la.maths

Class Matrix

java.lang.Object

la.la.Value

la.la.Value.Structured

la.maths.Vector

la.maths.Matrix

All Implemented Interfaces:

java.lang.Comparable<Value>, Value.Scannable

Direct Known Subclasses:

Matrix.Doubles, Matrix.GP2, Matrix.Ints

public abstract class Matrix
extends Vector

(Abstract) Matrix, for two-dimensional, rectangular, Vectors of Vectors; the new, principal operations are • nCols(), and • elt(r,c), plus as for all Vectors. Also see Matrix.GP2. And note that a non-rectangular, "jagged" Vector of Vectors can be created as, say, a Vector.GP of GPs.

Nested Class Summary

Nested Classes

Modifier and Type	Class and Description
static class	Matrix.Doubles Matrices of Reals (Cts, doubles).
static class	Matrix.GP2 A simple, general-purpose implementation of a Matrix.
static class	Matrix.Ints Matrices of Ints (ints).
class	Matrix.Jacobi The Jacobi algorithm finds the Eigen-Values and Eigen-Vectors of 'this' square, symmetric Matrix of Cts; it is usually invoked via eigen(), eigenValues(), or eigenVectors().

Nested classes/interfaces inherited from class la.maths.Vector

Vector.Derived, Vector.GP, Vector.Slice, Vector.Strings, Vector.Weighted

Nested classes/interfaces inherited from class la.la.Value

Value.Atomic, Value.Bool, Value.Char, Value.Chars, Value.Cts, Value.Defer, Value.Discrete, Value.Enum, Value.Inc_Or, Value.Int, Value.Lambda, Value.List, Value.Maybe, Value.Option, Value.Real, Value.Scannable, Value.Structured, Value.Triv, Value.Tuple

Field Summary

Fields inherited from class la.maths.Vector

empty

Fields inherited from class la.la.Value

C, comparator, CR, E, eight, ffalse, five, fiveR, four, fourR, half, negCR, negOne, negOneR, negTenR, nilVal, nine, None, one, oneR, PI, PIby2, point01, point1, point5, point9, quarter, seven, six, ten, tenR, three, threeR, triv, ttrue, two, twoR, zero, zeroR

Constructor Summary

Constructors

Constructor and Description
Matrix() The trivial constructor for use when the number of columns is not known before calling a Matrix constructor.
Matrix(int nC) Constructor for use when the number of columns, nC, is known before calling the constructor, and the Matrix is known to be proper (rectangular).

Method Summary

Modifier and Type	Method and Description
Matrix	asMatrix() Return 'this'; also see Vector.asMatrix().
static Matrix.Doubles	doubles(double[][] xs) Convenience function : double[][] → Matrix, where each elt(r,c) is an exact Real.
static Matrix.Doubles	doubles(double[][] xs, double AoM) Convenience function, doubles : double[][] × AoM → Matrix, where every element of the Matrix has the same AoM().
Value.Tuple	eigen() Return the Eigen-Values and Eigen-Vectors, (eVals, eVecs), of 'this' square, symmetric Matrix using the Matrix.Jacobi algorithm.
Vector	eigenValues() Return the Eigen-Values of 'this' square, symmetric Matrix.
Matrix	eigenVectors() Return the Eigen-Vectors of 'this' square, symmetric Matrix.
Vector	elt(int r) Row 'r' (top-level element r) of 'this' Matrix; a sub-class of Matrix might be able to provide a more efficient version than this default.
abstract Value	elt(int r, int c) The Matrix element at position c of elt(r), that is at column c of row r of 'this' Matrix.
Matrix	elts(int[] rs, int[] cs) Return the sub-Matrix with rows, rs, and columns, cs.
Type	eltType() The element Type of 'this' Matrix must be VECTOR.
static Matrix	fromLexical(Lexical lex) Read a Matrix of Cts from Lexical analyser, 'lex', such as a Lexical.forFile(fname), say.
Matrix	GJ_fp() GJ_fp, the Gauss-Jordan (GJ) elimination algorithm with full row and column pivoting (fp); also see inverse().
static Matrix	id(int N) Return the N×N identity Matrix of Ints; note, is static.
static Matrix.Ints	ints_UR(boolean incDiag, int[][] ns, int diagV) Make a symmetric Int Matrix from 'ns' which is an upper-right triangular array of array of int.
static Matrix.Ints	ints(int[][] ns) Convenience function, ints : int[][] → Matrix.
Matrix	inverse() Return the inverse of a (square, non-singular) Matrix by GJ_fp().
boolean	isRectangular() True, every Matrix is rectangular by definition.
boolean	isSquare() ?nElts()==nCols()?
double	logDeterminant() Return the log of the determinant of 'this' square, symmetric Matrix.
static void	main(java.lang.String[] argv) main() allows Vector to be (slightly) tested, in isolation.
int	nCols() The number of colums; note every Matrix is rectangular.
abstract int	nElts() The number of rows (top-level elements) in 'this' Matrix.
int	nElts(int r) Return nCols(); note that every Matrix is rectangular.
protected int	nEltsRaw(int r) Called in the rectangularity-check within setNcols(int) if the constructor Matrix() was used; it need not be implemented (overridden) if Matrix(nC) was used.
int	rowLength() Every row of 'this' Matrix has nCols() elements.
protected void	setNcols(int nC) If the trivial constructor, Matrix(), has been used, a sub-class of Matrix must override nEltsRaw(int) and call setNcols(nC) before nCols() (or nElts(r)) is used.
double	slowDeterminant() Calculate the determinant of 'this' Matrix by the slow, recursive (naive) method.
Matrix	times(Matrix M) Matrix-multiplication of two matrices of Cts, 'this' and M.
Vector	times(Vector v) 'this' Matrix, M, times the Vector of Cts, v, as a column-Vector, as in M v. Also see times(Matrix).
Matrix	transpose() Return the transpose of 'this' Matrix.
static Matrix.GP2	values(Value[][] vs) Convenience function, values : Value[][] → GP2.

Methods inherited from class la.maths.Vector

addMore, addMore, append, append, as_1xn, as_nx1, asDiagonal, asQ, asQrotn, bOp, closes, col, cols, combine, compareTo, constAoM, constNlAoM, constWt, csv, csv, delete, delete, dot, doubles, doubles, doubles, drop, drop, dropLast, elts, foldl, foldl, foldr, foldr, fromScannable, ints, ints, map, match, merge, nearlySymmetric, nearlySymmetric, nlAoM, nlAoM, norm_Cts, norm, normalised, normalised1, normalised2, opens, repeat, rotate, rotate, scaled, shape, singleton, slice, sorted, sorted, sorted, strings, sumSq, take, takeLast, toSeries, type, uniqueElts, unzip, unzip, uOp, values, weight, weight, weight, weightLike, weightLike, wt, wtMedian, wtNlAoM, wtNlAoM, wts, wts, zip, zip

Methods inherited from class la.la.Value.Structured

AoM, elts, frth, fst, print, separator, snd, thrd, toString

Methods inherited from class la.la.Value

apply, cons, cts, errMsg, error, force, isTuple, just, n, pair, quad, real, RTE, triple, tuple, UOE, x

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, wait, wait, wait

Constructor Detail

Matrix

public Matrix(int nC)

Constructor for use when the number of columns, nC, is known before calling the constructor, and the Matrix is known to be proper (rectangular). In this case setNcols(int) must not be called and nEltsRaw(int) need not be implemented -- also see Matrix().

Matrix

public Matrix()

The trivial constructor for use when the number of columns is not known before calling a Matrix constructor. Also see setNcols(int), and constructor Matrix(nC).

Method Detail

setNcols

protected void setNcols(int nC)

If the trivial constructor, Matrix(), has been used, a sub-class of Matrix must override nEltsRaw(int) and call setNcols(nC) before nCols() (or nElts(r)) is used.

nEltsRaw

protected int nEltsRaw(int r)

Called in the rectangularity-check within setNcols(int) if the constructor Matrix() was used; it need not be implemented (overridden) if Matrix(nC) was used. See nElts(int).

eltType

public Type eltType()

The element Type of 'this' Matrix must be VECTOR. Also see Vector.eltType().

Overrides:

eltType in class Vector

nElts

public abstract int nElts()

The number of rows (top-level elements) in 'this' Matrix. Also see nCols().

Specified by:

nElts in class Vector

nCols

public int nCols()

The number of colums; note every Matrix is rectangular.

Overrides:

nCols in class Vector

elt

public Vector elt(int r)

Row 'r' (top-level element r) of 'this' Matrix; a sub-class of Matrix might be able to provide a more efficient version than this default. Also see elt(r,c). Note that elt(r).wt(c) is 1.0, ∀ c.

Specified by:

elt in class Vector

nElts

public int nElts(int r)

Return nCols(); note that every Matrix is rectangular.

Overrides:

nElts in class Vector

elt

public abstract Value elt(int r,
                          int c)

The Matrix element at position c of elt(r), that is at column c of row r of 'this' Matrix.

Overrides:

elt in class Vector

elts

public Matrix elts(int[] rs,
                   int[] cs)

Return the sub-Matrix with rows, rs, and columns, cs. Also see Vector.slice(int, int).

isRectangular

public boolean isRectangular()

True, every Matrix is rectangular by definition.

Overrides:

isRectangular in class Vector

rowLength

public int rowLength()

Every row of 'this' Matrix has nCols() elements.

Overrides:

rowLength in class Vector

isSquare

public boolean isSquare()

?nElts()==nCols()?

Overrides:

isSquare in class Vector

asMatrix

public Matrix asMatrix()

Return 'this'; also see Vector.asMatrix().

Overrides:

asMatrix in class Vector

id

public static Matrix id(int N)

Return the N×N identity Matrix of Ints; note, is static.

times

public Vector times(Vector v)

'this' Matrix, M, times the Vector of Cts, v, as a column-Vector, as in M v. Also see times(Matrix).

times

public Matrix times(Matrix M)

Matrix-multiplication of two matrices of Cts, 'this' and M.

Overrides:

times in class Vector

transpose

public Matrix transpose()

Return the transpose of 'this' Matrix. Note, transpose().wt(r) is 1.0 for all r.

inverse

public Matrix inverse()

Return the inverse of a (square, non-singular) Matrix by GJ_fp().

GJ_fp

public Matrix GJ_fp()

GJ_fp, the Gauss-Jordan (GJ) elimination algorithm with full row and column pivoting (fp); also see inverse().

eigen

public Value.Tuple eigen()

Return the Eigen-Values and Eigen-Vectors, (eVals, eVecs), of 'this' square, symmetric Matrix using the Matrix.Jacobi algorithm. Also see eigenValues() and eigenVectors().

eigenValues

public Vector eigenValues()

Return the Eigen-Values of 'this' square, symmetric Matrix. Also see eigen().

eigenVectors

public Matrix eigenVectors()

Return the Eigen-Vectors of 'this' square, symmetric Matrix. Also see eigen().

logDeterminant

public double logDeterminant()

Return the log of the determinant of 'this' square, symmetric Matrix. This is the sum of the logs of its Eigen-values.

slowDeterminant

public double slowDeterminant()

Calculate the determinant of 'this' Matrix by the slow, recursive (naive) method. It is probably OK for 3x3 and smaller but it will be slow and may have numerical problems for a large Matrix. Also see logDeterminant().

fromLexical

public static Matrix fromLexical(Lexical lex)

Read a Matrix of Cts from Lexical analyser, 'lex', such as a Lexical.forFile(fname), say. Elements (columns) are separated by white space, rows by newline(s). AoM is guessed from the data.

values

public static Matrix.GP2 values(Value[][] vs)

Convenience function, values : Value[][] → GP2. Also see the 1D Vector.values(Value[]).

ints

public static Matrix.Ints ints(int[][] ns)

Convenience function, ints : int[][] → Matrix. Also see the 1D Vector.ints(int[]).

ints_UR

public static Matrix.Ints ints_UR(boolean incDiag,
                                  int[][] ns,
                                  int diagV)

Make a symmetric Int Matrix from 'ns' which is an upper-right triangular array of array of int. If 'incDiag' is false, 'ns' omits the main diagonal and all diagonal Values of the Matrix are taken to equal 'diagV'. If incDiag is true we don't care about diagV.

doubles

public static Matrix.Doubles doubles(double[][] xs,
                                     double AoM)

Convenience function, doubles : double[][] × AoM → Matrix, where every element of the Matrix has the same AoM(). Also see the 1D Vector.doubles(double[],double).

doubles

public static Matrix.Doubles doubles(double[][] xs)

Convenience function : double[][] → Matrix, where each elt(r,c) is an exact Real. Also see the 1D Vector.doubles(double[]).

main

public static void main(java.lang.String[] argv)

main() allows Vector to be (slightly) tested, in isolation.