linalg (n)  Linux Manuals
linalg: Linear Algebra
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 lowlevel procedures and highlevel 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 reimplementation of Hume's LA package and the desire to offer lowlevel procedures as found in the wellknown 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.

 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 ndimdimensional space, along the
ndirth direction.

 integer ndim
 Dimension of the vector (number of components)
 integer ndir
 Direction (0, ..., ndim1)

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

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

 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

 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: ncols1)

 ::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)

 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: ncols1)

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

Returns a single column of a matrix as a list

 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: nrows1)

 ::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)

 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: nrows1)

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

Returns a single element of a matrix/vector

 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

 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

 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: ncols1)

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

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

 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: nrows1)

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)

 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)

 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)

 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.

 string type

Type of check:

 •
 "shape"  the two objects have the same shape (for all elementwise 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 matrixmatrix or matrixvector 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.

 list matrix
 Matrix to be inspected
 float eps
 Tolerance for determining approximate equality (defaults to 1.0e8)

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.

 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

 list vector
 Vector, list of coefficients

 ::math::linearalgebra::norm_two vector

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

 list vector
 Vector, list of coefficients

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

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

 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.

 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)

 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.

 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.

 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)

 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)

 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
(xy)
Specialised variants are: sub_vect and sub_mat (slightly faster, but no check on the arguments)

 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)

 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*xs*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.

 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

 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.

 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)

 list vect1
 First vector
 list vect2
 Second vector

 ::math::linearalgebra::crossproduct vect1 vect2

Compute the cross product of two (threedimensional) vectors

 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.

 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.

 integer size
 Dimension of the matrix

 ::math::linearalgebra::mkDiagonal diag

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

 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.

 integer size
 Dimension of the matrix

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

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

 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 illconditioned with respect to
eigenvalue/eigenvector problems. Therefore they
are good candidates for testing the accuracy
of algorithms and implementations.

 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.

 integer size
 Dimension of the matrix

 ::math::linearalgebra::mkOnes size

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

 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.)

 integer size
 Dimension of the matrix

 ::math::linearalgebra::mkFrank size

Create a Frank matrix of size size. (Frank matrices are
fairly wellbehaved matrices)

 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.)

 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.

 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.

 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.

 list matrix
 Square matrix (matrix A)
 list bvect
 Vector or matrix whose columns are the individual bvectors

 ::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.

 list matrix
 Square matrix (matrix A)
 list bvect
 Vector or matrix whose columns are the individual bvectors

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

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

 list matrix
 Lower or uppertriangular matrix (matrix A)
 list bvect
 Vector or matrix whose columns are the individual bvectors
 string uplo
 Indicates whether the matrix is lowertriangular (L) or uppertriangular (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.

 list matrix
 Square matrix (matrix A; in band form)
 list bvect
 Vector or matrix whose columns are the individual bvectors

 ::math::linearalgebra::solveTriangularBand matrix bvect

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

 list matrix
 Uppertriangular matrix (matrix A)
 list bvect
 Vector or matrix whose columns are the individual bvectors

 ::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.

 list A
 Matrix to be decomposed
 float eps
 Tolerance (defaults to 2.3e16)

 ::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.

 list A
 Matrix whose eigenvalues must be determined
 float eps
 Tolerance (defaults to 2.3e16)

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

Determines the solution to a leastsqaures 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.

 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 2.3e16)

 ::math::linearalgebra::choleski matrix

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

 list matrix
 Matrix to be decomposed

 ::math::linearalgebra::orthonormalizeColumns matrix

Use the modified GramSchmidt method to orthogonalize and normalize
the columns of the given matrix and return the result.

 list matrix
 Matrix whose columns must be orthonormalized

 ::math::linearalgebra::orthonormalizeRows matrix

Use the modified GramSchmidt method to orthogonalize and normalize
the rows of the given matrix and return the result.

 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.

 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 n1.
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.

 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.

 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. bvectors

 ::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).

 list matrix
 Square matrix (matrix A)
 double tolerance
 The relative tolerance of the eigenvalue (default:1.e8).
 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.

 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.

 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 fullsize 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 offdiagonal 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+i1 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+$i1}] $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 wellknown 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 Tkenabled shell like wish, then the command
namespace import ::math::linearalgebra

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

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 Tcllib SF Trackers [http://sourceforge.net/tracker/?group_id=12883]. Please also report any ideas for enhancements you may have for either package and/or documentation.KEYWORDS
least squares, linear algebra, linear equations, math, matrices, matrix, vectorsCATEGORY
MathematicsCOPYRIGHT
Copyright (c) 20042008 Arjen Markus <arjenmarkus [at] users.sourceforge.net> Copyright (c) 2004 Ed Hume <http://www.hume.com/contact.us.htm> Copyright (c) 2008 Michael Buadin <relaxkmike [at] users.sourceforge.net>