math::linearalgebra(3tcl) Tcl Math Library math::linearalgebra(3tcl)
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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
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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 ordi-
nary 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 ordi-
nary 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 stan-
dard 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 prob-
lems. 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 permuta-
tion 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 permuta-
tion 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 ele-
ments 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(3tcl)