1math::linearalgebra(n) Tcl Math Library math::linearalgebra(n)
2______________________________________________________________________________
6
8 math::linearalgebra - Linear Algebra
9
11 package require Tcl ?8.4?
12 package require math::linearalgebra ?1.1?
14 ::math::linearalgebra::mkVector ndim value
16 ::math::linearalgebra::mkUnitVector ndim ndir
18 ::math::linearalgebra::mkMatrix nrows ncols value
20 ::math::linearalgebra::getrow matrix row ?imin? ?imax?
22 ::math::linearalgebra::setrow matrix row newvalues ?imin? ?imax?
24 ::math::linearalgebra::getcol matrix col ?imin? ?imax?
26 ::math::linearalgebra::setcol matrix col newvalues ?imin? ?imax?
28 ::math::linearalgebra::getelem matrix row col
30 ::math::linearalgebra::setelem matrix row ?col? newvalue
32 ::math::linearalgebra::swaprows matrix irow1 irow2 ?imin? ?imax?
34 ::math::linearalgebra::swapcols matrix icol1 icol2 ?imin? ?imax?
36 ::math::linearalgebra::show obj ?format? ?rowsep? ?colsep?
38 ::math::linearalgebra::dim obj
40 ::math::linearalgebra::shape obj
42 ::math::linearalgebra::conforming type obj1 obj2
44 ::math::linearalgebra::symmetric matrix ?eps?
46 ::math::linearalgebra::norm vector type
48 ::math::linearalgebra::norm_one vector
50 ::math::linearalgebra::norm_two vector
52 ::math::linearalgebra::norm_max vector ?index?
54 ::math::linearalgebra::normMatrix matrix type
56 ::math::linearalgebra::dotproduct vect1 vect2
58 ::math::linearalgebra::unitLengthVector vector
60 ::math::linearalgebra::normalizeStat mv
62 ::math::linearalgebra::axpy scale mv1 mv2
64 ::math::linearalgebra::add mv1 mv2
66 ::math::linearalgebra::sub mv1 mv2
68 ::math::linearalgebra::scale scale mv
70 ::math::linearalgebra::rotate c s vect1 vect2
72 ::math::linearalgebra::transpose matrix
74 ::math::linearalgebra::matmul mv1 mv2
76 ::math::linearalgebra::angle vect1 vect2
78 ::math::linearalgebra::crossproduct vect1 vect2
80 ::math::linearalgebra::matmul mv1 mv2
82 ::math::linearalgebra::mkIdentity size
84 ::math::linearalgebra::mkDiagonal diag
86 ::math::linearalgebra::mkRandom size
88 ::math::linearalgebra::mkTriangular size ?uplo? ?value?
90 ::math::linearalgebra::mkHilbert size
92 ::math::linearalgebra::mkDingdong size
94 ::math::linearalgebra::mkOnes size
96 ::math::linearalgebra::mkMoler size
98 ::math::linearalgebra::mkFrank size
100 ::math::linearalgebra::mkBorder size
102 ::math::linearalgebra::mkWilkinsonW+ size
104 ::math::linearalgebra::mkWilkinsonW- size
106 ::math::linearalgebra::solveGauss matrix bvect
108 ::math::linearalgebra::solvePGauss matrix bvect
110 ::math::linearalgebra::solveTriangular matrix bvect ?uplo?
112 ::math::linearalgebra::solveGaussBand matrix bvect
114 ::math::linearalgebra::solveTriangularBand matrix bvect
116 ::math::linearalgebra::determineSVD A eps
118 ::math::linearalgebra::eigenvectorsSVD A eps
120 ::math::linearalgebra::leastSquaresSVD A y qmin eps
122 ::math::linearalgebra::choleski matrix
124 ::math::linearalgebra::orthonormalizeColumns matrix
126 ::math::linearalgebra::orthonormalizeRows matrix
128 ::math::linearalgebra::dger matrix alpha x y ?scope?
130 ::math::linearalgebra::dgetrf matrix
132 ::math::linearalgebra::to_LA mv
134 ::math::linearalgebra::from_LA mv
136_________________________________________________________________
138
140 This package offers both low-level procedures and high-level algorithms
141 to deal with linear algebra problems:
142 · robust solution of linear equations or least squares problems
144 · determining eigenvectors and eigenvalues of symmetric matrices
146 · various decompositions of general matrices or matrices of a spe‐
148 cific form
149 · (limited) support for matrices in band storage, a common type of
151 sparse matrices
152 It arose as a re-implementation of Hume's LA package and the desire to
154 offer low-level procedures as found in the well-known BLAS library.
155 Matrices are implemented as lists of lists rather linear lists with
156 reserved elements, as in the original LA package, as it was found that
157 such an implementation is actually faster.
158 It is advisable, however, to use the procedures that are offered, such
160 as setrow and getrow, rather than rely on this representation explic‐
161 itly: that way it is to switch to a possibly even faster compiled
162 implementation that supports the same API.
163 Note: When using this package in combination with Tk, there may be a
165 naming conflict, as both this package and Tk define a command scale.
166 See the NAMING CONFLICT section below.
167
169 The package defines the following public procedures (several exist as
170 specialised procedures, see below):
171 Constructing matrices and vectors
173 ::math::linearalgebra::mkVector ndim value
175 Create a vector with ndim elements, each with the value value.
176 integer ndim
178 Dimension of the vector (number of components)
179 double value
181 Uniform value to be used (default: 0.0)
182 ::math::linearalgebra::mkUnitVector ndim ndir
185 Create a unit vector in ndim-dimensional space, along the ndir-
186 th direction.
187 integer ndim
189 Dimension of the vector (number of components)
190 integer ndir
192 Direction (0, ..., ndim-1)
193 ::math::linearalgebra::mkMatrix nrows ncols value
196 Create a matrix with nrows rows and ncols columns. All elements
197 have the value value.
198 integer nrows
200 Number of rows
201 integer ncols
203 Number of columns
204 double value
206 Uniform value to be used (default: 0.0)
207 ::math::linearalgebra::getrow matrix row ?imin? ?imax?
210 Returns a single row of a matrix as a list
211 list matrix
213 Matrix in question
214 integer row
216 Index of the row to return
217 integer imin
219 Minimum index of the column (default: 0)
220 integer imax
222 Maximum index of the column (default: ncols-1)
223 ::math::linearalgebra::setrow matrix row newvalues ?imin? ?imax?
226 Set a single row of a matrix to new values (this list must have
227 the same number of elements as the number of columns in the
228 matrix)
229 list matrix
231 name of the matrix in question
232 integer row
234 Index of the row to update
235 list newvalues
237 List of new values for the row
238 integer imin
240 Minimum index of the column (default: 0)
241 integer imax
243 Maximum index of the column (default: ncols-1)
244 ::math::linearalgebra::getcol matrix col ?imin? ?imax?
247 Returns a single column of a matrix as a list
248 list matrix
250 Matrix in question
251 integer col
253 Index of the column to return
254 integer imin
256 Minimum index of the row (default: 0)
257 integer imax
259 Maximum index of the row (default: nrows-1)
260 ::math::linearalgebra::setcol matrix col newvalues ?imin? ?imax?
263 Set a single column of a matrix to new values (this list must
264 have the same number of elements as the number of rows in the
265 matrix)
266 list matrix
268 name of the matrix in question
269 integer col
271 Index of the column to update
272 list newvalues
274 List of new values for the column
275 integer imin
277 Minimum index of the row (default: 0)
278 integer imax
280 Maximum index of the row (default: nrows-1)
281 ::math::linearalgebra::getelem matrix row col
284 Returns a single element of a matrix/vector
285 list matrix
287 Matrix or vector in question
288 integer row
290 Row of the element
291 integer col
293 Column of the element (not present for vectors)
294 ::math::linearalgebra::setelem matrix row ?col? newvalue
297 Set a single element of a matrix (or vector) to a new value
298 list matrix
300 name of the matrix in question
301 integer row
303 Row of the element
304 integer col
306 Column of the element (not present for vectors)
307 ::math::linearalgebra::swaprows matrix irow1 irow2 ?imin? ?imax?
310 Swap two rows in a matrix completely or only a selected part
311 list matrix
313 name of the matrix in question
314 integer irow1
316 Index of first row
317 integer irow2
319 Index of second row
320 integer imin
322 Minimum column index (default: 0)
323 integer imin
325 Maximum column index (default: ncols-1)
326 ::math::linearalgebra::swapcols matrix icol1 icol2 ?imin? ?imax?
329 Swap two columns in a matrix completely or only a selected part
330 list matrix
332 name of the matrix in question
333 integer irow1
335 Index of first column
336 integer irow2
338 Index of second column
339 integer imin
341 Minimum row index (default: 0)
342 integer imin
344 Maximum row index (default: nrows-1)
345 Querying matrices and vectors
347 ::math::linearalgebra::show obj ?format? ?rowsep? ?colsep?
349 Return a string representing the vector or matrix, for easy
350 printing. (There is currently no way to print fixed sets of
351 columns)
352 list obj
354 Matrix or vector in question
355 string format
357 Format for printing the numbers (default: %6.4f)
358 string rowsep
360 String to use for separating rows (default: newline)
361 string colsep
363 String to use for separating columns (default: space)
364 ::math::linearalgebra::dim obj
367 Returns the number of dimensions for the object (either 0 for a
368 scalar, 1 for a vector and 2 for a matrix)
369 any obj
371 Scalar, vector, or matrix
372 ::math::linearalgebra::shape obj
375 Returns the number of elements in each dimension for the object
376 (either an empty list for a scalar, a single number for a vector
377 and a list of the number of rows and columns for a matrix)
378 any obj
380 Scalar, vector, or matrix
381 ::math::linearalgebra::conforming type obj1 obj2
384 Checks if two objects (vector or matrix) have conforming shapes,
385 that is if they can be applied in an operation like addition or
386 matrix multiplication.
387 string type
389 Type of check:
390 · "shape" - the two objects have the same shape (for
392 all element-wise operations)
393 · "rows" - the two objects have the same number of
395 rows (for use as A and b in a system of linear
396 equations Ax = b
397 · "matmul" - the first object has the same number of
399 columns as the number of rows of the second
400 object. Useful for matrix-matrix or matrix-vector
401 multiplication.
402 list obj1
404 First vector or matrix (left operand)
405 list obj2
407 Second vector or matrix (right operand)
408 ::math::linearalgebra::symmetric matrix ?eps?
411 Checks if the given (square) matrix is symmetric. The argument
412 eps is the tolerance.
413 list matrix
415 Matrix to be inspected
416 float eps
418 Tolerance for determining approximate equality (defaults
419 to 1.0e-8)
420 Basic operations
422 ::math::linearalgebra::norm vector type
424 Returns the norm of the given vector. The type argument can be:
425 1, 2, inf or max, respectively the sum of absolute values, the
426 ordinary Euclidean norm or the max norm.
427 list vector
429 Vector, list of coefficients
430 string type
432 Type of norm (default: 2, the Euclidean norm)
433 ::math::linearalgebra::norm_one vector
435 Returns the L1 norm of the given vector, the sum of absolute
436 values
437 list vector
439 Vector, list of coefficients
440 ::math::linearalgebra::norm_two vector
442 Returns the L2 norm of the given vector, the ordinary Euclidean
443 norm
444 list vector
446 Vector, list of coefficients
447 ::math::linearalgebra::norm_max vector ?index?
449 Returns the Linf norm of the given vector, the maximum absolute
450 coefficient
451 list vector
453 Vector, list of coefficients
454 integer index
456 (optional) if non zero, returns a list made of the maxi‐
457 mum value and the index where that maximum was found. if
458 zero, returns the maximum value.
459 ::math::linearalgebra::normMatrix matrix type
462 Returns the norm of the given matrix. The type argument can be:
463 1, 2, inf or max, respectively the sum of absolute values, the
464 ordinary Euclidean norm or the max norm.
465 list matrix
467 Matrix, list of row vectors
468 string type
470 Type of norm (default: 2, the Euclidean norm)
471 ::math::linearalgebra::dotproduct vect1 vect2
474 Determine the inproduct or dot product of two vectors. These
475 must have the same shape (number of dimensions)
476 list vect1
478 First vector, list of coefficients
479 list vect2
481 Second vector, list of coefficients
482 ::math::linearalgebra::unitLengthVector vector
485 Return a vector in the same direction with length 1.
486 list vector
488 Vector to be normalized
489 ::math::linearalgebra::normalizeStat mv
492 Normalize the matrix or vector in a statistical sense: the mean
493 of the elements of the columns of the result is zero and the
494 standard deviation is 1.
495 list mv
497 Vector or matrix to be normalized in the above sense
498 ::math::linearalgebra::axpy scale mv1 mv2
501 Return a vector or matrix that results from a "daxpy" operation,
502 that is: compute a*x+y (a a scalar and x and y both vectors or
503 matrices of the same shape) and return the result.
504 Specialised variants are: axpy_vect and axpy_mat (slightly
506 faster, but no check on the arguments)
507 double scale
509 The scale factor for the first vector/matrix (a)
510 list mv1
512 First vector or matrix (x)
513 list mv2
515 Second vector or matrix (y)
516 ::math::linearalgebra::add mv1 mv2
519 Return a vector or matrix that is the sum of the two arguments
520 (x+y)
521 Specialised variants are: add_vect and add_mat (slightly faster,
523 but no check on the arguments)
524 list mv1
526 First vector or matrix (x)
527 list mv2
529 Second vector or matrix (y)
530 ::math::linearalgebra::sub mv1 mv2
533 Return a vector or matrix that is the difference of the two
534 arguments (x-y)
535 Specialised variants are: sub_vect and sub_mat (slightly faster,
537 but no check on the arguments)
538 list mv1
540 First vector or matrix (x)
541 list mv2
543 Second vector or matrix (y)
544 ::math::linearalgebra::scale scale mv
547 Scale a vector or matrix and return the result, that is: compute
548 a*x.
549 Specialised variants are: scale_vect and scale_mat (slightly
551 faster, but no check on the arguments)
552 double scale
554 The scale factor for the vector/matrix (a)
555 list mv
557 Vector or matrix (x)
558 ::math::linearalgebra::rotate c s vect1 vect2
561 Apply a planar rotation to two vectors and return the result as
562 a list of two vectors: c*x-s*y and s*x+c*y. In algorithms you
563 can often easily determine the cosine and sine of the angle, so
564 it is more efficient to pass that information directly.
565 double c
567 The cosine of the angle
568 double s
570 The sine of the angle
571 list vect1
573 First vector (x)
574 list vect2
576 Seocnd vector (x)
577 ::math::linearalgebra::transpose matrix
580 Transpose a matrix
581 list matrix
583 Matrix to be transposed
584 ::math::linearalgebra::matmul mv1 mv2
587 Multiply a vector/matrix with another vector/matrix. The result
588 is a matrix, if both x and y are matrices or both are vectors,
589 in which case the "outer product" is computed. If one is a vec‐
590 tor and the other is a matrix, then the result is a vector.
591 list mv1
593 First vector/matrix (x)
594 list mv2
596 Second vector/matrix (y)
597 ::math::linearalgebra::angle vect1 vect2
600 Compute the angle between two vectors (in radians)
601 list vect1
603 First vector
604 list vect2
606 Second vector
607 ::math::linearalgebra::crossproduct vect1 vect2
610 Compute the cross product of two (three-dimensional) vectors
611 list vect1
613 First vector
614 list vect2
616 Second vector
617 ::math::linearalgebra::matmul mv1 mv2
620 Multiply a vector/matrix with another vector/matrix. The result
621 is a matrix, if both x and y are matrices or both are vectors,
622 in which case the "outer product" is computed. If one is a vec‐
623 tor and the other is a matrix, then the result is a vector.
624 list mv1
626 First vector/matrix (x)
627 list mv2
629 Second vector/matrix (y)
630 Common matrices and test matrices
632 ::math::linearalgebra::mkIdentity size
634 Create an identity matrix of dimension size.
635 integer size
637 Dimension of the matrix
638 ::math::linearalgebra::mkDiagonal diag
641 Create a diagonal matrix whose diagonal elements are the ele‐
642 ments of the vector diag.
643 list diag
645 Vector whose elements are used for the diagonal
646 ::math::linearalgebra::mkRandom size
649 Create a square matrix whose elements are uniformly distributed
650 random numbers between 0 and 1 of dimension size.
651 integer size
653 Dimension of the matrix
654 ::math::linearalgebra::mkTriangular size ?uplo? ?value?
657 Create a triangular matrix with non-zero elements in the upper
658 or lower part, depending on argument uplo.
659 integer size
661 Dimension of the matrix
662 string uplo
664 Fill the upper (U) or lower part (L)
665 double value
667 Value to fill the matrix with
668 ::math::linearalgebra::mkHilbert size
671 Create a Hilbert matrix of dimension size. Hilbert matrices are
672 very ill-conditioned with respect to eigenvalue/eigenvector
673 problems. Therefore they are good candidates for testing the
674 accuracy of algorithms and implementations.
675 integer size
677 Dimension of the matrix
678 ::math::linearalgebra::mkDingdong size
681 Create a "dingdong" matrix of dimension size. Dingdong matrices
682 are imprecisely represented, but have the property of being very
683 stable in such algorithms as Gauss elimination.
684 integer size
686 Dimension of the matrix
687 ::math::linearalgebra::mkOnes size
690 Create a square matrix of dimension size whose entries are all
691 1.
692 integer size
694 Dimension of the matrix
695 ::math::linearalgebra::mkMoler size
698 Create a Moler matrix of size size. (Moler matrices have a very
699 simple Choleski decomposition. It has one small eigenvalue and
700 it can easily upset elimination methods for systems of linear
701 equations.)
702 integer size
704 Dimension of the matrix
705 ::math::linearalgebra::mkFrank size
708 Create a Frank matrix of size size. (Frank matrices are fairly
709 well-behaved matrices)
710 integer size
712 Dimension of the matrix
713 ::math::linearalgebra::mkBorder size
716 Create a bordered matrix of size size. (Bordered matrices have a
717 very low rank and can upset certain specialised algorithms.)
718 integer size
720 Dimension of the matrix
721 ::math::linearalgebra::mkWilkinsonW+ size
724 Create a Wilkinson W+ of size size. This kind of matrix has
725 pairs of eigenvalues that are very close together. Usually the
726 order (size) is odd.
727 integer size
729 Dimension of the matrix
730 ::math::linearalgebra::mkWilkinsonW- size
733 Create a Wilkinson W- of size size. This kind of matrix has
734 pairs of eigenvalues with opposite signs, when the order (size)
735 is odd.
736 integer size
738 Dimension of the matrix
739 Common algorithms
741 ::math::linearalgebra::solveGauss matrix bvect
743 Solve a system of linear equations (Ax=b) using Gauss elimina‐
744 tion. Returns the solution (x) as a vector or matrix of the
745 same shape as bvect.
746 list matrix
748 Square matrix (matrix A)
749 list bvect
751 Vector or matrix whose columns are the individual b-vec‐
752 tors
753 ::math::linearalgebra::solvePGauss matrix bvect
755 Solve a system of linear equations (Ax=b) using Gauss elimina‐
756 tion with partial pivoting. Returns the solution (x) as a vector
757 or matrix of the same shape as bvect.
758 list matrix
760 Square matrix (matrix A)
761 list bvect
763 Vector or matrix whose columns are the individual b-vec‐
764 tors
765 ::math::linearalgebra::solveTriangular matrix bvect ?uplo?
768 Solve a system of linear equations (Ax=b) by backward substitu‐
769 tion. The matrix is supposed to be upper-triangular.
770 list matrix
772 Lower or upper-triangular matrix (matrix A)
773 list bvect
775 Vector or matrix whose columns are the individual b-vec‐
776 tors
777 string uplo
779 Indicates whether the matrix is lower-triangular (L) or
780 upper-triangular (U). Defaults to "U".
781 ::math::linearalgebra::solveGaussBand matrix bvect
783 Solve a system of linear equations (Ax=b) using Gauss elimina‐
784 tion, where the matrix is stored as a band matrix (cf. STORAGE).
785 Returns the solution (x) as a vector or matrix of the same shape
786 as bvect.
787 list matrix
789 Square matrix (matrix A; in band form)
790 list bvect
792 Vector or matrix whose columns are the individual b-vec‐
793 tors
794 ::math::linearalgebra::solveTriangularBand matrix bvect
797 Solve a system of linear equations (Ax=b) by backward substitu‐
798 tion. The matrix is supposed to be upper-triangular and stored
799 in band form.
800 list matrix
802 Upper-triangular matrix (matrix A)
803 list bvect
805 Vector or matrix whose columns are the individual b-vec‐
806 tors
807 ::math::linearalgebra::determineSVD A eps
810 Determines the Singular Value Decomposition of a matrix: A = U S
811 Vtrans. Returns a list with the matrix U, the vector of singu‐
812 lar values S and the matrix V.
813 list A Matrix to be decomposed
815 float eps
817 Tolerance (defaults to 2.3e-16)
818 ::math::linearalgebra::eigenvectorsSVD A eps
821 Determines the eigenvectors and eigenvalues of a real symmetric
822 matrix, using SVD. Returns a list with the matrix of normalized
823 eigenvectors and their eigenvalues.
824 list A Matrix whose eigenvalues must be determined
826 float eps
828 Tolerance (defaults to 2.3e-16)
829 ::math::linearalgebra::leastSquaresSVD A y qmin eps
832 Determines the solution to a least-sqaures problem Ax ~ y via
833 singular value decomposition. The result is the vector x.
834 Note that if you add a column of 1s to the matrix, then this
836 column will represent a constant like in: y = a*x1 + b*x2 + c.
837 To force the intercept to be zero, simply leave it out.
838 list A Matrix of independent variables
840 list y List of observed values
842 float qmin
844 Minimum singular value to be considered (defaults to 0.0)
845 float eps
847 Tolerance (defaults to 2.3e-16)
848 ::math::linearalgebra::choleski matrix
851 Determine the Choleski decomposition of a symmetric positive
852 semidefinite matrix (this condition is not checked!). The result
853 is the lower-triangular matrix L such that L Lt = matrix.
854 list matrix
856 Matrix to be decomposed
857 ::math::linearalgebra::orthonormalizeColumns matrix
860 Use the modified Gram-Schmidt method to orthogonalize and nor‐
861 malize the columns of the given matrix and return the result.
862 list matrix
864 Matrix whose columns must be orthonormalized
865 ::math::linearalgebra::orthonormalizeRows matrix
868 Use the modified Gram-Schmidt method to orthogonalize and nor‐
869 malize the rows of the given matrix and return the result.
870 list matrix
872 Matrix whose rows must be orthonormalized
873 ::math::linearalgebra::dger matrix alpha x y ?scope?
876 Perform the rank 1 operation A + alpha*x*y' inline (that is: the
877 matrix A is adjusted). For convenience the new matrix is also
878 returned as the result.
879 list matrix
881 Matrix whose rows must be adjusted
882 double alpha
884 Scale factor
885 list x A column vector
887 list y A column vector
889 list scope
891 If not provided, the operation is performed on all
892 rows/columns of A if provided, it is expected to be the
893 list {imin imax jmin jmax} where:
894 · imin Minimum row index
896 · imax Maximum row index
898 · jmin Minimum column index
900 · jmax Maximum column index
902 ::math::linearalgebra::dgetrf matrix
905 Computes an LU factorization of a general matrix, using partial,
906 pivoting with row interchanges. Returns the permutation vector.
907 The factorization has the form
909 P * A = L * U
911 where P is a permutation matrix, L is lower triangular with unit
913 diagonal elements, and U is upper triangular.
914 list matrix
916 On entry, the matrix to be factored. On exit, the fac‐
917 tors L and U from the factorization P*A = L*U; the unit
918 diagonal elements of L are not stored.
919 Compability with the LA package
921 ::math::linearalgebra::to_LA mv
923 Transforms a vector or matrix into the format used by the origi‐
924 nal LA package.
925 list mv
927 Matrix or vector
928 ::math::linearalgebra::from_LA mv
930 Transforms a vector or matrix from the format used by the origi‐
931 nal LA package into the format used by the present implementa‐
932 tion.
933 list mv
935 Matrix or vector as used by the LA package
936
938 While most procedures assume that the matrices are given in full form,
939 the procedures solveGaussBand and solveTriangularBand assume that the
940 matrices are stored as band matrices. This common type of "sparse"
941 matrices is related to ordinary matrices as follows:
942 · "A" is a full-size matrix with N rows and M columns.
944 · "B" is a band matrix, with m upper and lower diagonals and n
946 rows.
947 · "B" can be stored in an ordinary matrix of (2m+1) columns (one
949 for each off-diagonal and the main diagonal) and n rows.
950 · Element i,j (i = -m,...,m; j =1,...,n) of "B" corresponds to
952 element k,j of "A" where k = M+i-1 and M is at least (!) n, the
953 number of rows in "B".
954 · To set element (i,j) of matrix "B" use:
956 setelem B $j [expr {$N+$i-1}] $value
958 (There is no convenience procedure for this yet)
961
963 There is a difference between the original LA package by Hume and the
964 current implementation. Whereas the LA package uses a linear list, the
965 current package uses lists of lists to represent matrices. It turns out
966 that with this representation, the algorithms are faster and easier to
967 implement.
968 The LA package was used as a model and in fact the implementation of,
970 for instance, the SVD algorithm was taken from that package. The set of
971 procedures was expanded using ideas from the well-known BLAS library
972 and some algorithms were updated from the second edition of J.C. Nash's
973 book, Compact Numerical Methods for Computers, (Adam Hilger, 1990) that
974 inspired the LA package.
975 Two procedures are provided to make the transition between the two
977 implementations easier: to_LA and from_LA. They are described above.
978
980 Odds and ends: the following algorithms have not been implemented yet:
981 · determineQR
983 · certainlyPositive, diagonallyDominant
985
987 If you load this package in a Tk-enabled shell like wish, then the com‐
988 mand
989 namespace import ::math::linearalgebra
990 results in an error message about "scale". This is due to the fact that
991 Tk defines all its commands in the global namespace. The solution is to
992 import the linear algebra commands in a namespace that is not the
993 global one:
994 package require math::linearalgebra
996 namespace eval compute {
997 namespace import ::math::linearalgebra::*
998 ... use the linear algebra version of scale ...
999 }
1000 To use Tk's scale command in that same namespace you can rename it:
1002 namespace eval compute {
1004 rename ::scale scaleTk
1005 scaleTk .scale ...
1006 }
1007
1010 This document, and the package it describes, will undoubtedly contain
1011 bugs and other problems. Please report such in the category math ::
1012 linearalgebra of the Tcllib SF Trackers [http://source‐
1013 forge.net/tracker/?group_id=12883]. Please also report any ideas for
1014 enhancements you may have for either package and/or documentation.
1015
1017 least squares, linear algebra, linear equations, math, matrices, vec‐
1018 tors
1019
1021 Copyright (c) 2004-2008 Arjen Markus <arjenmarkus@users.sourceforge.net>
1022 Copyright (c) 2004 Ed Hume <http://www.hume.com/contact.us.htm>
1023 Copyright (c) 2008 Michael Buadin <relaxkmike@users.sourceforge.net>
1024math 1.1 math::linearalgebra(n)