math::linearalgebra(n)         Tcl Math Library         math::linearalgebra(n)



______________________________________________________________________________

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:

       o      robust solution of linear equations or least squares problems

       o      determining eigenvectors and eigenvalues of symmetric matrices

       o      various decompositions of general matrices or matrices of a
              specific form

       o      (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:

                     o      "shape" - the two objects have the same shape (for
                            all element-wise operations)

                     o      "rows" - the two objects have the same number of
                            rows (for use as A and b in a system of linear
                            equations Ax = b

                     o      "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:

                     o      imin Minimum row index

                     o      imax Maximum row index

                     o      jmin Minimum column index

                     o      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:

       o      "A" is a full-size matrix with N rows and M columns.

       o      "B" is a band matrix, with m upper and lower diagonals and n
              rows.

       o      "B" can be stored in an ordinary matrix of (2m+1) columns (one
              for each off-diagonal and the main diagonal) and n rows.

       o      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".

       o      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:

       o      determineQR

       o      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>




math                                  1.1               math::linearalgebra(n)
