math::linearalgebra - Linear Algebra
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.
The package defines the following public procedures (several exist as specialised procedures, see below):
Constructing matrices and vectors
Create a vector with ndim elements, each with the value value.
Dimension of the vector (number of components)
Uniform value to be used (default: 0.0)
Create a unit vector in ndim-dimensional space, along the ndir-th direction.
Dimension of the vector (number of components)
Direction (0, ..., ndim-1)
Create a matrix with nrows rows and ncols columns. All elements have the value value.
Number of rows
Number of columns
Uniform value to be used (default: 0.0)
Returns a single row of a matrix as a list
Matrix in question
Index of the row to return
Minimum index of the column (default: 0)
Maximum index of the column (default: ncols-1)
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)
name of the matrix in question
Index of the row to update
List of new values for the row
Minimum index of the column (default: 0)
Maximum index of the column (default: ncols-1)
Returns a single column of a matrix as a list
Matrix in question
Index of the column to return
Minimum index of the row (default: 0)
Maximum index of the row (default: nrows-1)
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)
name of the matrix in question
Index of the column to update
List of new values for the column
Minimum index of the row (default: 0)
Maximum index of the row (default: nrows-1)
Returns a single element of a matrix/vector
Matrix or vector in question
Row of the element
Column of the element (not present for vectors)
Set a single element of a matrix (or vector) to a new value
name of the matrix in question
Row of the element
Column of the element (not present for vectors)
Swap two rows in a matrix completely or only a selected part
name of the matrix in question
Index of first row
Index of second row
Minimum column index (default: 0)
Maximum column index (default: ncols-1)
Swap two columns in a matrix completely or only a selected part
name of the matrix in question
Index of first column
Index of second column
Minimum row index (default: 0)
Maximum row index (default: nrows-1)
Querying matrices and vectors
Return a string representing the vector or matrix, for easy printing. (There is currently no way to print fixed sets of columns)
Matrix or vector in question
Format for printing the numbers (default: %6.4f)
String to use for separating rows (default: newline)
String to use for separating columns (default: space)
Returns the number of dimensions for the object (either 0 for a scalar, 1 for a vector and 2 for a matrix)
Scalar, vector, or matrix
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)
Scalar, vector, or matrix
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.
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.
First vector or matrix (left operand)
Second vector or matrix (right operand)
Checks if the given (square) matrix is symmetric. The argument eps is the tolerance.
Matrix to be inspected
Tolerance for determining approximate equality (defaults to 1.0e-8)
Basic operations
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.
Vector, list of coefficients
Type of norm (default: 2, the Euclidean norm)
Returns the L1 norm of the given vector, the sum of absolute values
Vector, list of coefficients
Returns the L2 norm of the given vector, the ordinary Euclidean norm
Vector, list of coefficients
Returns the Linf norm of the given vector, the maximum absolute coefficient
Vector, list of coefficients
(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.
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.
Matrix, list of row vectors
Type of norm (default: 2, the Euclidean norm)
Determine the inproduct or dot product of two vectors. These must have the same shape (number of dimensions)
First vector, list of coefficients
Second vector, list of coefficients
Return a vector in the same direction with length 1.
Vector to be normalized
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.
Vector or matrix to be normalized in the above sense
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)
The scale factor for the first vector/matrix (a)
First vector or matrix (x)
Second vector or matrix (y)
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)
First vector or matrix (x)
Second vector or matrix (y)
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)
First vector or matrix (x)
Second vector or matrix (y)
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)
The scale factor for the vector/matrix (a)
Vector or matrix (x)
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.
The cosine of the angle
The sine of the angle
First vector (x)
Seocnd vector (x)
Transpose a matrix
Matrix to be transposed
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.
First vector/matrix (x)
Second vector/matrix (y)
Compute the angle between two vectors (in radians)
First vector
Second vector
Compute the cross product of two (three-dimensional) vectors
First vector
Second vector
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.
First vector/matrix (x)
Second vector/matrix (y)
Common matrices and test matrices
Create an identity matrix of dimension size.
Dimension of the matrix
Create a diagonal matrix whose diagonal elements are the elements of the vector diag.
Vector whose elements are used for the diagonal
Create a square matrix whose elements are uniformly distributed random numbers between 0 and 1 of dimension size.
Dimension of the matrix
Create a triangular matrix with non-zero elements in the upper or lower part, depending on argument uplo.
Dimension of the matrix
Fill the upper (U) or lower part (L)
Value to fill the matrix with
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.
Dimension of the matrix
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.
Dimension of the matrix
Create a square matrix of dimension size whose entries are all 1.
Dimension of the matrix
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.)
Dimension of the matrix
Create a Frank matrix of size size. (Frank matrices are fairly well-behaved matrices)
Dimension of the matrix
Create a bordered matrix of size size. (Bordered matrices have a very low rank and can upset certain specialised algorithms.)
Dimension of the matrix
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.
Dimension of the matrix
Create a Wilkinson W- of size size. This kind of matrix has pairs of eigenvalues with opposite signs, when the order (size) is odd.
Dimension of the matrix
Common algorithms
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.
Square matrix (matrix A)
Vector or matrix whose columns are the individual b-vectors
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.
Square matrix (matrix A)
Vector or matrix whose columns are the individual b-vectors
Solve a system of linear equations (Ax=b) by backward substitution. The matrix is supposed to be upper-triangular.
Lower or upper-triangular matrix (matrix A)
Vector or matrix whose columns are the individual b-vectors
Indicates whether the matrix is lower-triangular (L) or upper-triangular (U). Defaults to "U".
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.
Square matrix (matrix A; in band form)
Vector or matrix whose columns are the individual b-vectors
Solve a system of linear equations (Ax=b) by backward substitution. The matrix is supposed to be upper-triangular and stored in band form.
Upper-triangular matrix (matrix A)
Vector or matrix whose columns are the individual b-vectors
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.
Matrix to be decomposed
Tolerance (defaults to 2.3e-16)
Determines the eigenvectors and eigenvalues of a real symmetric matrix, using SVD. Returns a list with the matrix of normalized eigenvectors and their eigenvalues.
Matrix whose eigenvalues must be determined
Tolerance (defaults to 2.3e-16)
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.
Matrix of independent variables
List of observed values
Minimum singular value to be considered (defaults to 0.0)
Tolerance (defaults to 2.3e-16)
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.
Matrix to be decomposed
Use the modified Gram-Schmidt method to orthogonalize and normalize the columns of the given matrix and return the result.
Matrix whose columns must be orthonormalized
Use the modified Gram-Schmidt method to orthogonalize and normalize the rows of the given matrix and return the result.
Matrix whose rows must be orthonormalized
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.
Matrix whose rows must be adjusted
Scale factor
A column vector
A column vector
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
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.
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.
Returns the determinant of the given matrix, based on PA=LU decomposition, i.e. Gauss partial pivotal.
Square matrix (matrix A)
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
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).
Square matrix (matrix A)
The relative tolerance of the eigenvalue (default:1.e-8).
The maximum number of iterations (default:10).
Compability with the LA package Two procedures are provided for compatibility with Hume's LA package:
Transforms a vector or matrix into the format used by the original LA package.
Matrix or vector
Transforms a vector or matrix from the format used by the original LA package into the format used by the present implementation.
Matrix or vector as used by the LA package
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.
Odds and ends: the following algorithms have not been implemented yet:
determineQR
certainlyPositive, diagonallyDominant
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 ...
}
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.
Mathematics
Copyright © 2004-2008 Arjen Markus <arjenmarkus@users.sourceforge.net>
Copyright © 2004 Ed Hume <http://www.hume.com/contact.us.htm>
Copyright © 2008 Michael Buadin <relaxkmike@users.sourceforge.net>