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# encoding: utf-8
#
# = matrix.rb
#
# An implementation of Matrix and Vector classes.
#
# See classes Matrix and Vector for documentation.
#
# Current Maintainer:: Marc-André Lafortune
# Original Author:: Keiju ISHITSUKA
# Original Documentation:: Gavin Sinclair (sourced from <i>Ruby in a Nutshell</i> (Matsumoto, O'Reilly))
##
require "e2mmap.rb"
module ExceptionForMatrix # :nodoc:
extend Exception2MessageMapper
def_e2message(TypeError, "wrong argument type %s (expected %s)")
def_e2message(ArgumentError, "Wrong # of arguments(%d for %d)")
def_exception("ErrDimensionMismatch", "\#{self.name} dimension mismatch")
def_exception("ErrNotRegular", "Not Regular Matrix")
def_exception("ErrOperationNotDefined", "Operation(%s) can\\'t be defined: %s op %s")
def_exception("ErrOperationNotImplemented", "Sorry, Operation(%s) not implemented: %s op %s")
end
#
# The +Matrix+ class represents a mathematical matrix. It provides methods for creating
# matrices, operating on them arithmetically and algebraically,
# and determining their mathematical properties (trace, rank, inverse, determinant).
#
# == Method Catalogue
#
# To create a matrix:
# * Matrix[*rows]
# * Matrix.[](*rows)
# * Matrix.rows(rows, copy = true)
# * Matrix.columns(columns)
# * Matrix.build(row_size, column_size, &block)
# * Matrix.diagonal(*values)
# * Matrix.scalar(n, value)
# * Matrix.identity(n)
# * Matrix.unit(n)
# * Matrix.I(n)
# * Matrix.zero(n)
# * Matrix.row_vector(row)
# * Matrix.column_vector(column)
#
# To access Matrix elements/columns/rows/submatrices/properties:
# * #[](i, j)
# * #row_size
# * #column_size
# * #row(i)
# * #column(j)
# * #collect
# * #map
# * #each
# * #each_with_index
# * #find_index
# * #minor(*param)
#
# Properties of a matrix:
# * #diagonal?
# * #empty?
# * #hermitian?
# * #lower_triangular?
# * #normal?
# * #orthogonal?
# * #permutation?
# * #real?
# * #regular?
# * #singular?
# * #square?
# * #symmetric?
# * #unitary?
# * #upper_triangular?
# * #zero?
#
# Matrix arithmetic:
# * #*(m)
# * #+(m)
# * #-(m)
# * #/(m)
# * #inverse
# * #inv
# * #**
#
# Matrix functions:
# * #determinant
# * #det
# * #rank
# * #round
# * #trace
# * #tr
# * #transpose
# * #t
#
# Matrix decompositions:
# * #eigen
# * #eigensystem
# * #lup
# * #lup_decomposition
#
# Complex arithmetic:
# * conj
# * conjugate
# * imag
# * imaginary
# * real
# * rect
# * rectangular
#
# Conversion to other data types:
# * #coerce(other)
# * #row_vectors
# * #column_vectors
# * #to_a
#
# String representations:
# * #to_s
# * #inspect
#
class Matrix
include Enumerable
include ExceptionForMatrix
autoload :EigenvalueDecomposition, "matrix/eigenvalue_decomposition"
autoload :LUPDecomposition, "matrix/lup_decomposition"
# instance creations
private_class_method :new
attr_reader :rows
protected :rows
#
# Creates a matrix where each argument is a row.
# Matrix[ [25, 93], [-1, 66] ]
# => 25 93
# -1 66
#
def Matrix.[](*rows)
rows(rows, false)
end
#
# Creates a matrix where +rows+ is an array of arrays, each of which is a row
# of 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
#
def Matrix.rows(rows, copy = true)
rows = convert_to_array(rows)
rows.map! do |row|
convert_to_array(row, copy)
end
size = (rows[0] || []).size
rows.each do |row|
Matrix.Raise ErrDimensionMismatch, "row size differs (#{row.size} should be #{size})" unless row.size == size
end
new rows, size
end
#
# Creates a matrix using +columns+ as an array of column vectors.
# Matrix.columns([[25, 93], [-1, 66]])
# => 25 -1
# 93 66
#
def Matrix.columns(columns)
rows(columns, false).transpose
end
#
# Creates a matrix of size +row_size+ x +column_size+.
# It fills the values by calling the given block,
# passing the current row and column.
# Returns an enumerator if no block is given.
#
# m = Matrix.build(2, 4) {|row, col| col - row }
# => Matrix[[0, 1, 2, 3], [-1, 0, 1, 2]]
# m = Matrix.build(3) { rand }
# => a 3x3 matrix with random elements
#
def Matrix.build(row_size, column_size = row_size)
row_size = CoercionHelper.coerce_to_int(row_size)
column_size = CoercionHelper.coerce_to_int(column_size)
raise ArgumentError if row_size < 0 || column_size < 0
return to_enum :build, row_size, column_size unless block_given?
rows = Array.new(row_size) do |i|
Array.new(column_size) do |j|
yield i, j
end
end
new rows, column_size
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
#
def Matrix.diagonal(*values)
size = values.size
rows = Array.new(size) {|j|
row = Array.new(size, 0)
row[j] = values[j]
row
}
new rows
end
#
# Creates an +n+ by +n+ diagonal matrix where each diagonal element is
# +value+.
# Matrix.scalar(2, 5)
# => 5 0
# 0 5
#
def Matrix.scalar(n, value)
diagonal(*Array.new(n, value))
end
#
# Creates an +n+ by +n+ identity matrix.
# Matrix.identity(2)
# => 1 0
# 0 1
#
def Matrix.identity(n)
scalar(n, 1)
end
class << Matrix
alias unit identity
alias I identity
end
#
# Creates a zero matrix.
# Matrix.zero(2)
# => 0 0
# 0 0
#
def Matrix.zero(row_size, column_size = row_size)
rows = Array.new(row_size){Array.new(column_size, 0)}
new rows, column_size
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
#
def Matrix.row_vector(row)
row = convert_to_array(row)
new [row]
end
#
# 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
#
def Matrix.column_vector(column)
column = convert_to_array(column)
new [column].transpose, 1
end
#
# Creates a empty matrix of +row_size+ x +column_size+.
# At least one of +row_size+ or +column_size+ must be 0.
#
# m = Matrix.empty(2, 0)
# m == Matrix[ [], [] ]
# => true
# n = Matrix.empty(0, 3)
# n == Matrix.columns([ [], [], [] ])
# => true
# m * n
# => Matrix[[0, 0, 0], [0, 0, 0]]
#
def Matrix.empty(row_size = 0, column_size = 0)
Matrix.Raise ArgumentError, "One size must be 0" if column_size != 0 && row_size != 0
Matrix.Raise ArgumentError, "Negative size" if column_size < 0 || row_size < 0
new([[]]*row_size, column_size)
end
#
# Matrix.new is private; use Matrix.rows, columns, [], etc... to create.
#
def initialize(rows, column_size = rows[0].size)
# No checking is done at this point. rows must be an Array of Arrays.
# column_size must be the size of the first row, if there is one,
# otherwise it *must* be specified and can be any integer >= 0
@rows = rows
@column_size = column_size
end
def new_matrix(rows, column_size = rows[0].size) # :nodoc:
self.class.send(:new, rows, column_size) # bypass privacy of Matrix.new
end
private :new_matrix
#
# Returns element (+i+,+j+) of the matrix. That is: row +i+, column +j+.
#
def [](i, j)
@rows.fetch(i){return nil}[j]
end
alias element []
alias component []
def []=(i, j, v)
@rows[i][j] = v
end
alias set_element []=
alias set_component []=
private :[]=, :set_element, :set_component
#
# Returns the number of rows.
#
def row_size
@rows.size
end
#
# Returns the number of columns.
#
attr_reader :column_size
#
# Returns row vector number +i+ of the matrix as a Vector (starting at 0 like
# an array). When a block is given, the elements of that vector are iterated.
#
def row(i, &block) # :yield: e
if block_given?
@rows.fetch(i){return self}.each(&block)
self
else
Vector.elements(@rows.fetch(i){return nil})
end
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.
#
def column(j) # :yield: e
if block_given?
return self if j >= column_size || j < -column_size
row_size.times do |i|
yield @rows[i][j]
end
self
else
return nil if j >= column_size || j < -column_size
col = Array.new(row_size) {|i|
@rows[i][j]
}
Vector.elements(col, false)
end
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 { |e| e**2 }
# => 1 4
# 9 16
#
def collect(&block) # :yield: e
return to_enum(:collect) unless block_given?
rows = @rows.collect{|row| row.collect(&block)}
new_matrix rows, column_size
end
alias map collect
#
# Yields all elements of the matrix, starting with those of the first row,
# or returns an Enumerator is no block given.
# Elements can be restricted by passing an argument:
# * :all (default): yields all elements
# * :diagonal: yields only elements on the diagonal
# * :off_diagonal: yields all elements except on the diagonal
# * :lower: yields only elements on or below the diagonal
# * :strict_lower: yields only elements below the diagonal
# * :strict_upper: yields only elements above the diagonal
# * :upper: yields only elements on or above the diagonal
#
# Matrix[ [1,2], [3,4] ].each { |e| puts e }
# # => prints the numbers 1 to 4
# Matrix[ [1,2], [3,4] ].each(:strict_lower).to_a # => [3]
#
def each(which = :all) # :yield: e
return to_enum :each, which unless block_given?
last = column_size - 1
case which
when :all
block = Proc.new
@rows.each do |row|
row.each(&block)
end
when :diagonal
@rows.each_with_index do |row, row_index|
yield row.fetch(row_index){return self}
end
when :off_diagonal
@rows.each_with_index do |row, row_index|
column_size.times do |col_index|
yield row[col_index] unless row_index == col_index
end
end
when :lower
@rows.each_with_index do |row, row_index|
0.upto([row_index, last].min) do |col_index|
yield row[col_index]
end
end
when :strict_lower
@rows.each_with_index do |row, row_index|
[row_index, column_size].min.times do |col_index|
yield row[col_index]
end
end
when :strict_upper
@rows.each_with_index do |row, row_index|
(row_index+1).upto(last) do |col_index|
yield row[col_index]
end
end
when :upper
@rows.each_with_index do |row, row_index|
row_index.upto(last) do |col_index|
yield row[col_index]
end
end
else
Matrix.Raise ArgumentError, "expected #{which.inspect} to be one of :all, :diagonal, :off_diagonal, :lower, :strict_lower, :strict_upper or :upper"
end
self
end
#
# Same as #each, but the row index and column index in addition to the element
#
# Matrix[ [1,2], [3,4] ].each_with_index do |e, row, col|
# puts "#{e} at #{row}, #{col}"
# end
# # => Prints:
# # 1 at 0, 0
# # 2 at 0, 1
# # 3 at 1, 0
# # 4 at 1, 1
#
def each_with_index(which = :all) # :yield: e, row, column
return to_enum :each_with_index, which unless block_given?
last = column_size - 1
case which
when :all
@rows.each_with_index do |row, row_index|
row.each_with_index do |e, col_index|
yield e, row_index, col_index
end
end
when :diagonal
@rows.each_with_index do |row, row_index|
yield row.fetch(row_index){return self}, row_index, row_index
end
when :off_diagonal
@rows.each_with_index do |row, row_index|
column_size.times do |col_index|
yield row[col_index], row_index, col_index unless row_index == col_index
end
end
when :lower
@rows.each_with_index do |row, row_index|
0.upto([row_index, last].min) do |col_index|
yield row[col_index], row_index, col_index
end
end
when :strict_lower
@rows.each_with_index do |row, row_index|
[row_index, column_size].min.times do |col_index|
yield row[col_index], row_index, col_index
end
end
when :strict_upper
@rows.each_with_index do |row, row_index|
(row_index+1).upto(last) do |col_index|
yield row[col_index], row_index, col_index
end
end
when :upper
@rows.each_with_index do |row, row_index|
row_index.upto(last) do |col_index|
yield row[col_index], row_index, col_index
end
end
else
Matrix.Raise ArgumentError, "expected #{which.inspect} to be one of :all, :diagonal, :off_diagonal, :lower, :strict_lower, :strict_upper or :upper"
end
self
end
SELECTORS = {all: true, diagonal: true, off_diagonal: true, lower: true, strict_lower: true, strict_upper: true, upper: true}.freeze
#
# :call-seq:
# index(value, selector = :all) -> [row, column]
# index(selector = :all){ block } -> [row, column]
# index(selector = :all) -> an_enumerator
#
# The index method is specialized to return the index as [row, column]
# It also accepts an optional +selector+ argument, see #each for details.
#
# Matrix[ [1,2], [3,4] ].index(&:even?) # => [0, 1]
# Matrix[ [1,1], [1,1] ].index(1, :strict_lower) # => [1, 0]
#
def index(*args)
raise ArgumentError, "wrong number of arguments(#{args.size} for 0-2)" if args.size > 2
which = (args.size == 2 || SELECTORS.include?(args.last)) ? args.pop : :all
return to_enum :find_index, which, *args unless block_given? || args.size == 1
if args.size == 1
value = args.first
each_with_index(which) do |e, row_index, col_index|
return row_index, col_index if e == value
end
else
each_with_index(which) do |e, row_index, col_index|
return row_index, col_index if yield e
end
end
nil
end
alias_method :find_index, :index
#
# Returns a section of the matrix. The parameters are either:
# * start_row, nrows, start_col, ncols; OR
# * row_range, col_range
#
# Matrix.diagonal(9, 5, -3).minor(0..1, 0..2)
# => 9 0 0
# 0 5 0
#
# Like Array#[], negative indices count backward from the end of the
# row or column (-1 is the last element). Returns nil if the starting
# row or column is greater than row_size or column_size respectively.
#
def minor(*param)
case param.size
when 2
row_range, col_range = param
from_row = row_range.first
from_row += row_size if from_row < 0
to_row = row_range.end
to_row += row_size if to_row < 0
to_row += 1 unless row_range.exclude_end?
size_row = to_row - from_row
from_col = col_range.first
from_col += column_size if from_col < 0
to_col = col_range.end
to_col += column_size if to_col < 0
to_col += 1 unless col_range.exclude_end?
size_col = to_col - from_col
when 4
from_row, size_row, from_col, size_col = param
return nil if size_row < 0 || size_col < 0
from_row += row_size if from_row < 0
from_col += column_size if from_col < 0
else
Matrix.Raise ArgumentError, param.inspect
end
return nil if from_row > row_size || from_col > column_size || from_row < 0 || from_col < 0
rows = @rows[from_row, size_row].collect{|row|
row[from_col, size_col]
}
new_matrix rows, [column_size - from_col, size_col].min
end
#--
# TESTING -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
#++
#
# Returns +true+ is this is a diagonal matrix.
# Raises an error if matrix is not square.
#
def diagonal?
Matrix.Raise ErrDimensionMismatch unless square?
each(:off_diagonal).all?(&:zero?)
end
#
# Returns +true+ if this is an empty matrix, i.e. if the number of rows
# or the number of columns is 0.
#
def empty?
column_size == 0 || row_size == 0
end
#
# Returns +true+ is this is an hermitian matrix.
# Raises an error if matrix is not square.
#
def hermitian?
Matrix.Raise ErrDimensionMismatch unless square?
each_with_index(:upper).all? do |e, row, col|
e == rows[col][row].conj
end
end
#
# Returns +true+ is this is a lower triangular matrix.
#
def lower_triangular?
each(:strict_upper).all?(&:zero?)
end
#
# Returns +true+ is this is a normal matrix.
# Raises an error if matrix is not square.
#
def normal?
Matrix.Raise ErrDimensionMismatch unless square?
rows.each_with_index do |row_i, i|
rows.each_with_index do |row_j, j|
s = 0
rows.each_with_index do |row_k, k|
s += row_i[k] * row_j[k].conj - row_k[i].conj * row_k[j]
end
return false unless s == 0
end
end
true
end
#
# Returns +true+ is this is an orthogonal matrix
# Raises an error if matrix is not square.
#
def orthogonal?
Matrix.Raise ErrDimensionMismatch unless square?
rows.each_with_index do |row, i|
column_size.times do |j|
s = 0
row_size.times do |k|
s += row[k] * rows[k][j]
end
return false unless s == (i == j ? 1 : 0)
end
end
true
end
#
# Returns +true+ is this is a permutation matrix
# Raises an error if matrix is not square.
#
def permutation?
Matrix.Raise ErrDimensionMismatch unless square?
cols = Array.new(column_size)
rows.each_with_index do |row, i|
found = false
row.each_with_index do |e, j|
if e == 1
return false if found || cols[j]
found = cols[j] = true
elsif e != 0
return false
end
end
return false unless found
end
true
end
#
# Returns +true+ if all entries of the matrix are real.
#
def real?
all?(&:real?)
end
#
# Returns +true+ if this is a regular (i.e. non-singular) matrix.
#
def regular?
not singular?
end
#
# Returns +true+ is this is a singular matrix.
#
def singular?
determinant == 0
end
#
# Returns +true+ is this is a square matrix.
#
def square?
column_size == row_size
end
#
# Returns +true+ is this is a symmetric matrix.
# Raises an error if matrix is not square.
#
def symmetric?
Matrix.Raise ErrDimensionMismatch unless square?
each_with_index(:strict_upper) do |e, row, col|
return false if e != rows[col][row]
end
true
end
#
# Returns +true+ is this is a unitary matrix
# Raises an error if matrix is not square.
#
def unitary?
Matrix.Raise ErrDimensionMismatch unless square?
rows.each_with_index do |row, i|
column_size.times do |j|
s = 0
row_size.times do |k|
s += row[k].conj * rows[k][j]
end
return false unless s == (i == j ? 1 : 0)
end
end
true
end
#
# Returns +true+ is this is an upper triangular matrix.
#
def upper_triangular?
each(:strict_lower).all?(&:zero?)
end
#
# Returns +true+ is this is a matrix with only zero elements
#
def zero?
all?(&:zero?)
end
#--
# OBJECT METHODS -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
#++
#
# Returns +true+ if and only if the two matrices contain equal elements.
#
def ==(other)
return false unless Matrix === other &&
column_size == other.column_size # necessary for empty matrices
rows == other.rows
end
def eql?(other)
return false unless Matrix === other &&
column_size == other.column_size # necessary for empty matrices
rows.eql? other.rows
end
#
# Returns a clone of the matrix, so that the contents of each do not reference
# identical objects.
# There should be no good reason to do this since Matrices are immutable.
#
def clone
new_matrix @rows.map(&:dup), column_size
end
#
# Returns a hash-code for the matrix.
#
def hash
@rows.hash
end
#--
# ARITHMETIC -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
#++
#
# Matrix multiplication.
# Matrix[[2,4], [6,8]] * Matrix.identity(2)
# => 2 4
# 6 8
#
def *(m) # m is matrix or vector or number
case(m)
when Numeric
rows = @rows.collect {|row|
row.collect {|e| e * m }
}
return new_matrix rows, column_size
when Vector
m = self.class.column_vector(m)
r = self * m
return r.column(0)
when Matrix
Matrix.Raise ErrDimensionMismatch if column_size != m.row_size
rows = Array.new(row_size) {|i|
Array.new(m.column_size) {|j|
(0 ... column_size).inject(0) do |vij, k|
vij + self[i, k] * m[k, j]
end
}
}
return new_matrix rows, m.column_size
else
return apply_through_coercion(m, __method__)
end
end
#
# Matrix addition.
# Matrix.scalar(2,5) + Matrix[[1,0], [-4,7]]
# => 6 0
# -4 12
#
def +(m)
case m
when Numeric
Matrix.Raise ErrOperationNotDefined, "+", self.class, m.class
when Vector
m = self.class.column_vector(m)
when Matrix
else
return apply_through_coercion(m, __method__)
end
Matrix.Raise ErrDimensionMismatch unless row_size == m.row_size and column_size == m.column_size
rows = Array.new(row_size) {|i|
Array.new(column_size) {|j|
self[i, j] + m[i, j]
}
}
new_matrix rows, column_size
end
#
# Matrix subtraction.
# Matrix[[1,5], [4,2]] - Matrix[[9,3], [-4,1]]
# => -8 2
# 8 1
#
def -(m)
case m
when Numeric
Matrix.Raise ErrOperationNotDefined, "-", self.class, m.class
when Vector
m = self.class.column_vector(m)
when Matrix
else
return apply_through_coercion(m, __method__)
end
Matrix.Raise ErrDimensionMismatch unless row_size == m.row_size and column_size == m.column_size
rows = Array.new(row_size) {|i|
Array.new(column_size) {|j|
self[i, j] - m[i, j]
}
}
new_matrix rows, column_size
end
#
# Matrix division (multiplication by the inverse).
# Matrix[[7,6], [3,9]] / Matrix[[2,9], [3,1]]
# => -7 1
# -3 -6
#
def /(other)
case other
when Numeric
rows = @rows.collect {|row|
row.collect {|e| e / other }
}
return new_matrix rows, column_size
when Matrix
return self * other.inverse
else
return apply_through_coercion(other, __method__)
end
end
#
# Returns the inverse of the matrix.
# Matrix[[-1, -1], [0, -1]].inverse
# => -1 1
# 0 -1
#
def inverse
Matrix.Raise ErrDimensionMismatch unless square?
self.class.I(row_size).send(:inverse_from, self)
end
alias inv inverse
def inverse_from(src) # :nodoc:
last = row_size - 1
a = src.to_a
0.upto(last) do |k|
i = k
akk = a[k][k].abs
(k+1).upto(last) do |j|
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]
0.upto(last) do |ii|
next if ii == k
q = a[ii][k].quo(akk)
a[ii][k] = 0
(k + 1).upto(last) do |j|
a[ii][j] -= a[k][j] * q
end
0.upto(last) do |j|
@rows[ii][j] -= @rows[k][j] * q
end
end
(k+1).upto(last) do |j|
a[k][j] = a[k][j].quo(akk)
end
0.upto(last) do |j|
@rows[k][j] = @rows[k][j].quo(akk)
end
end
self
end
private :inverse_from
#
# Matrix exponentiation.
# Equivalent to multiplying the matrix by itself N times.
# Non integer exponents will be handled by diagonalizing the matrix.
#
# Matrix[[7,6], [3,9]] ** 2
# => 67 96
# 48 99
#
def ** (other)
case other
when Integer
x = self
if other <= 0
x = self.inverse
return self.class.identity(self.column_size) if other == 0
other = -other
end
z = nil
loop do
z = z ? z * x : x if other[0] == 1
return z if (other >>= 1).zero?
x *= x
end
when Numeric
v, d, v_inv = eigensystem
v * self.class.diagonal(*d.each(:diagonal).map{|e| e ** other}) * v_inv
else
Matrix.Raise ErrOperationNotDefined, "**", self.class, other.class
end
end
#--
# MATRIX FUNCTIONS -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
#++
#
# Returns the determinant of the matrix.
#
# Beware that using Float values can yield erroneous results
# because of their lack of precision.
# Consider using exact types like Rational or BigDecimal instead.
#
# Matrix[[7,6], [3,9]].determinant
# => 45
#
def determinant
Matrix.Raise ErrDimensionMismatch unless square?
m = @rows
case row_size
# Up to 4x4, give result using Laplacian expansion by minors.
# This will typically be faster, as well as giving good results
# in case of Floats
when 0
+1
when 1
+ m[0][0]
when 2
+ m[0][0] * m[1][1] - m[0][1] * m[1][0]
when 3
m0, m1, m2 = m
+ m0[0] * m1[1] * m2[2] - m0[0] * m1[2] * m2[1] \
- m0[1] * m1[0] * m2[2] + m0[1] * m1[2] * m2[0] \
+ m0[2] * m1[0] * m2[1] - m0[2] * m1[1] * m2[0]
when 4
m0, m1, m2, m3 = m
+ m0[0] * m1[1] * m2[2] * m3[3] - m0[0] * m1[1] * m2[3] * m3[2] \
- m0[0] * m1[2] * m2[1] * m3[3] + m0[0] * m1[2] * m2[3] * m3[1] \
+ m0[0] * m1[3] * m2[1] * m3[2] - m0[0] * m1[3] * m2[2] * m3[1] \
- m0[1] * m1[0] * m2[2] * m3[3] + m0[1] * m1[0] * m2[3] * m3[2] \
+ m0[1] * m1[2] * m2[0] * m3[3] - m0[1] * m1[2] * m2[3] * m3[0] \
- m0[1] * m1[3] * m2[0] * m3[2] + m0[1] * m1[3] * m2[2] * m3[0] \
+ m0[2] * m1[0] * m2[1] * m3[3] - m0[2] * m1[0] * m2[3] * m3[1] \
- m0[2] * m1[1] * m2[0] * m3[3] + m0[2] * m1[1] * m2[3] * m3[0] \
+ m0[2] * m1[3] * m2[0] * m3[1] - m0[2] * m1[3] * m2[1] * m3[0] \
- m0[3] * m1[0] * m2[1] * m3[2] + m0[3] * m1[0] * m2[2] * m3[1] \
+ m0[3] * m1[1] * m2[0] * m3[2] - m0[3] * m1[1] * m2[2] * m3[0] \
- m0[3] * m1[2] * m2[0] * m3[1] + m0[3] * m1[2] * m2[1] * m3[0]
else
# For bigger matrices, use an efficient and general algorithm.
# Currently, we use the Gauss-Bareiss algorithm
determinant_bareiss
end
end
alias_method :det, :determinant
#
# Private. Use Matrix#determinant
#
# Returns the determinant of the matrix, using
# Bareiss' multistep integer-preserving gaussian elimination.
# It has the same computational cost order O(n^3) as standard Gaussian elimination.
# Intermediate results are fraction free and of lower complexity.
# A matrix of Integers will have thus intermediate results that are also Integers,
# with smaller bignums (if any), while a matrix of Float will usually have
# intermediate results with better precision.
#
def determinant_bareiss
size = row_size
last = size - 1
a = to_a
no_pivot = Proc.new{ return 0 }
sign = +1
pivot = 1
size.times do |k|
previous_pivot = pivot
if (pivot = a[k][k]) == 0
switch = (k+1 ... size).find(no_pivot) {|row|
a[row][k] != 0
}
a[switch], a[k] = a[k], a[switch]
pivot = a[k][k]
sign = -sign
end
(k+1).upto(last) do |i|
ai = a[i]
(k+1).upto(last) do |j|
ai[j] = (pivot * ai[j] - ai[k] * a[k][j]) / previous_pivot
end
end
end
sign * pivot
end
private :determinant_bareiss
#
# deprecated; use Matrix#determinant
#
def determinant_e
warn "#{caller(1)[0]}: warning: Matrix#determinant_e is deprecated; use #determinant"
rank
end
alias det_e determinant_e
#
# Returns the rank of the matrix.
# Beware that using Float values can yield erroneous results
# because of their lack of precision.
# Consider using exact types like Rational or BigDecimal instead.
#
# Matrix[[7,6], [3,9]].rank
# => 2
#
def rank
# We currently use Bareiss' multistep integer-preserving gaussian elimination
# (see comments on determinant)
a = to_a
last_column = column_size - 1
last_row = row_size - 1
pivot_row = 0
previous_pivot = 1
0.upto(last_column) do |k|
switch_row = (pivot_row .. last_row).find {|row|
a[row][k] != 0
}
if switch_row
a[switch_row], a[pivot_row] = a[pivot_row], a[switch_row] unless pivot_row == switch_row
pivot = a[pivot_row][k]
(pivot_row+1).upto(last_row) do |i|
ai = a[i]
(k+1).upto(last_column) do |j|
ai[j] = (pivot * ai[j] - ai[k] * a[pivot_row][j]) / previous_pivot
end
end
pivot_row += 1
previous_pivot = pivot
end
end
pivot_row
end
#
# deprecated; use Matrix#rank
#
def rank_e
warn "#{caller(1)[0]}: warning: Matrix#rank_e is deprecated; use #rank"
rank
end
# Returns a matrix with entries rounded to the given precision
# (see Float#round)
#
def round(ndigits=0)
map{|e| e.round(ndigits)}
end
#
# Returns the trace (sum of diagonal elements) of the matrix.
# Matrix[[7,6], [3,9]].trace
# => 16
#
def trace
Matrix.Raise ErrDimensionMismatch unless square?
(0...column_size).inject(0) do |tr, i|
tr + @rows[i][i]
end
end
alias tr trace
#
# Returns the transpose of the matrix.
# Matrix[[1,2], [3,4], [5,6]]
# => 1 2
# 3 4
# 5 6
# Matrix[[1,2], [3,4], [5,6]].transpose
# => 1 3 5
# 2 4 6
#
def transpose
return self.class.empty(column_size, 0) if row_size.zero?
new_matrix @rows.transpose, row_size
end
alias t transpose
#--
# DECOMPOSITIONS -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
#++
#
# Returns the Eigensystem of the matrix; see +EigenvalueDecomposition+.
# m = Matrix[[1, 2], [3, 4]]
# v, d, v_inv = m.eigensystem
# d.diagonal? # => true
# v.inv == v_inv # => true
# (v * d * v_inv).round(5) == m # => true
#
def eigensystem
EigenvalueDecomposition.new(self)
end
alias eigen eigensystem
#
# Returns the LUP decomposition of the matrix; see +LUPDecomposition+.
# a = Matrix[[1, 2], [3, 4]]
# l, u, p = a.lup
# l.lower_triangular? # => true
# u.upper_triangular? # => true
# p.permutation? # => true
# l * u == a * p # => true
# a.lup.solve([2, 5]) # => Vector[(1/1), (1/2)]
#
def lup
LUPDecomposition.new(self)
end
alias lup_decomposition lup
#--
# COMPLEX ARITHMETIC -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
#++
#
# Returns the conjugate of the matrix.
# Matrix[[Complex(1,2), Complex(0,1), 0], [1, 2, 3]]
# => 1+2i i 0
# 1 2 3
# Matrix[[Complex(1,2), Complex(0,1), 0], [1, 2, 3]].conjugate
# => 1-2i -i 0
# 1 2 3
#
def conjugate
collect(&:conjugate)
end
alias conj conjugate
#
# Returns the imaginary part of the matrix.
# Matrix[[Complex(1,2), Complex(0,1), 0], [1, 2, 3]]
# => 1+2i i 0
# 1 2 3
# Matrix[[Complex(1,2), Complex(0,1), 0], [1, 2, 3]].imaginary
# => 2i i 0
# 0 0 0
#
def imaginary
collect(&:imaginary)
end
alias imag imaginary
#
# Returns the real part of the matrix.
# Matrix[[Complex(1,2), Complex(0,1), 0], [1, 2, 3]]
# => 1+2i i 0
# 1 2 3
# Matrix[[Complex(1,2), Complex(0,1), 0], [1, 2, 3]].real
# => 1 0 0
# 1 2 3
#
def real
collect(&:real)
end
#
# Returns an array containing matrices corresponding to the real and imaginary
# parts of the matrix
#
# m.rect == [m.real, m.imag] # ==> true for all matrices m
#
def rect
[real, imag]
end
alias rectangular rect
#--
# CONVERTING -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
#++
#
# 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.
#
def coerce(other)
case other
when Numeric
return Scalar.new(other), self
else
raise TypeError, "#{self.class} can't be coerced into #{other.class}"
end
end
#
# Returns an array of the row vectors of the matrix. See Vector.
#
def row_vectors
Array.new(row_size) {|i|
row(i)
}
end
#
# Returns an array of the column vectors of the matrix. See Vector.
#
def column_vectors
Array.new(column_size) {|i|
column(i)
}
end
#
# Returns an array of arrays that describe the rows of the matrix.
#
def to_a
@rows.collect(&:dup)
end
def elements_to_f
warn "#{caller(1)[0]}: warning: Matrix#elements_to_f is deprecated, use map(&:to_f)"
map(&:to_f)
end
def elements_to_i
warn "#{caller(1)[0]}: warning: Matrix#elements_to_i is deprecated, use map(&:to_i)"
map(&:to_i)
end
def elements_to_r
warn "#{caller(1)[0]}: warning: Matrix#elements_to_r is deprecated, use map(&:to_r)"
map(&:to_r)
end
#--
# PRINTING -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
#++
#
# Overrides Object#to_s
#
def to_s
if empty?
"#{self.class}.empty(#{row_size}, #{column_size})"
else
"#{self.class}[" + @rows.collect{|row|
"[" + row.collect{|e| e.to_s}.join(", ") + "]"
}.join(", ")+"]"
end
end
#
# Overrides Object#inspect
#
def inspect
if empty?
"#{self.class}.empty(#{row_size}, #{column_size})"
else
"#{self.class}#{@rows.inspect}"
end
end
# Private helper modules
module ConversionHelper # :nodoc:
#
# Converts the obj to an Array. If copy is set to true
# a copy of obj will be made if necessary.
#
def convert_to_array(obj, copy = false) # :nodoc:
case obj
when Array
copy ? obj.dup : obj
when Vector
obj.to_a
else
begin
converted = obj.to_ary
rescue Exception => e
raise TypeError, "can't convert #{obj.class} into an Array (#{e.message})"
end
raise TypeError, "#{obj.class}#to_ary should return an Array" unless converted.is_a? Array
converted
end
end
private :convert_to_array
end
extend ConversionHelper
module CoercionHelper # :nodoc:
#
# Applies the operator +oper+ with argument +obj+
# through coercion of +obj+
#
def apply_through_coercion(obj, oper)
coercion = obj.coerce(self)
raise TypeError unless coercion.is_a?(Array) && coercion.length == 2
coercion[0].public_send(oper, coercion[1])
rescue
raise TypeError, "#{obj.inspect} can't be coerced into #{self.class}"
end
private :apply_through_coercion
#
# Helper method to coerce a value into a specific class.
# Raises a TypeError if the coercion fails or the returned value
# is not of the right class.
# (from Rubinius)
#
def self.coerce_to(obj, cls, meth) # :nodoc:
return obj if obj.kind_of?(cls)
begin
ret = obj.__send__(meth)
rescue Exception => e
raise TypeError, "Coercion error: #{obj.inspect}.#{meth} => #{cls} failed:\n" \
"(#{e.message})"
end
raise TypeError, "Coercion error: obj.#{meth} did NOT return a #{cls} (was #{ret.class})" unless ret.kind_of? cls
ret
end
def self.coerce_to_int(obj)
coerce_to(obj, Integer, :to_int)
end
end
include CoercionHelper
# Private CLASS
class Scalar < Numeric # :nodoc:
include ExceptionForMatrix
include CoercionHelper
def initialize(value)
@value = value
end
# ARITHMETIC
def +(other)
case other
when Numeric
Scalar.new(@value + other)
when Vector, Matrix
Scalar.Raise ErrOperationNotDefined, "+", @value.class, other.class
else
apply_through_coercion(other, __method__)
end
end
def -(other)
case other
when Numeric
Scalar.new(@value - other)
when Vector, Matrix
Scalar.Raise ErrOperationNotDefined, "-", @value.class, other.class
else
apply_through_coercion(other, __method__)
end
end
def *(other)
case other
when Numeric
Scalar.new(@value * other)
when Vector, Matrix
other.collect{|e| @value * e}
else
apply_through_coercion(other, __method__)
end
end
def / (other)
case other
when Numeric
Scalar.new(@value / other)
when Vector
Scalar.Raise ErrOperationNotDefined, "/", @value.class, other.class
when Matrix
self * other.inverse
else
apply_through_coercion(other, __method__)
end
end
def ** (other)
case other
when Numeric
Scalar.new(@value ** other)
when Vector
Scalar.Raise ErrOperationNotDefined, "**", @value.class, other.class
when Matrix
#other.powered_by(self)
Scalar.Raise ErrOperationNotImplemented, "**", @value.class, other.class
else
apply_through_coercion(other, __method__)
end
end
end
end
#
# 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)
#
# To access elements:
# * #[](i)
#
# To enumerate the elements:
# * #each2(v)
# * #collect2(v)
#
# Vector arithmetic:
# * #*(x) "is matrix or number"
# * #+(v)
# * #-(v)
#
# Vector functions:
# * #inner_product(v)
# * #collect
# * #magnitude
# * #map
# * #map2(v)
# * #norm
# * #normalize
# * #r
# * #size
#
# Conversion to other data types:
# * #covector
# * #to_a
# * #coerce(other)
#
# String representations:
# * #to_s
# * #inspect
#
class Vector
include ExceptionForMatrix
include Enumerable
include Matrix::CoercionHelper
extend Matrix::ConversionHelper
#INSTANCE CREATION
private_class_method :new
attr_reader :elements
protected :elements
#
# Creates a Vector from a list of elements.
# Vector[7, 4, ...]
#
def Vector.[](*array)
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.
#
def Vector.elements(array, copy = true)
new convert_to_array(array, copy)
end
#
# Vector.new is private; use Vector[] or Vector.elements to create.
#
def initialize(array)
# No checking is done at this point.
@elements = array
end
# ACCESSING
#
# Returns element number +i+ (starting at zero) of the vector.
#
def [](i)
@elements[i]
end
alias element []
alias component []
def []=(i, v)
@elements[i]= v
end
alias set_element []=
alias set_component []=
private :[]=, :set_element, :set_component
#
# Returns the number of elements in the vector.
#
def size
@elements.size
end
#--
# ENUMERATIONS -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
#++
#
# Iterate over the elements of this vector
#
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.
#
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
#
# Collects (as in Enumerable#collect) over the elements of this vector and +v+
# in conjunction.
#
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
#--
# COMPARING -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
#++
#
# Returns +true+ iff the two vectors have the same elements in the same order.
#
def ==(other)
return false unless Vector === other
@elements == other.elements
end
def eql?(other)
return false unless Vector === other
@elements.eql? other.elements
end
#
# Return a copy of the vector.
#
def clone
self.class.elements(@elements)
end
#
# Return a hash-code for the vector.
#
def hash
@elements.hash
end
#--
# ARITHMETIC -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
#++
#
# Multiplies the vector by +x+, where +x+ is a number or another vector.
#
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.
#
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.
#
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 division.
#
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 FUNCTIONS -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
#++
#
# Returns the inner product of this vector with the other.
# Vector[4,7].inner_product Vector[10,1] => 47
#
def inner_product(v)
Vector.Raise ErrDimensionMismatch if size != v.size
p = 0
each2(v) {|v1, v2|
p += v1 * v2
}
p
end
#
# Like Array#collect.
#
def collect(&block) # :yield: e
return to_enum(:collect) unless block_given?
els = @elements.collect(&block)
self.class.elements(els, false)
end
alias map collect
#
# Returns the modulus (Pythagorean distance) of the vector.
# Vector[5,8,2].r => 9.643650761
#
def magnitude
Math.sqrt(@elements.inject(0) {|v, e| v + e.abs2})
end
alias r magnitude
alias norm magnitude
#
# Like Vector#collect2, but returns a Vector instead of an Array.
#
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
class ZeroVectorError < StandardError
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
#
def normalize
n = magnitude
raise ZeroVectorError, "Zero vectors can not be normalized" if n == 0
self / n
end
#--
# CONVERTING
#++
#
# Creates a single-row matrix from this vector.
#
def covector
Matrix.row_vector(self)
end
#
# Returns the elements of the vector in an array.
#
def to_a
@elements.dup
end
def elements_to_f
warn "#{caller(1)[0]}: warning: Vector#elements_to_f is deprecated"
map(&:to_f)
end
def elements_to_i
warn "#{caller(1)[0]}: warning: Vector#elements_to_i is deprecated"
map(&:to_i)
end
def elements_to_r
warn "#{caller(1)[0]}: warning: Vector#elements_to_r is deprecated"
map(&:to_r)
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.
#
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
#--
# PRINTING -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
#++
#
# Overrides Object#to_s
#
def to_s
"Vector[" + @elements.join(", ") + "]"
end
#
# Overrides Object#inspect
#
def inspect
"Vector" + @elements.inspect
end
end
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