2.3.8/lib/matrix.rb, line 1754
def Vector.elements(array, copy = true)
new convert_to_array(array, copy)
end
Creates a vector from an Array. The optional second argument specifies whether the array itself or a copy is used internally.
# File tmp/rubies/ruby-2.3.8/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
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 tmp/rubies/ruby-2.3.8/lib/matrix.rb, line 1774
def initialize(array)
# No checking is done at this point.
@elements = array
end
Vector.new is private; use Vector[] or Vector.elements to create.
# File tmp/rubies/ruby-2.3.8/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
Multiplies the vector by x, where x is a number or a matrix.
# File tmp/rubies/ruby-2.3.8/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
Vector addition.
# File tmp/rubies/ruby-2.3.8/lib/matrix.rb, line 1988
def +@
self
end
# File tmp/rubies/ruby-2.3.8/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
Vector subtraction.
# File tmp/rubies/ruby-2.3.8/lib/matrix.rb, line 1992
def -@
collect {|e| -e }
end
# File tmp/rubies/ruby-2.3.8/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
Vector division.
# File tmp/rubies/ruby-2.3.8/lib/matrix.rb, line 1892
def ==(other)
return false unless Vector === other
@elements == other.elements
end
Returns true iff the two vectors have the same elements in the same order.
# File tmp/rubies/ruby-2.3.8/lib/matrix.rb, line 1784
def [](i)
@elements[i]
end
Returns element number i (starting at zero) of the vector.
# File tmp/rubies/ruby-2.3.8/lib/matrix.rb, line 1790
def []=(i, v)
@elements[i]= v
end
# File tmp/rubies/ruby-2.3.8/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
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 tmp/rubies/ruby-2.3.8/lib/matrix.rb, line 1905
def clone
self.class.elements(@elements)
end
Returns a copy of the vector.
# File tmp/rubies/ruby-2.3.8/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
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 tmp/rubies/ruby-2.3.8/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
Like Array#collect.
# File tmp/rubies/ruby-2.3.8/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
Collects (as in Enumerable#collect) over the elements of this vector and v in conjunction.
# File tmp/rubies/ruby-2.3.8/lib/matrix.rb, line 2113
def covector
Matrix.row_vector(self)
end
Creates a single-row matrix from this vector.
# File tmp/rubies/ruby-2.3.8/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
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 tmp/rubies/ruby-2.3.8/lib/matrix.rb, line 1818
def each(&block)
return to_enum(:each) unless block_given?
@elements.each(&block)
self
end
Iterate over the elements of this vector
# File tmp/rubies/ruby-2.3.8/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
Iterate over the elements of this vector and v in conjunction.
# File tmp/rubies/ruby-2.3.8/lib/matrix.rb, line 2124
def elements_to_f
warn "#{caller(1)[0]}: warning: Vector#elements_to_f is deprecated"
map(&:to_f)
end
# File tmp/rubies/ruby-2.3.8/lib/matrix.rb, line 2129
def elements_to_i
warn "#{caller(1)[0]}: warning: Vector#elements_to_i is deprecated"
map(&:to_i)
end
# File tmp/rubies/ruby-2.3.8/lib/matrix.rb, line 2134
def elements_to_r
warn "#{caller(1)[0]}: warning: Vector#elements_to_r is deprecated"
map(&:to_r)
end
# File tmp/rubies/ruby-2.3.8/lib/matrix.rb, line 1897
def eql?(other)
return false unless Vector === other
@elements.eql? other.elements
end
# File tmp/rubies/ruby-2.3.8/lib/matrix.rb, line 1912
def hash
@elements.hash
end
Returns a hash-code for the vector.
# File tmp/rubies/ruby-2.3.8/lib/matrix.rb, line 1881
def independent?(*vs)
self.class.independent?(self, *vs)
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 tmp/rubies/ruby-2.3.8/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
Returns the inner product of this vector with the other.
Vector[4,7].inner_product Vector[10,1] => 47
# File tmp/rubies/ruby-2.3.8/lib/matrix.rb, line 2169
def inspect
"Vector" + @elements.inspect
end
Overrides Object#inspect
# File tmp/rubies/ruby-2.3.8/lib/matrix.rb, line 2063
def magnitude
Math.sqrt(@elements.inject(0) {|v, e| v + e.abs2})
end
Returns the modulus (Pythagorean distance) of the vector.
Vector[5,8,2].r => 9.643650761
# File tmp/rubies/ruby-2.3.8/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
Like Vector#collect2, but returns a Vector instead of an Array.
# File tmp/rubies/ruby-2.3.8/lib/matrix.rb, line 2086
def normalize
n = magnitude
raise ZeroVectorError, "Zero vectors can not be normalized" if n == 0
self / n
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 tmp/rubies/ruby-2.3.8/lib/matrix.rb, line 1800
def round(ndigits=0)
map{|e| e.round(ndigits)}
end
Returns a vector with entries rounded to the given precision (see Float#round)
# File tmp/rubies/ruby-2.3.8/lib/matrix.rb, line 1807
def size
@elements.size
end
Returns the number of elements in the vector.
# File tmp/rubies/ruby-2.3.8/lib/matrix.rb, line 2120
def to_a
@elements.dup
end
Returns the elements of the vector in an array.
# File tmp/rubies/ruby-2.3.8/lib/matrix.rb, line 2162
def to_s
"Vector[" + @elements.join(", ") + "]"
end
Overrides Object#to_s