# 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:

- `Vector.[](*array)`

- `Vector.elements`(array, copy = true)

- `Vector.basis`(size: n, index: k)

- `Vector.zero(n)`

To access elements:

- `[](i)`

To set elements:

- `[]=`(i, v)

To enumerate the elements:

- `each2(v)`

- `collect2(v)`

Properties of vectors:

- `angle_with(v)`

- `Vector.independent?(*vs)`

- `independent?(*vs)`

- `zero?`

`Vector` arithmetic:

- `*(x)` “is matrix or number”

- `+(v)`

- `-(v)`

- #/(v)

- `+`@

- `-`@

`Vector` functions:

- `inner_product(v)`, dot(v)

- `cross_product(v)`, cross(v)

- `collect`

- `collect!`

- `magnitude`

- `map`

- `map!`

- `map2(v)`

- `norm`

- `normalize`

- `r`

- `round`

- `size`

Conversion to other data types:

- `covector`

- `to_a`

- `coerce(other)`

`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 lib/matrix.rb, line 1865
    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 lib/matrix.rb, line 1882
    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 lib/matrix.rb, line 1873
    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 lib/matrix.rb, line 2029
    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

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

    # File lib/matrix.rb, line 1904
    def initialize(array)
      # No checking is done at this point.
      @elements = array
    end

zero(size) click to toggle source

Return a zero vector.

    Vector.zero(3) => Vector[0, 0, 0]

    # File lib/matrix.rb, line 1895
    def Vector.zero(size)
      raise ArgumentError, "invalid size (#{size} for 0..)" if size < 0
      array = Array.new(size, 0)
      new convert_to_array(array, false)
    end

### Public Instance Methods

*(x) click to toggle source

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

    # File lib/matrix.rb, line 2103
    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 lib/matrix.rb, line 2120
    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 lib/matrix.rb, line 2168
    def +@
      self
    end

-(v) click to toggle source

`Vector` subtraction.

    # File lib/matrix.rb, line 2138
    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 lib/matrix.rb, line 2172
    def -@
      collect {|e| -e }
    end

/(x) click to toggle source

`Vector` division.

    # File lib/matrix.rb, line 2156
    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 lib/matrix.rb, line 2079
    def ==(other)
      return false unless Vector === other
      @elements == other.elements
    end

vector[range] click to toggle source
vector[integer]

Returns element or elements of the vector.

    # File lib/matrix.rb, line 1918
    def [](i)
      @elements[i]
    end

Also aliased as: element, component
vector[range] = new_vector click to toggle source
vector[range] = row_matrix
vector[range] = new_element
vector[integer] = new_element

`Set` element or elements of vector.

    # File lib/matrix.rb, line 1933
    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

Also aliased as: set_element, set_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 lib/matrix.rb, line 2288
    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
      dot = inner_product(v)
      if dot.abs >= prod
        dot.positive? ? 0 : Math::PI
      else
        Math.acos(dot / prod)
      end
    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 lib/matrix.rb, line 2348
    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() { |e| ... } click to toggle source

Like `Array#collect`.

    # File lib/matrix.rb, line 2232
    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
collect!(&block) click to toggle source

Like `Array#collect!`

    # File lib/matrix.rb, line 2242
    def collect!(&block)
      return to_enum(:collect!) unless block_given?
      raise FrozenError, "can't modify frozen Vector" if frozen?
      @elements.collect!(&block)
      self
    end

Also aliased as: map!
collect2(v) { |e1, e2| ... } click to toggle source

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

    # File lib/matrix.rb, line 2007
    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)
Alias for: []
covector() click to toggle source

Creates a single-row matrix from this vector.

    # File lib/matrix.rb, line 2308
    def covector
      Matrix.row_vector(self)
    end

cross(*vs)
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 lib/matrix.rb, line 2207
    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)
Alias for: inner_product
each(&block) click to toggle source

Iterate over the elements of this vector

    # File lib/matrix.rb, line 1984
    def each(&block)
      return to_enum(:each) unless block_given?
      @elements.each(&block)
      self
    end

each2(v) { |e1, e2| ... } click to toggle source

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

    # File lib/matrix.rb, line 1993
    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)
Alias for: []
elements_to_f() click to toggle source

    # File lib/matrix.rb, line 2326
    def elements_to_f
      warn "Vector#elements_to_f is deprecated", uplevel: 1
      map(&:to_f)
    end

elements_to_i() click to toggle source

    # File lib/matrix.rb, line 2331
    def elements_to_i
      warn "Vector#elements_to_i is deprecated", uplevel: 1
      map(&:to_i)
    end

elements_to_r() click to toggle source

    # File lib/matrix.rb, line 2336
    def elements_to_r
      warn "Vector#elements_to_r is deprecated", uplevel: 1
      map(&:to_r)
    end

eql?(other) click to toggle source

    # File lib/matrix.rb, line 2084
    def eql?(other)
      return false unless Vector === other
      @elements.eql? other.elements
    end

freeze() click to toggle source
Calls superclass method `Object#freeze`

    # File lib/matrix.rb, line 2058
    def freeze
      @elements.freeze
      super
    end

hash() click to toggle source

Returns a hash-code for the vector.

    # File lib/matrix.rb, line 2092
    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 lib/matrix.rb, line 2047
    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 lib/matrix.rb, line 2184
    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 lib/matrix.rb, line 2371
    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 lib/matrix.rb, line 2254
    def magnitude
      Math.sqrt(@elements.inject(0) {|v, e| v + e.abs2})
    end

Also aliased as: r, norm
map()
Alias for: collect
map!(&block)
Alias for: collect!
map2(v) { |e1, e2| ... } click to toggle source

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

    # File lib/matrix.rb, line 2263
    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()
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 lib/matrix.rb, line 2277
    def normalize
      n = magnitude
      raise ZeroVectorError, "Zero vectors can not be normalized" if n == 0
      self / n
    end

r()
Alias for: magnitude
round(ndigits=0) click to toggle source

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

    # File lib/matrix.rb, line 1966
    def round(ndigits=0)
      map{|e| e.round(ndigits)}
    end

size() click to toggle source

Returns the number of elements in the vector.

    # File lib/matrix.rb, line 1973
    def size
      @elements.size
    end

to_a() click to toggle source

Returns the elements of the vector in an array.

    # File lib/matrix.rb, line 2315
    def to_a
      @elements.dup
    end

to_matrix() click to toggle source

Return a single-column matrix from this vector

    # File lib/matrix.rb, line 2322
    def to_matrix
      Matrix.column_vector(self)
    end

to_s() click to toggle source

Overrides `Object#to_s`

    # File lib/matrix.rb, line 2364
    def to_s
      "Vector[" + @elements.join(", ") + "]"
    end

zero?() click to toggle source

Returns `true` iff all elements are zero.

    # File lib/matrix.rb, line 2054
    def zero?
      all?(&:zero?)
    end

### Private Instance Methods

initialize_copy(v) click to toggle source

Called for dup & clone.

Calls superclass method

    # File lib/matrix.rb, line 2066
            def initialize_copy(v)
      super
      @elements = @elements.dup unless frozen?
    end

set_component(i, v)
Alias for: []=
set_element(i, v)
Alias for: []=
set_range(range, value) click to toggle source

    # File lib/matrix.rb, line 1951
            def set_range(range, value)
      if value.is_a?(Vector)
        raise ArgumentError, "vector to be set has wrong size" unless range.size == value.size
        @elements[range] = value.elements
      elsif value.is_a?(Matrix)
        Matrix.Raise ErrDimensionMismatch unless value.row_count == 1
        @elements[range] = value.row(0).elements
      else
        @elements[range] = Array.new(range.size, value)
      end
    end

set_value(index, value) click to toggle source

    # File lib/matrix.rb, line 1947
            def set_value(index, value)
      @elements[index] = value
    end