NAME

math::linearalgebra -
Linear Algebra

SYNOPSIS

package require Tcl  ? 8.4 ? 
package require math::linearalgebra  ? 1.1 ? 
::math::linearalgebra::mkVector ndim value
::math::linearalgebra::mkUnitVector ndim ndir
::math::linearalgebra::mkMatrix nrows ncols value
::math::linearalgebra::getrow matrix row  ? imin ?   ? imax ? 
::math::linearalgebra::setrow matrix row newvalues  ? imin ?   ? imax ? 
::math::linearalgebra::getcol matrix col  ? imin ?   ? imax ? 
::math::linearalgebra::setcol matrix col newvalues  ? imin ?   ? imax ? 
::math::linearalgebra::getelem matrix row col
::math::linearalgebra::setelem matrix row  ? col ?  newvalue
::math::linearalgebra::swaprows matrix irow1 irow2  ? imin ?   ? imax ? 
::math::linearalgebra::swapcols matrix icol1 icol2  ? imin ?   ? imax ? 
::math::linearalgebra::show obj  ? format ?   ? rowsep ?   ? colsep ? 
::math::linearalgebra::dim obj
::math::linearalgebra::shape obj
::math::linearalgebra::conforming type obj1 obj2
::math::linearalgebra::symmetric matrix  ? eps ? 
::math::linearalgebra::norm vector type
::math::linearalgebra::norm_one vector
::math::linearalgebra::norm_two vector
::math::linearalgebra::norm_max vector  ? index ? 
::math::linearalgebra::normMatrix matrix type
::math::linearalgebra::dotproduct vect1 vect2
::math::linearalgebra::unitLengthVector vector
::math::linearalgebra::normalizeStat mv
::math::linearalgebra::axpy scale mv1 mv2
::math::linearalgebra::add mv1 mv2
::math::linearalgebra::sub mv1 mv2
::math::linearalgebra::scale scale mv
::math::linearalgebra::rotate c s vect1 vect2
::math::linearalgebra::transpose matrix
::math::linearalgebra::matmul mv1 mv2
::math::linearalgebra::angle vect1 vect2
::math::linearalgebra::crossproduct vect1 vect2
::math::linearalgebra::matmul mv1 mv2
::math::linearalgebra::mkIdentity size
::math::linearalgebra::mkDiagonal diag
::math::linearalgebra::mkRandom size
::math::linearalgebra::mkTriangular size  ? uplo ?   ? value ? 
::math::linearalgebra::mkHilbert size
::math::linearalgebra::mkDingdong size
::math::linearalgebra::mkOnes size
::math::linearalgebra::mkMoler size
::math::linearalgebra::mkFrank size
::math::linearalgebra::mkBorder size
::math::linearalgebra::mkWilkinsonW+ size
::math::linearalgebra::mkWilkinsonW- size
::math::linearalgebra::solveGauss matrix bvect
::math::linearalgebra::solvePGauss matrix bvect
::math::linearalgebra::solveTriangular matrix bvect  ? uplo ? 
::math::linearalgebra::solveGaussBand matrix bvect
::math::linearalgebra::solveTriangularBand matrix bvect
::math::linearalgebra::determineSVD A eps
::math::linearalgebra::eigenvectorsSVD A eps
::math::linearalgebra::leastSquaresSVD A y qmin eps
::math::linearalgebra::choleski matrix
::math::linearalgebra::orthonormalizeColumns matrix
::math::linearalgebra::orthonormalizeRows matrix
::math::linearalgebra::dger matrix alpha x y  ? scope ? 
::math::linearalgebra::dgetrf matrix
::math::linearalgebra::det matrix
::math::linearalgebra::largesteigen matrix tolerance maxiter
::math::linearalgebra::to_LA mv
::math::linearalgebra::from_LA mv

DESCRIPTION

This package offers both low-level procedures and high-level algorithms to deal with linear algebra problems:

• robust solution of linear equations or least squares problems
• determining eigenvectors and eigenvalues of symmetric matrices
• various decompositions of general matrices or matrices of a specific form
• (limited) support for matrices in band storage, a common type of sparse matrices

It arose as a re-implementation of Hume's LA package and the desire to offer low-level procedures as found in the well-known BLAS library. Matrices are implemented as lists of lists rather linear lists with reserved elements, as in the original LA package, as it was found that such an implementation is actually faster.

It is advisable, however, to use the procedures that are offered, such as setrow and getrow, rather than rely on this representation explicitly: that way it is to switch to a possibly even faster compiled implementation that supports the same API.

Note: When using this package in combination with Tk, there may be a naming conflict, as both this package and Tk define a command scale. See the NAMING CONFLICT section below.

PROCEDURES

The package defines the following public procedures (several exist as specialised procedures, see below):

Constructing matrices and vectors

::math::linearalgebra::mkVector ndim value

Create a vector with ndim elements, each with the value value.

Type	Name	Mode
integer	ndim
	Dimension of the vector (number of components)
double	value
	Uniform value to be used (default: 0.0)

::math::linearalgebra::mkUnitVector ndim ndir

Create a unit vector in ndim-dimensional space, along the ndir-th direction.

Type	Name	Mode
integer	ndim
	Dimension of the vector (number of components)
integer	ndir
	Direction (0, ..., ndim-1)

::math::linearalgebra::mkMatrix nrows ncols value

Create a matrix with nrows rows and ncols columns. All elements have the value value.

Type	Name	Mode
integer	nrows
	Number of rows
integer	ncols
	Number of columns
double	value
	Uniform value to be used (default: 0.0)

::math::linearalgebra::getrow matrix row ? imin ? ? imax ?

Returns a single row of a matrix as a list

Type	Name	Mode
list	matrix
	Matrix in question
integer	row
	Index of the row to return
integer	imin
	Minimum index of the column (default: 0)
integer	imax
	Maximum index of the column (default: ncols-1)

::math::linearalgebra::setrow matrix row newvalues ? imin ? ? imax ?

Set a single row of a matrix to new values (this list must have the same number of elements as the number of columns in the matrix)

Type	Name	Mode
list	matrix
	name of the matrix in question
integer	row
	Index of the row to update
list	newvalues
	List of new values for the row
integer	imin
	Minimum index of the column (default: 0)
integer	imax
	Maximum index of the column (default: ncols-1)

::math::linearalgebra::getcol matrix col ? imin ? ? imax ?

Returns a single column of a matrix as a list

Type	Name	Mode
list	matrix
	Matrix in question
integer	col
	Index of the column to return
integer	imin
	Minimum index of the row (default: 0)
integer	imax
	Maximum index of the row (default: nrows-1)

::math::linearalgebra::setcol matrix col newvalues ? imin ? ? imax ?

Set a single column of a matrix to new values (this list must have the same number of elements as the number of rows in the matrix)

Type	Name	Mode
list	matrix
	name of the matrix in question
integer	col
	Index of the column to update
list	newvalues
	List of new values for the column
integer	imin
	Minimum index of the row (default: 0)
integer	imax
	Maximum index of the row (default: nrows-1)

::math::linearalgebra::getelem matrix row col

Returns a single element of a matrix/vector

Type	Name	Mode
list	matrix
	Matrix or vector in question
integer	row
	Row of the element
integer	col
	Column of the element (not present for vectors)

::math::linearalgebra::setelem matrix row ? col ? newvalue

Set a single element of a matrix (or vector) to a new value

Type	Name	Mode
list	matrix
	name of the matrix in question
integer	row
	Row of the element
integer	col
	Column of the element (not present for vectors)

::math::linearalgebra::swaprows matrix irow1 irow2 ? imin ? ? imax ?

Swap two rows in a matrix completely or only a selected part

Type	Name	Mode
list	matrix
	name of the matrix in question
integer	irow1
	Index of first row
integer	irow2
	Index of second row
integer	imin
	Minimum column index (default: 0)
integer	imin
	Maximum column index (default: ncols-1)

::math::linearalgebra::swapcols matrix icol1 icol2 ? imin ? ? imax ?

Swap two columns in a matrix completely or only a selected part

Type	Name	Mode
list	matrix
	name of the matrix in question
integer	irow1
	Index of first column
integer	irow2
	Index of second column
integer	imin
	Minimum row index (default: 0)
integer	imin
	Maximum row index (default: nrows-1)

Querying matrices and vectors

::math::linearalgebra::show obj ? format ? ? rowsep ? ? colsep ?

Return a string representing the vector or matrix, for easy printing. (There is currently no way to print fixed sets of columns)

Type	Name	Mode
list	obj
	Matrix or vector in question
string	format
	Format for printing the numbers (default: %6.4f)
string	rowsep
	String to use for separating rows (default: newline)
string	colsep
	String to use for separating columns (default: space)

::math::linearalgebra::dim obj

Returns the number of dimensions for the object (either 0 for a scalar, 1 for a vector and 2 for a matrix)

Type	Name	Mode
any	obj
	Scalar, vector, or matrix

::math::linearalgebra::shape obj

Returns the number of elements in each dimension for the object (either an empty list for a scalar, a single number for a vector and a list of the number of rows and columns for a matrix)

Type	Name	Mode
any	obj
	Scalar, vector, or matrix

::math::linearalgebra::conforming type obj1 obj2

Checks if two objects (vector or matrix) have conforming shapes, that is if they can be applied in an operation like addition or matrix multiplication.

Type	Name	Mode
string	type
	Type of check: "shape" - the two objects have the same shape (for all element-wise operations) "rows" - the two objects have the same number of rows (for use as A and b in a system of linear equations Ax = b "matmul" - the first object has the same number of columns as the number of rows of the second object. Useful for matrix-matrix or matrix-vector multiplication.
list	obj1
	First vector or matrix (left operand)
list	obj2
	Second vector or matrix (right operand)

::math::linearalgebra::symmetric matrix ? eps ?

Checks if the given (square) matrix is symmetric. The argument eps is the tolerance.

Type	Name	Mode
list	matrix
	Matrix to be inspected
float	eps
	Tolerance for determining approximate equality (defaults to 1.0e-8)

Basic operations

::math::linearalgebra::norm vector type

Returns the norm of the given vector. The type argument can be: 1, 2, inf or max, respectively the sum of absolute values, the ordinary Euclidean norm or the max norm.

Type	Name	Mode
list	vector
	Vector, list of coefficients
string	type
	Type of norm (default: 2, the Euclidean norm)

::math::linearalgebra::norm_one vector

Returns the L1 norm of the given vector, the sum of absolute values

Type	Name	Mode
list	vector
	Vector, list of coefficients

::math::linearalgebra::norm_two vector

Returns the L2 norm of the given vector, the ordinary Euclidean norm

Type	Name	Mode
list	vector
	Vector, list of coefficients

::math::linearalgebra::norm_max vector ? index ?

Returns the Linf norm of the given vector, the maximum absolute coefficient

Type	Name	Mode
list	vector
	Vector, list of coefficients
integer	index
	(optional) if non zero, returns a list made of the maximum value and the index where that maximum was found. if zero, returns the maximum value.

::math::linearalgebra::normMatrix matrix type

Returns the norm of the given matrix. The type argument can be: 1, 2, inf or max, respectively the sum of absolute values, the ordinary Euclidean norm or the max norm.

Type	Name	Mode
list	matrix
	Matrix, list of row vectors
string	type
	Type of norm (default: 2, the Euclidean norm)

::math::linearalgebra::dotproduct vect1 vect2

Determine the inproduct or dot product of two vectors. These must have the same shape (number of dimensions)

Type	Name	Mode
list	vect1
	First vector, list of coefficients
list	vect2
	Second vector, list of coefficients

::math::linearalgebra::unitLengthVector vector

Return a vector in the same direction with length 1.

Type	Name	Mode
list	vector
	Vector to be normalized

::math::linearalgebra::normalizeStat mv

Normalize the matrix or vector in a statistical sense: the mean of the elements of the columns of the result is zero and the standard deviation is 1.

Type	Name	Mode
list	mv
	Vector or matrix to be normalized in the above sense

::math::linearalgebra::axpy scale mv1 mv2

Return a vector or matrix that results from a "daxpy" operation, that is: compute a*x+y (a a scalar and x and y both vectors or matrices of the same shape) and return the result.
Specialised variants are: axpy_vect and axpy_mat (slightly faster, but no check on the arguments)

Type	Name	Mode
double	scale
	The scale factor for the first vector/matrix (a)
list	mv1
	First vector or matrix (x)
list	mv2
	Second vector or matrix (y)

::math::linearalgebra::add mv1 mv2

Return a vector or matrix that is the sum of the two arguments (x+y)
Specialised variants are: add_vect and add_mat (slightly faster, but no check on the arguments)

Type	Name	Mode
list	mv1
	First vector or matrix (x)
list	mv2
	Second vector or matrix (y)

::math::linearalgebra::sub mv1 mv2

Return a vector or matrix that is the difference of the two arguments (x-y)
Specialised variants are: sub_vect and sub_mat (slightly faster, but no check on the arguments)

Type	Name	Mode
list	mv1
	First vector or matrix (x)
list	mv2
	Second vector or matrix (y)

::math::linearalgebra::scale scale mv

Scale a vector or matrix and return the result, that is: compute a*x.
Specialised variants are: scale_vect and scale_mat (slightly faster, but no check on the arguments)

Type	Name	Mode
double	scale
	The scale factor for the vector/matrix (a)
list	mv
	Vector or matrix (x)

::math::linearalgebra::rotate c s vect1 vect2

Apply a planar rotation to two vectors and return the result as a list of two vectors: c*x-s*y and s*x+c*y. In algorithms you can often easily determine the cosine and sine of the angle, so it is more efficient to pass that information directly.

Type	Name	Mode
double	c
	The cosine of the angle
double	s
	The sine of the angle
list	vect1
	First vector (x)
list	vect2
	Seocnd vector (x)

::math::linearalgebra::transpose matrix

Transpose a matrix

Type	Name	Mode
list	matrix
	Matrix to be transposed

::math::linearalgebra::matmul mv1 mv2

Multiply a vector/matrix with another vector/matrix. The result is a matrix, if both x and y are matrices or both are vectors, in which case the "outer product" is computed. If one is a vector and the other is a matrix, then the result is a vector.

Type	Name	Mode
list	mv1
	First vector/matrix (x)
list	mv2
	Second vector/matrix (y)

::math::linearalgebra::angle vect1 vect2

Compute the angle between two vectors (in radians)

Type	Name	Mode
list	vect1
	First vector
list	vect2
	Second vector

::math::linearalgebra::crossproduct vect1 vect2

Compute the cross product of two (three-dimensional) vectors

Type	Name	Mode
list	vect1
	First vector
list	vect2
	Second vector

::math::linearalgebra::matmul mv1 mv2

Multiply a vector/matrix with another vector/matrix. The result is a matrix, if both x and y are matrices or both are vectors, in which case the "outer product" is computed. If one is a vector and the other is a matrix, then the result is a vector.

Type	Name	Mode
list	mv1
	First vector/matrix (x)
list	mv2
	Second vector/matrix (y)

Common matrices and test matrices

::math::linearalgebra::mkIdentity size

Create an identity matrix of dimension size.

Type	Name	Mode
integer	size
	Dimension of the matrix

::math::linearalgebra::mkDiagonal diag

Create a diagonal matrix whose diagonal elements are the elements of the vector diag.

Type	Name	Mode
list	diag
	Vector whose elements are used for the diagonal

::math::linearalgebra::mkRandom size

Create a square matrix whose elements are uniformly distributed random numbers between 0 and 1 of dimension size.

Type	Name	Mode
integer	size
	Dimension of the matrix

::math::linearalgebra::mkTriangular size ? uplo ? ? value ?

Create a triangular matrix with non-zero elements in the upper or lower part, depending on argument uplo.

Type	Name	Mode
integer	size
	Dimension of the matrix
string	uplo
	Fill the upper (U) or lower part (L)
double	value
	Value to fill the matrix with

::math::linearalgebra::mkHilbert size

Create a Hilbert matrix of dimension size. Hilbert matrices are very ill-conditioned with respect to eigenvalue/eigenvector problems. Therefore they are good candidates for testing the accuracy of algorithms and implementations.

Type	Name	Mode
integer	size
	Dimension of the matrix

::math::linearalgebra::mkDingdong size

Create a "dingdong" matrix of dimension size. Dingdong matrices are imprecisely represented, but have the property of being very stable in such algorithms as Gauss elimination.

Type	Name	Mode
integer	size
	Dimension of the matrix

::math::linearalgebra::mkOnes size

Create a square matrix of dimension size whose entries are all 1.

Type	Name	Mode
integer	size
	Dimension of the matrix

::math::linearalgebra::mkMoler size

Create a Moler matrix of size size. (Moler matrices have a very simple Choleski decomposition. It has one small eigenvalue and it can easily upset elimination methods for systems of linear equations.)

Type	Name	Mode
integer	size
	Dimension of the matrix

::math::linearalgebra::mkFrank size

Create a Frank matrix of size size. (Frank matrices are fairly well-behaved matrices)

Type	Name	Mode
integer	size
	Dimension of the matrix

::math::linearalgebra::mkBorder size

Create a bordered matrix of size size. (Bordered matrices have a very low rank and can upset certain specialised algorithms.)

Type	Name	Mode
integer	size
	Dimension of the matrix

::math::linearalgebra::mkWilkinsonW+ size

Create a Wilkinson W+ of size size. This kind of matrix has pairs of eigenvalues that are very close together. Usually the order (size) is odd.

Type	Name	Mode
integer	size
	Dimension of the matrix

::math::linearalgebra::mkWilkinsonW- size

Create a Wilkinson W- of size size. This kind of matrix has pairs of eigenvalues with opposite signs, when the order (size) is odd.

Type	Name	Mode
integer	size
	Dimension of the matrix

Common algorithms

::math::linearalgebra::solveGauss matrix bvect

Solve a system of linear equations (Ax=b) using Gauss elimination. Returns the solution (x) as a vector or matrix of the same shape as bvect.

Type	Name	Mode
list	matrix
	Square matrix (matrix A)
list	bvect
	Vector or matrix whose columns are the individual b-vectors

::math::linearalgebra::solvePGauss matrix bvect

Solve a system of linear equations (Ax=b) using Gauss elimination with partial pivoting. Returns the solution (x) as a vector or matrix of the same shape as bvect.

Type	Name	Mode
list	matrix
	Square matrix (matrix A)
list	bvect
	Vector or matrix whose columns are the individual b-vectors

::math::linearalgebra::solveTriangular matrix bvect ? uplo ?

Solve a system of linear equations (Ax=b) by backward substitution. The matrix is supposed to be upper-triangular.

Type	Name	Mode
list	matrix
	Lower or upper-triangular matrix (matrix A)
list	bvect
	Vector or matrix whose columns are the individual b-vectors
string	uplo
	Indicates whether the matrix is lower-triangular (L) or upper-triangular (U). Defaults to "U".

::math::linearalgebra::solveGaussBand matrix bvect

Solve a system of linear equations (Ax=b) using Gauss elimination, where the matrix is stored as a band matrix (cf. STORAGE). Returns the solution (x) as a vector or matrix of the same shape as bvect.

Type	Name	Mode
list	matrix
	Square matrix (matrix A; in band form)
list	bvect
	Vector or matrix whose columns are the individual b-vectors

::math::linearalgebra::solveTriangularBand matrix bvect

Solve a system of linear equations (Ax=b) by backward substitution. The matrix is supposed to be upper-triangular and stored in band form.

Type	Name	Mode
list	matrix
	Upper-triangular matrix (matrix A)
list	bvect
	Vector or matrix whose columns are the individual b-vectors

::math::linearalgebra::determineSVD A eps

Determines the Singular Value Decomposition of a matrix: A = U S Vtrans. Returns a list with the matrix U, the vector of singular values S and the matrix V.

Type	Name	Mode
list	A
	Matrix to be decomposed
float	eps
	Tolerance (defaults to 2.3e-16)

::math::linearalgebra::eigenvectorsSVD A eps

Determines the eigenvectors and eigenvalues of a real symmetric matrix, using SVD. Returns a list with the matrix of normalized eigenvectors and their eigenvalues.

Type	Name	Mode
list	A
	Matrix whose eigenvalues must be determined
float	eps
	Tolerance (defaults to 2.3e-16)

::math::linearalgebra::leastSquaresSVD A y qmin eps

Determines the solution to a least-sqaures problem Ax ~ y via singular value decomposition. The result is the vector x.
Note that if you add a column of 1s to the matrix, then this column will represent a constant like in: y = a*x1 + b*x2 + c. To force the intercept to be zero, simply leave it out.

Type	Name	Mode
list	A
	Matrix of independent variables
list	y
	List of observed values
float	qmin
	Minimum singular value to be considered (defaults to 0.0)
float	eps
	Tolerance (defaults to 2.3e-16)

::math::linearalgebra::choleski matrix

Determine the Choleski decomposition of a symmetric positive semidefinite matrix (this condition is not checked!). The result is the lower-triangular matrix L such that L Lt = matrix.

Type	Name	Mode
list	matrix
	Matrix to be decomposed

::math::linearalgebra::orthonormalizeColumns matrix

Use the modified Gram-Schmidt method to orthogonalize and normalize the columns of the given matrix and return the result.

Type	Name	Mode
list	matrix
	Matrix whose columns must be orthonormalized

::math::linearalgebra::orthonormalizeRows matrix

Use the modified Gram-Schmidt method to orthogonalize and normalize the rows of the given matrix and return the result.

Type	Name	Mode
list	matrix
	Matrix whose rows must be orthonormalized

::math::linearalgebra::dger matrix alpha x y ? scope ?

Perform the rank 1 operation A + alpha*x*y' inline (that is: the matrix A is adjusted). For convenience the new matrix is also returned as the result.

Type	Name	Mode
list	matrix
	Matrix whose rows must be adjusted
double	alpha
	Scale factor
list	x
	A column vector
list	y
	A column vector
list	scope
	If not provided, the operation is performed on all rows/columns of A if provided, it is expected to be the list {imin imax jmin jmax} where: imin Minimum row index imax Maximum row index jmin Minimum column index jmax Maximum column index

::math::linearalgebra::dgetrf matrix

Computes an LU factorization of a general matrix, using partial, pivoting with row interchanges. Returns the permutation vector.
The factorization has the form

   P * A = L * U

where P is a permutation matrix, L is lower triangular with unit diagonal elements, and U is upper triangular. Returns the permutation vector, as a list of length n-1. The last entry of the permutation is not stored, since it is implicitely known, with value n (the last row is not swapped with any other row). At index #i of the permutation is stored the index of the row #j which is swapped with row #i at step #i. That means that each index of the permutation gives the permutation at each step, not the cumulated permutation matrix, which is the product of permutations.

Type	Name	Mode
list	matrix
	On entry, the matrix to be factored. On exit, the factors L and U from the factorization P*A = L*U; the unit diagonal elements of L are not stored.

::math::linearalgebra::det matrix

Returns the determinant of the given matrix, based on PA=LU decomposition, i.e. Gauss partial pivotal.

Type	Name	Mode
list	matrix
	Square matrix (matrix A)
list	ipiv
	The pivots (optionnal). If the pivots are not provided, a PA=LU decomposition is performed. If the pivots are provided, we assume that it contains the pivots and that the matrix A contains the L and U factors, as provided by dgterf. b-vectors

::math::linearalgebra::largesteigen matrix tolerance maxiter

Returns a list made of the largest eigenvalue (in magnitude) and associated eigenvector. Uses iterative Power Method as provided as algorithm #7.3.3 of Golub & Van Loan. This algorithm is used here for a dense matrix (but is usually used for sparse matrices).

Type	Name	Mode
list	matrix
	Square matrix (matrix A)
double	tolerance
	The relative tolerance of the eigenvalue (default:1.e-8).
integer	maxiter
	The maximum number of iterations (default:10).

Compability with the LA package Two procedures are provided for compatibility with Hume's LA package:

::math::linearalgebra::to_LA mv

Transforms a vector or matrix into the format used by the original LA package.

Type	Name	Mode
list	mv
	Matrix or vector

::math::linearalgebra::from_LA mv

Transforms a vector or matrix from the format used by the original LA package into the format used by the present implementation.

Type	Name	Mode
list	mv
	Matrix or vector as used by the LA package

STORAGE

While most procedures assume that the matrices are given in full form, the procedures solveGaussBand and solveTriangularBand assume that the matrices are stored as band matrices. This common type of "sparse" matrices is related to ordinary matrices as follows:

• "A" is a full-size matrix with N rows and M columns.
• "B" is a band matrix, with m upper and lower diagonals and n rows.
• "B" can be stored in an ordinary matrix of (2m+1) columns (one for each off-diagonal and the main diagonal) and n rows.
• Element i,j (i = -m,...,m; j =1,...,n) of "B" corresponds to element k,j of "A" where k = M+i-1 and M is at least (!) n, the number of rows in "B".
• To set element (i,j) of matrix "B" use:

      setelem B $j [expr {$N+$i-1}] $value
  
  

(There is no convenience procedure for this yet)

REMARKS ON THE IMPLEMENTATION

There is a difference between the original LA package by Hume and the current implementation. Whereas the LA package uses a linear list, the current package uses lists of lists to represent matrices. It turns out that with this representation, the algorithms are faster and easier to implement.

The LA package was used as a model and in fact the implementation of, for instance, the SVD algorithm was taken from that package. The set of procedures was expanded using ideas from the well-known BLAS library and some algorithms were updated from the second edition of J.C. Nash's book, Compact Numerical Methods for Computers, (Adam Hilger, 1990) that inspired the LA package.

Two procedures are provided to make the transition between the two implementations easier: to_LA and from_LA. They are described above.

TODO

Odds and ends: the following algorithms have not been implemented yet:

• determineQR
• certainlyPositive, diagonallyDominant

NAMING CONFLICT

If you load this package in a Tk-enabled shell like wish, then the command

namespace import ::math::linearalgebra

results in an error message about "scale". This is due to the fact that Tk defines all its commands in the global namespace. The solution is to import the linear algebra commands in a namespace that is not the global one:

package require math::linearalgebra
namespace eval compute {
    namespace import ::math::linearalgebra::*
    ... use the linear algebra version of scale ...
}

To use Tk's scale command in that same namespace you can rename it:

namespace eval compute {
    rename ::scale scaleTk
    scaleTk .scale ...
}

BUGS, IDEAS, FEEDBACK

This document, and the package it describes, will undoubtedly contain bugs and other problems. Please report such in the category math :: linearalgebra of the http://sourceforge.net/tracker/?group_id=12883. Please also report any ideas for enhancements you may have for either package and/or documentation.

KEYWORDS

math, linear algebra, vectors, matrices, least squares, matrix, linear equations