Object
The Vector
class represents a mathematical vector, which is useful in its own right, and also constitutes a row or column of a Matrix
.
To create a Vector:
Vector.elements
(array, copy = true)
Vector.basis
(size: n, index: k)
To access elements:
To set elements:
[]=
(i, v)
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:
Creates a Vector
from a list of elements.
Vector[7, 4, ...]
# File matrix.rb, line 1898 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 matrix.rb, line 1915 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 matrix.rb, line 1906 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 matrix.rb, line 2062 def Vector.independent?(*vs) vs.each do |v| raise TypeError, "expected Vector, got #{v.class}" unless v.is_a?(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
Vector.new
is private; use Vector[] or Vector.elements
to create.
# File matrix.rb, line 1937 def initialize(array) # No checking is done at this point. @elements = array end
Multiplies the vector by x
, where x
is a number or a matrix.
# File matrix.rb, line 2136 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 raise ErrOperationNotDefined, ["*", self.class, x.class] else apply_through_coercion(x, __method__) end end
Vector
addition.
# File matrix.rb, line 2153 def +(v) case v when 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 matrix.rb, line 2171 def -(v) case v when 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
division.
# File matrix.rb, line 2189 def /(x) case x when Numeric els = @elements.collect{|e| e / x} self.class.elements(els, false) when Matrix, 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 matrix.rb, line 2112 def ==(other) return false unless Vector === other @elements == other.elements end
Returns element or elements of the vector.
# File matrix.rb, line 1951 def [](i) @elements[i] end
Set element or elements of vector.
# File matrix.rb, line 1966 def []=(i, v) raise FrozenError, "can't modify frozen Vector" if frozen? if i.is_a?(Range) range = Matrix::CoercionHelper.check_range(i, size, :vector) set_range(range, v) else index = Matrix::CoercionHelper.check_int(i, size, :index) set_value(index, v) end 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 matrix.rb, line 2321 def angle_with(v) raise TypeError, "Expected a Vector, got a #{v.class}" unless v.is_a?(Vector) raise ErrDimensionMismatch if size != v.size prod = magnitude * v.magnitude raise ZeroVectorError, "Can't get angle of zero vector" if prod == 0 dot = inner_product(v) if dot.abs >= prod dot.positive? ? 0 : Math::PI else Math.acos(dot / prod) 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 matrix.rb, line 2381 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 matrix.rb, line 2265 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 matrix.rb, line 2275 def collect!(&block) return to_enum(:collect!) unless block_given? raise FrozenError, "can't modify frozen Vector" if frozen? @elements.collect!(&block) self end
Collects (as in Enumerable#collect) over the elements of this vector and v
in conjunction.
# File matrix.rb, line 2040 def collect2(v) # :yield: e1, e2 raise TypeError, "Integer is not like Vector" if v.kind_of?(Integer) 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 matrix.rb, line 2341 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 matrix.rb, line 2240 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 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 matrix.rb, line 2017 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 matrix.rb, line 2026 def each2(v) # :yield: e1, e2 raise TypeError, "Integer is not like Vector" if v.kind_of?(Integer) 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 matrix.rb, line 2359 def elements_to_f warn "Vector#elements_to_f is deprecated", uplevel: 1 map(&:to_f) end
# File matrix.rb, line 2364 def elements_to_i warn "Vector#elements_to_i is deprecated", uplevel: 1 map(&:to_i) end
# File matrix.rb, line 2369 def elements_to_r warn "Vector#elements_to_r is deprecated", uplevel: 1 map(&:to_r) end
# File matrix.rb, line 2117 def eql?(other) return false unless Vector === other @elements.eql? other.elements end
Returns a hash-code for the vector.
# File matrix.rb, line 2125 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 matrix.rb, line 2080 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 matrix.rb, line 2217 def inner_product(v) raise ErrDimensionMismatch if size != v.size p = 0 each2(v) {|v1, v2| p += v1 * v2.conj } p end
Overrides Object#inspect
# File matrix.rb, line 2404 def inspect "Vector" + @elements.inspect end
Returns the modulus (Pythagorean distance) of the vector.
Vector[5,8,2].r => 9.643650761
# File matrix.rb, line 2287 def magnitude Math.sqrt(@elements.inject(0) {|v, e| v + e.abs2}) end
Like Vector#collect2
, but returns a Vector
instead of an Array.
# File matrix.rb, line 2296 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
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 2310 def normalize n = magnitude raise ZeroVectorError, "Zero vectors can not be normalized" if n == 0 self / n end
Returns a vector with entries rounded to the given precision (see Float#round)
# File matrix.rb, line 1999 def round(ndigits=0) map{|e| e.round(ndigits)} end
Returns the number of elements in the vector.
# File matrix.rb, line 2006 def size @elements.size end
Returns the elements of the vector in an array.
# File matrix.rb, line 2348 def to_a @elements.dup end
Return a single-column matrix from this vector
# File matrix.rb, line 2355 def to_matrix Matrix.column_vector(self) end