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.
- 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.
- 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.
- 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: 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)
- 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
- 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)
- 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
- 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: ncols-1)
- ::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: 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)
- 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 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.
- 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.
- 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 (x-y)
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*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.
- 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 (three-dimensional)
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 non-zero 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 ill-conditioned 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 well-behaved 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
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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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
- 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.
- 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. 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).
- 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.
- ::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 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)
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
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, vectors
CATEGORY¶
Mathematics
COPYRIGHT¶
Copyright (c) 2004-2008 Arjen Markus <arjenmarkus@users.sourceforge.net>
Copyright (c) 2004 Ed Hume <http://www.hume.com/contact.us.htm>
Copyright (c) 2008 Michael Buadin <relaxkmike@users.sourceforge.net>