Class | Matrix |
In: |
lib/matrix.rb
|
Parent: | Object |
The Matrix class represents a mathematical matrix, and provides methods for creating special-case matrices (zero, identity, diagonal, singular, vector), operating on them arithmetically and algebraically, and determining their mathematical properties (trace, rank, inverse, determinant).
Note that although matrices should theoretically be rectangular, this is not enforced by the class.
Also note that the determinant of integer matrices may be incorrectly calculated unless you also require ‘mathn‘. This may be fixed in the future.
To create a matrix:
To access Matrix elements/columns/rows/submatrices/properties:
Properties of a matrix:
Matrix arithmetic:
Matrix functions:
Conversion to other data types:
String representations:
identity | -> | unit |
identity | -> | I |
Creates a single-column matrix where the values of that column are as given in column.
Matrix.column_vector([4,5,6]) => 4 5 6
# File lib/matrix.rb, line 233 def Matrix.column_vector(column) case column when Vector Matrix.columns([column.to_a]) when Array Matrix.columns([column]) else Matrix.columns([[column]]) end end
Creates a matrix using columns as an array of column vectors.
Matrix.columns([[25, 93], [-1, 66]]) => 25 -1 93 66
# File lib/matrix.rb, line 144 def Matrix.columns(columns) rows = (0 .. columns[0].size - 1).collect { |i| (0 .. columns.size - 1).collect { |j| columns[j][i] } } Matrix.rows(rows, false) end
Creates a matrix where the diagonal elements are composed of values.
Matrix.diagonal(9, 5, -3) => 9 0 0 0 5 0 0 0 -3
# File lib/matrix.rb, line 162 def Matrix.diagonal(*values) size = values.size rows = (0 .. size - 1).collect { |j| row = Array.new(size).fill(0, 0, size) row[j] = values[j] row } rows(rows, false) end
This method is used by the other methods that create matrices, and is of no use to general users.
# File lib/matrix.rb, line 248 def initialize(init_method, *argv) self.send(init_method, *argv) end
Creates a single-row matrix where the values of that row are as given in row.
Matrix.row_vector([4,5,6]) => 4 5 6
# File lib/matrix.rb, line 214 def Matrix.row_vector(row) case row when Vector Matrix.rows([row.to_a], false) when Array Matrix.rows([row.dup], false) else Matrix.rows([[row]], false) end end
Creates a matrix where rows is an array of arrays, each of which is a row to the matrix. If the optional argument copy is false, use the given arrays as the internal structure of the matrix without copying.
Matrix.rows([[25, 93], [-1, 66]]) => 25 93 -1 66
# File lib/matrix.rb, line 133 def Matrix.rows(rows, copy = true) new(:init_rows, rows, copy) end
Matrix multiplication.
Matrix[[2,4], [6,8]] * Matrix.identity(2) => 2 4 6 8
# File lib/matrix.rb, line 451 def *(m) # m is matrix or vector or number case(m) when Numeric rows = @rows.collect { |row| row.collect { |e| e * m } } return Matrix.rows(rows, false) when Vector m = Matrix.column_vector(m) r = self * m return r.column(0) when Matrix Matrix.Raise ErrDimensionMismatch if column_size != m.row_size rows = (0 .. row_size - 1).collect { |i| (0 .. m.column_size - 1).collect { |j| vij = 0 0.upto(column_size - 1) do |k| vij += self[i, k] * m[k, j] end vij } } return Matrix.rows(rows, false) else x, y = m.coerce(self) return x * y end end
Matrix exponentiation. Defined for integer powers only. Equivalent to multiplying the matrix by itself N times.
Matrix[[7,6], [3,9]] ** 2 => 67 96 48 99
# File lib/matrix.rb, line 644 def ** (other) if other.kind_of?(Integer) x = self if other <= 0 x = self.inverse return Matrix.identity(self.column_size) if other == 0 other = -other end z = x n = other - 1 while n != 0 while (div, mod = n.divmod(2) mod == 0) x = x * x n = div end z *= x n -= 1 end z elsif other.kind_of?(Float) || defined?(Rational) && other.kind_of?(Rational) Matrix.Raise ErrOperationNotDefined, "**" else Matrix.Raise ErrOperationNotDefined, "**" end end
Matrix addition.
Matrix.scalar(2,5) + Matrix[[1,0], [-4,7]] => 6 0 -4 12
# File lib/matrix.rb, line 494 def +(m) case m when Numeric Matrix.Raise ErrOperationNotDefined, "+" when Vector m = Matrix.column_vector(m) when Matrix else x, y = m.coerce(self) return x + y end Matrix.Raise ErrDimensionMismatch unless row_size == m.row_size and column_size == m.column_size rows = (0 .. row_size - 1).collect { |i| (0 .. column_size - 1).collect { |j| self[i, j] + m[i, j] } } Matrix.rows(rows, false) end
Matrix subtraction.
Matrix[[1,5], [4,2]] - Matrix[[9,3], [-4,1]] => -8 2 8 1
# File lib/matrix.rb, line 524 def -(m) case m when Numeric Matrix.Raise ErrOperationNotDefined, "-" when Vector m = Matrix.column_vector(m) when Matrix else x, y = m.coerce(self) return x - y end Matrix.Raise ErrDimensionMismatch unless row_size == m.row_size and column_size == m.column_size rows = (0 .. row_size - 1).collect { |i| (0 .. column_size - 1).collect { |j| self[i, j] - m[i, j] } } Matrix.rows(rows, false) end
Matrix division (multiplication by the inverse).
Matrix[[7,6], [3,9]] / Matrix[[2,9], [3,1]] => -7 1 -3 -6
# File lib/matrix.rb, line 554 def /(other) case other when Numeric rows = @rows.collect { |row| row.collect { |e| e / other } } return Matrix.rows(rows, false) when Matrix return self * other.inverse else x, y = other.coerce(self) rerurn x / y end end
Returns true if and only if the two matrices contain equal elements.
# File lib/matrix.rb, line 400 def ==(other) return false unless Matrix === other other.compare_by_row_vectors(@rows) end
Returns a matrix that is the result of iteration of the given block over all elements of the matrix.
Matrix[ [1,2], [3,4] ].collect { |i| i**2 } => 1 4 9 16
# File lib/matrix.rb, line 327 def collect # :yield: e rows = @rows.collect{|row| row.collect{|e| yield e}} Matrix.rows(rows, false) end
Returns column vector number j of the matrix as a Vector (starting at 0 like an array). When a block is given, the elements of that vector are iterated.
# File lib/matrix.rb, line 305 def column(j) # :yield: e if block_given? 0.upto(row_size - 1) do |i| yield @rows[i][j] end else col = (0 .. row_size - 1).collect { |i| @rows[i][j] } Vector.elements(col, false) end end
Not really intended for general consumption.
# File lib/matrix.rb, line 410 def compare_by_row_vectors(rows) return false unless @rows.size == rows.size 0.upto(@rows.size - 1) do |i| return false unless @rows[i] == rows[i] end true end
Returns the determinant of the matrix. If the matrix is not square, the result is 0.
Matrix[[7,6], [3,9]].determinant => 63
# File lib/matrix.rb, line 681 def determinant return 0 unless square? size = row_size - 1 a = to_a det = 1 k = 0 begin if (akk = a[k][k]) == 0 i = k begin return 0 if (i += 1) > size end while a[i][k] == 0 a[i], a[k] = a[k], a[i] akk = a[k][k] det *= -1 end (k + 1).upto(size) do |i| q = a[i][k] / akk (k + 1).upto(size) do |j| a[i][j] -= a[k][j] * q end end det *= akk end while (k += 1) <= size det end
Overrides Object#inspect
# File lib/matrix.rb, line 870 def inspect "Matrix"+@rows.inspect end
Not for public consumption?
# File lib/matrix.rb, line 588 def inverse_from(src) size = row_size - 1 a = src.to_a for k in 0..size i = k akk = a[k][k].abs for j in (k+1)..size v = a[j][k].abs if v > akk i = j akk = v end end Matrix.Raise ErrNotRegular if akk == 0 if i != k a[i], a[k] = a[k], a[i] @rows[i], @rows[k] = @rows[k], @rows[i] end akk = a[k][k] for i in 0 .. size next if i == k q = a[i][k] / akk a[i][k] = 0 (k + 1).upto(size) do |j| a[i][j] -= a[k][j] * q end 0.upto(size) do |j| @rows[i][j] -= @rows[k][j] * q end end (k + 1).upto(size) do |j| a[k][j] /= akk end 0.upto(size) do |j| @rows[k][j] /= akk end end self end
Returns a section of the matrix. The parameters are either:
Matrix.diagonal(9, 5, -3).minor(0..1, 0..2) => 9 0 0 0 5 0
# File lib/matrix.rb, line 342 def minor(*param) case param.size when 2 from_row = param[0].first size_row = param[0].end - from_row size_row += 1 unless param[0].exclude_end? from_col = param[1].first size_col = param[1].end - from_col size_col += 1 unless param[1].exclude_end? when 4 from_row = param[0] size_row = param[1] from_col = param[2] size_col = param[3] else Matrix.Raise ArgumentError, param.inspect end rows = @rows[from_row, size_row].collect{ |row| row[from_col, size_col] } Matrix.rows(rows, false) end
Returns the rank of the matrix. Beware that using Float values, with their usual lack of precision, can affect the value returned by this method. Use Rational values instead if this is important to you.
Matrix[[7,6], [3,9]].rank => 2
# File lib/matrix.rb, line 720 def rank if column_size > row_size a = transpose.to_a a_column_size = row_size a_row_size = column_size else a = to_a a_column_size = column_size a_row_size = row_size end rank = 0 k = 0 begin if (akk = a[k][k]) == 0 i = k exists = true begin if (i += 1) > a_column_size - 1 exists = false break end end while a[i][k] == 0 if exists a[i], a[k] = a[k], a[i] akk = a[k][k] else i = k exists = true begin if (i += 1) > a_row_size - 1 exists = false break end end while a[k][i] == 0 if exists k.upto(a_column_size - 1) do |j| a[j][k], a[j][i] = a[j][i], a[j][k] end akk = a[k][k] else next end end end (k + 1).upto(a_row_size - 1) do |i| q = a[i][k] / akk (k + 1).upto(a_column_size - 1) do |j| a[i][j] -= a[k][j] * q end end rank += 1 end while (k += 1) <= a_column_size - 1 return rank end
Returns true if this is a regular matrix.
# File lib/matrix.rb, line 374 def regular? square? and rank == column_size end
Returns true is this is a singular (i.e. non-regular) matrix.
# File lib/matrix.rb, line 381 def singular? not regular? end
Returns true is this is a square matrix. See note in column_size about this being unreliable, though.
# File lib/matrix.rb, line 389 def square? column_size == row_size end
Overrides Object#to_s
# File lib/matrix.rb, line 860 def to_s "Matrix[" + @rows.collect{ |row| "[" + row.collect{|e| e.to_s}.join(", ") + "]" }.join(", ")+"]" end