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# NAME

math::linearalgebra - Linear Algebra

# Table Of Contents

# SYNOPSIS

package require Tcl ?8.4?

package require math::linearalgebra ?1.1.5?

**::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 questioninteger

*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 questioninteger

*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 questioninteger

*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 questioninteger

*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 questioninteger

*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

`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.

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.

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 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 Tcllib Trackers. Please also
report any ideas for enhancements you may have for either package and/or
documentation.

When proposing code changes, please provide *unified diffs*, i.e the output of
**diff -u**.

Note further that *attachments* are strongly preferred over inlined patches.
Attachments can be made by going to the **Edit** form of the ticket
immediately after its creation, and then using the left-most button in the
secondary navigation bar.

# KEYWORDS

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

# CATEGORY

Mathematics

# COPYRIGHT

Copyright © 2004-2008 Arjen Markus

Copyright © 2004 Ed Hume http://www\.hume\.com/contact\.us\.htm

Copyright © 2008 Michael Buadin