Class: Matrix
- Inherits:
-
Object
- Object
- Matrix
- Extended by:
- ConversionHelper
- Includes:
- Enumerable, ExceptionForMatrix, CoercionHelper
- Defined in:
- lib/matrix.rb
Overview
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
Defined Under Namespace
Modules: CoercionHelper, ConversionHelper Classes: Scalar
Constant Summary collapse
- SELECTORS =
{all: true, diagonal: true, off_diagonal: true, lower: true, strict_lower: true, strict_upper: true, upper: true}.freeze
Instance Attribute Summary collapse
-
#column_size ⇒ Object
readonly
Returns the number of columns.
Class Method Summary collapse
-
.[](*rows) ⇒ Object
Creates a matrix where each argument is a row.
-
.build(row_size, column_size = row_size) ⇒ Object
Creates a matrix of size
row_size
xcolumn_size
. -
.column_vector(column) ⇒ Object
Creates a single-column matrix where the values of that column are as given in
column
. -
.columns(columns) ⇒ Object
Creates a matrix using
columns
as an array of column vectors. -
.diagonal(*values) ⇒ Object
Creates a matrix where the diagonal elements are composed of
values
. -
.empty(row_size = 0, column_size = 0) ⇒ Object
Creates a empty matrix of
row_size
xcolumn_size
. -
.identity(n) ⇒ Object
(also: unit, I)
Creates an
n
byn
identity matrix. -
.row_vector(row) ⇒ Object
Creates a single-row matrix where the values of that row are as given in
row
. -
.rows(rows, copy = true) ⇒ Object
Creates a matrix where
rows
is an array of arrays, each of which is a row of the matrix. -
.scalar(n, value) ⇒ Object
Creates an
n
byn
diagonal matrix where each diagonal element isvalue
. -
.zero(row_size, column_size = row_size) ⇒ Object
Creates a zero matrix.
Instance Method Summary collapse
-
#*(m) ⇒ Object
Matrix multiplication.
-
#**(other) ⇒ Object
Matrix exponentiation.
-
#+(m) ⇒ Object
Matrix addition.
-
#-(m) ⇒ Object
Matrix subtraction.
-
#/(other) ⇒ Object
Matrix division (multiplication by the inverse).
-
#==(other) ⇒ Object
Returns
true
if and only if the two matrices contain equal elements. -
#[](i, j) ⇒ Object
(also: #element, #component)
Returns element (
i
,j
) of the matrix. -
#clone ⇒ Object
Returns a clone of the matrix, so that the contents of each do not reference identical objects.
-
#coerce(other) ⇒ Object
The coerce method provides support for Ruby type coercion.
-
#collect(&block) ⇒ Object
(also: #map)
Returns a matrix that is the result of iteration of the given block over all elements of the matrix.
-
#column(j) ⇒ Object
Returns column vector number
j
of the matrix as a Vector (starting at 0 like an array). -
#column_vectors ⇒ Object
Returns an array of the column vectors of the matrix.
-
#conjugate ⇒ Object
(also: #conj)
Returns the conjugate of the matrix.
-
#determinant ⇒ Object
(also: #det)
Returns the determinant of the matrix.
-
#determinant_e ⇒ Object
(also: #det_e)
deprecated; use Matrix#determinant.
-
#diagonal? ⇒ Boolean
Returns
true
is this is a diagonal matrix. -
#each(which = :all) ⇒ Object
Yields all elements of the matrix, starting with those of the first row, or returns an Enumerator is no block given.
-
#each_with_index(which = :all) ⇒ Object
Same as #each, but the row index and column index in addition to the element.
-
#eigensystem ⇒ Object
(also: #eigen)
Returns the Eigensystem of the matrix; see
EigenvalueDecomposition
. - #elements_to_f ⇒ Object
- #elements_to_i ⇒ Object
- #elements_to_r ⇒ Object
-
#empty? ⇒ Boolean
Returns
true
if this is an empty matrix, i.e. - #eql?(other) ⇒ Boolean
-
#hash ⇒ Object
Returns a hash-code for the matrix.
-
#hermitian? ⇒ Boolean
Returns
true
is this is an hermitian matrix. -
#imaginary ⇒ Object
(also: #imag)
Returns the imaginary part of the matrix.
-
#index(*args) ⇒ Object
(also: #find_index)
:call-seq: index(value, selector = :all) -> [row, column] index(selector = :all){ block } -> [row, column] index(selector = :all) -> an_enumerator.
-
#initialize(rows, column_size = rows[0].size) ⇒ Matrix
constructor
Matrix.new is private; use Matrix.rows, columns, [], etc...
-
#inspect ⇒ Object
Overrides Object#inspect.
-
#inverse ⇒ Object
(also: #inv)
Returns the inverse of the matrix.
-
#lower_triangular? ⇒ Boolean
Returns
true
is this is a lower triangular matrix. -
#lup ⇒ Object
(also: #lup_decomposition)
Returns the LUP decomposition of the matrix; see
LUPDecomposition
. -
#minor(*param) ⇒ Object
Returns a section of the matrix.
-
#normal? ⇒ Boolean
Returns
true
is this is a normal matrix. -
#orthogonal? ⇒ Boolean
Returns
true
is this is an orthogonal matrix Raises an error if matrix is not square. -
#permutation? ⇒ Boolean
Returns
true
is this is a permutation matrix Raises an error if matrix is not square. -
#rank ⇒ Object
Returns the rank of the matrix.
-
#rank_e ⇒ Object
deprecated; use Matrix#rank.
-
#real ⇒ Object
Returns the real part of the matrix.
-
#real? ⇒ Boolean
Returns
true
if all entries of the matrix are real. -
#rect ⇒ Object
(also: #rectangular)
Returns an array containing matrices corresponding to the real and imaginary parts of the matrix.
-
#regular? ⇒ Boolean
Returns
true
if this is a regular (i.e. non-singular) matrix. -
#round(ndigits = 0) ⇒ Object
Returns a matrix with entries rounded to the given precision (see Float#round).
-
#row(i, &block) ⇒ Object
Returns row vector number
i
of the matrix as a Vector (starting at 0 like an array). -
#row_size ⇒ Object
Returns the number of rows.
-
#row_vectors ⇒ Object
Returns an array of the row vectors of the matrix.
-
#singular? ⇒ Boolean
Returns
true
is this is a singular matrix. -
#square? ⇒ Boolean
Returns
true
is this is a square matrix. -
#symmetric? ⇒ Boolean
Returns
true
is this is a symmetric matrix. -
#to_a ⇒ Object
Returns an array of arrays that describe the rows of the matrix.
-
#to_s ⇒ Object
Overrides Object#to_s.
-
#trace ⇒ Object
(also: #tr)
Returns the trace (sum of diagonal elements) of the matrix.
-
#transpose ⇒ Object
(also: #t)
Returns the transpose of the matrix.
-
#unitary? ⇒ Boolean
Returns
true
is this is a unitary matrix Raises an error if matrix is not square. -
#upper_triangular? ⇒ Boolean
Returns
true
is this is an upper triangular matrix. -
#zero? ⇒ Boolean
Returns
true
is this is a matrix with only zero elements.
Methods included from CoercionHelper
Constructor Details
#initialize(rows, column_size = rows[0].size) ⇒ Matrix
Matrix.new is private; use Matrix.rows, columns, [], etc... to create.
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# File 'lib/matrix.rb', line 298 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 |
Instance Attribute Details
#column_size ⇒ Object (readonly)
Returns the number of columns.
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# File 'lib/matrix.rb', line 337 def column_size @column_size end |
Class Method Details
.[](*rows) ⇒ Object
Creates a matrix where each argument is a row.
Matrix[ [25, 93], [-1, 66] ]
=> 25 93
-1 66
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# File 'lib/matrix.rb', line 140 def Matrix.[](*rows) Matrix.rows(rows, false) end |
.build(row_size, column_size = row_size) ⇒ Object
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
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# File 'lib/matrix.rb', line 185 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 |
.column_vector(column) ⇒ Object
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
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# File 'lib/matrix.rb', line 270 def Matrix.column_vector(column) column = convert_to_array(column) new [column].transpose, 1 end |
.columns(columns) ⇒ Object
Creates a matrix using columns
as an array of column vectors.
Matrix.columns([[25, 93], [-1, 66]])
=> 25 -1
93 66
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# File 'lib/matrix.rb', line 170 def Matrix.columns(columns) Matrix.rows(columns, false).transpose end |
.diagonal(*values) ⇒ Object
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
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# File 'lib/matrix.rb', line 205 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 |
.empty(row_size = 0, column_size = 0) ⇒ Object
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]]
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# File 'lib/matrix.rb', line 288 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 |
.identity(n) ⇒ Object Also known as: unit, I
Creates an n
by n
identity matrix.
Matrix.identity(2)
=> 1 0
0 1
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# File 'lib/matrix.rb', line 232 def Matrix.identity(n) Matrix.scalar(n, 1) end |
.row_vector(row) ⇒ Object
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
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# File 'lib/matrix.rb', line 257 def Matrix.row_vector(row) row = convert_to_array(row) new [row] end |
.rows(rows, copy = true) ⇒ Object
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
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# File 'lib/matrix.rb', line 152 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 |
.scalar(n, value) ⇒ Object
Creates an n
by n
diagonal matrix where each diagonal element is value
.
Matrix.scalar(2, 5)
=> 5 0
0 5
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# File 'lib/matrix.rb', line 222 def Matrix.scalar(n, value) Matrix.diagonal(*Array.new(n, value)) end |
.zero(row_size, column_size = row_size) ⇒ Object
Creates a zero matrix.
Matrix.zero(2)
=> 0 0
0 0
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# File 'lib/matrix.rb', line 246 def Matrix.zero(row_size, column_size = row_size) rows = Array.new(row_size){Array.new(column_size, 0)} new rows, column_size end |
Instance Method Details
#*(m) ⇒ Object
Matrix multiplication.
Matrix[[2,4], [6,8]] * Matrix.identity(2)
=> 2 4
6 8
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# File 'lib/matrix.rb', line 803 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 = Matrix.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 |
#**(other) ⇒ Object
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
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# File 'lib/matrix.rb', line 970 def ** (other) case other when Integer x = self if other <= 0 x = self.inverse return Matrix.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 * Matrix.diagonal(*d.each(:diagonal).map{|e| e ** other}) * v_inv else Matrix.Raise ErrOperationNotDefined, "**", self.class, other.class end end |
#+(m) ⇒ Object
Matrix addition.
Matrix.scalar(2,5) + Matrix[[1,0], [-4,7]]
=> 6 0
-4 12
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# File 'lib/matrix.rb', line 836 def +(m) case m when Numeric Matrix.Raise ErrOperationNotDefined, "+", self.class, m.class when Vector m = Matrix.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 |
#-(m) ⇒ Object
Matrix subtraction.
Matrix[[1,5], [4,2]] - Matrix[[9,3], [-4,1]]
=> -8 2
8 1
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# File 'lib/matrix.rb', line 863 def -(m) case m when Numeric Matrix.Raise ErrOperationNotDefined, "-", self.class, m.class when Vector m = Matrix.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 |
#/(other) ⇒ Object
Matrix division (multiplication by the inverse).
Matrix[[7,6], [3,9]] / Matrix[[2,9], [3,1]]
=> -7 1
-3 -6
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# File 'lib/matrix.rb', line 890 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 |
#==(other) ⇒ Object
Returns true
if and only if the two matrices contain equal elements.
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# File 'lib/matrix.rb', line 765 def ==(other) return false unless Matrix === other && column_size == other.column_size # necessary for empty matrices rows == other.rows end |
#[](i, j) ⇒ Object Also known as: element, component
Returns element (i
,j
) of the matrix. That is: row i
, column j
.
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# File 'lib/matrix.rb', line 314 def [](i, j) @rows.fetch(i){return nil}[j] end |
#clone ⇒ Object
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.
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# File 'lib/matrix.rb', line 782 def clone new_matrix @rows.map(&:dup), column_size end |
#coerce(other) ⇒ Object
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.
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# File 'lib/matrix.rb', line 1275 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 |
#collect(&block) ⇒ Object Also known as: map
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
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# File 'lib/matrix.rb', line 380 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 |
#column(j) ⇒ Object
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.
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# File 'lib/matrix.rb', line 357 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 |
#column_vectors ⇒ Object
Returns an array of the column vectors of the matrix. See Vector.
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# File 'lib/matrix.rb', line 1296 def column_vectors Array.new(column_size) {|i| column(i) } end |
#conjugate ⇒ Object Also known as: conj
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
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# File 'lib/matrix.rb', line 1221 def conjugate collect(&:conjugate) end |
#determinant ⇒ Object Also known as: det
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
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# File 'lib/matrix.rb', line 1007 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 end end |
#determinant_e ⇒ Object Also known as: det_e
deprecated; use Matrix#determinant
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# File 'lib/matrix.rb', line 1089 def determinant_e warn "#{caller(1)[0]}: warning: Matrix#determinant_e is deprecated; use #determinant" rank end |
#diagonal? ⇒ Boolean
Returns true
is this is a diagonal matrix. Raises an error if matrix is not square.
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# File 'lib/matrix.rb', line 598 def diagonal? Matrix.Raise ErrDimensionMismatch unless square? each(:off_diagonal).all?(&:zero?) end |
#each(which = :all) ⇒ Object
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]
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# File 'lib/matrix.rb', line 403 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 |
#each_with_index(which = :all) ⇒ Object
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
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# File 'lib/matrix.rb', line 464 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 |
#eigensystem ⇒ Object Also known as: eigen
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
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# File 'lib/matrix.rb', line 1188 def eigensystem EigenvalueDecomposition.new(self) end |
#elements_to_f ⇒ Object
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# File 'lib/matrix.rb', line 1309 def elements_to_f warn "#{caller(1)[0]}: warning: Matrix#elements_to_f is deprecated, use map(&:to_f)" map(&:to_f) end |
#elements_to_i ⇒ Object
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# File 'lib/matrix.rb', line 1314 def elements_to_i warn "#{caller(1)[0]}: warning: Matrix#elements_to_i is deprecated, use map(&:to_i)" map(&:to_i) end |
#elements_to_r ⇒ Object
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# File 'lib/matrix.rb', line 1319 def elements_to_r warn "#{caller(1)[0]}: warning: Matrix#elements_to_r is deprecated, use map(&:to_r)" map(&:to_r) end |
#empty? ⇒ Boolean
Returns true
if this is an empty matrix, i.e. if the number of rows or the number of columns is 0.
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# File 'lib/matrix.rb', line 607 def empty? column_size == 0 || row_size == 0 end |
#eql?(other) ⇒ Boolean
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# File 'lib/matrix.rb', line 771 def eql?(other) return false unless Matrix === other && column_size == other.column_size # necessary for empty matrices rows.eql? other.rows end |
#hash ⇒ Object
Returns a hash-code for the matrix.
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# File 'lib/matrix.rb', line 789 def hash @rows.hash end |
#hermitian? ⇒ Boolean
Returns true
is this is an hermitian matrix. Raises an error if matrix is not square.
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# File 'lib/matrix.rb', line 615 def hermitian? Matrix.Raise ErrDimensionMismatch unless square? each_with_index(:strict_upper).all? do |e, row, col| e == rows[col][row].conj end end |
#imaginary ⇒ Object Also known as: imag
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
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# File 'lib/matrix.rb', line 1235 def imaginary collect(&:imaginary) end |
#index(*args) ⇒ Object Also known as: find_index
: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]
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# File 'lib/matrix.rb', line 527 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 |
#inspect ⇒ Object
Overrides Object#inspect
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# File 'lib/matrix.rb', line 1344 def inspect if empty? "Matrix.empty(#{row_size}, #{column_size})" else "Matrix#{@rows.inspect}" end end |
#inverse ⇒ Object Also known as: inv
Returns the inverse of the matrix.
Matrix[[-1, -1], [0, -1]].inverse
=> -1 1
0 -1
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# File 'lib/matrix.rb', line 910 def inverse Matrix.Raise ErrDimensionMismatch unless square? Matrix.I(row_size).send(:inverse_from, self) end |
#lower_triangular? ⇒ Boolean
Returns true
is this is a lower triangular matrix.
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# File 'lib/matrix.rb', line 625 def lower_triangular? each(:strict_upper).all?(&:zero?) end |
#lup ⇒ Object Also known as: lup_decomposition
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)]
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# File 'lib/matrix.rb', line 1203 def lup LUPDecomposition.new(self) end |
#minor(*param) ⇒ Object
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.
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# File 'lib/matrix.rb', line 557 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 |
#normal? ⇒ Boolean
Returns true
is this is a normal matrix. Raises an error if matrix is not square.
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# File 'lib/matrix.rb', line 633 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 |
#orthogonal? ⇒ Boolean
Returns true
is this is an orthogonal matrix Raises an error if matrix is not square.
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# File 'lib/matrix.rb', line 651 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 |
#permutation? ⇒ Boolean
Returns true
is this is a permutation matrix Raises an error if matrix is not square.
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# File 'lib/matrix.rb', line 669 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 |
#rank ⇒ Object
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
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# File 'lib/matrix.rb', line 1104 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 |
#rank_e ⇒ Object
deprecated; use Matrix#rank
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# File 'lib/matrix.rb', line 1135 def rank_e warn "#{caller(1)[0]}: warning: Matrix#rank_e is deprecated; use #rank" rank end |
#real ⇒ Object
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
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# File 'lib/matrix.rb', line 1249 def real collect(&:real) end |
#real? ⇒ Boolean
Returns true
if all entries of the matrix are real.
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# File 'lib/matrix.rb', line 690 def real? all?(&:real?) end |
#rect ⇒ Object Also known as: rectangular
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
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# File 'lib/matrix.rb', line 1259 def rect [real, imag] end |
#regular? ⇒ Boolean
Returns true
if this is a regular (i.e. non-singular) matrix.
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# File 'lib/matrix.rb', line 697 def regular? not singular? end |
#round(ndigits = 0) ⇒ Object
Returns a matrix with entries rounded to the given precision (see Float#round)
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# File 'lib/matrix.rb', line 1143 def round(ndigits=0) map{|e| e.round(ndigits)} end |
#row(i, &block) ⇒ Object
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.
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# File 'lib/matrix.rb', line 343 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 |
#row_size ⇒ Object
Returns the number of rows.
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# File 'lib/matrix.rb', line 330 def row_size @rows.size end |
#row_vectors ⇒ Object
Returns an array of the row vectors of the matrix. See Vector.
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# File 'lib/matrix.rb', line 1287 def row_vectors Array.new(row_size) {|i| row(i) } end |
#singular? ⇒ Boolean
Returns true
is this is a singular matrix.
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# File 'lib/matrix.rb', line 704 def singular? determinant == 0 end |
#square? ⇒ Boolean
Returns true
is this is a square matrix.
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# File 'lib/matrix.rb', line 711 def square? column_size == row_size end |
#symmetric? ⇒ Boolean
Returns true
is this is a symmetric matrix. Raises an error if matrix is not square.
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# File 'lib/matrix.rb', line 719 def symmetric? Matrix.Raise ErrDimensionMismatch unless square? each_with_index(:strict_upper).all? do |e, row, col| e == rows[col][row] end end |
#to_a ⇒ Object
Returns an array of arrays that describe the rows of the matrix.
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# File 'lib/matrix.rb', line 1305 def to_a @rows.collect(&:dup) end |
#to_s ⇒ Object
Overrides Object#to_s
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# File 'lib/matrix.rb', line 1331 def to_s if empty? "Matrix.empty(#{row_size}, #{column_size})" else "Matrix[" + @rows.collect{|row| "[" + row.collect{|e| e.to_s}.join(", ") + "]" }.join(", ")+"]" end end |
#trace ⇒ Object Also known as: tr
Returns the trace (sum of diagonal elements) of the matrix.
Matrix[[7,6], [3,9]].trace
=> 16
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# File 'lib/matrix.rb', line 1152 def trace Matrix.Raise ErrDimensionMismatch unless square? (0...column_size).inject(0) do |tr, i| tr + @rows[i][i] end end |
#transpose ⇒ Object Also known as: t
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
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# File 'lib/matrix.rb', line 1170 def transpose return Matrix.empty(column_size, 0) if row_size.zero? new_matrix @rows.transpose, row_size end |
#unitary? ⇒ Boolean
Returns true
is this is a unitary matrix Raises an error if matrix is not square.
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# File 'lib/matrix.rb', line 730 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 |
#upper_triangular? ⇒ Boolean
Returns true
is this is an upper triangular matrix.
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# File 'lib/matrix.rb', line 747 def upper_triangular? each(:strict_lower).all?(&:zero?) end |
#zero? ⇒ Boolean
Returns true
is this is a matrix with only zero elements
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# File 'lib/matrix.rb', line 754 def zero? all?(&:zero?) end |