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 ?1.1.5?
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::det matrix
134 ::math::linearalgebra::largesteigen matrix tolerance maxiter
136 ::math::linearalgebra::to_LA mv
138 ::math::linearalgebra::from_LA mv
140______________________________________________________________________________
142
144 This package offers both low-level procedures and high-level algorithms
145 to deal with linear algebra problems:
146 · robust solution of linear equations or least squares problems
148 · determining eigenvectors and eigenvalues of symmetric matrices
150 · various decompositions of general matrices or matrices of a spe‐
152 cific form
153 · (limited) support for matrices in band storage, a common type of
155 sparse matrices
156 It arose as a re-implementation of Hume's LA package and the desire to
158 offer low-level procedures as found in the well-known BLAS library.
159 Matrices are implemented as lists of lists rather linear lists with
160 reserved elements, as in the original LA package, as it was found that
161 such an implementation is actually faster.
162 It is advisable, however, to use the procedures that are offered, such
164 as setrow and getrow, rather than rely on this representation explic‐
165 itly: that way it is to switch to a possibly even faster compiled
166 implementation that supports the same API.
167 Note: When using this package in combination with Tk, there may be a
169 naming conflict, as both this package and Tk define a command scale.
170 See the NAMING CONFLICT section below.
171
173 The package defines the following public procedures (several exist as
174 specialised procedures, see below):
175 Constructing matrices and vectors
177 ::math::linearalgebra::mkVector ndim value
179 Create a vector with ndim elements, each with the value value.
180 integer ndim
182 Dimension of the vector (number of components)
183 double value
185 Uniform value to be used (default: 0.0)
186 ::math::linearalgebra::mkUnitVector ndim ndir
189 Create a unit vector in ndim-dimensional space, along the ndir-
190 th direction.
191 integer ndim
193 Dimension of the vector (number of components)
194 integer ndir
196 Direction (0, ..., ndim-1)
197 ::math::linearalgebra::mkMatrix nrows ncols value
200 Create a matrix with nrows rows and ncols columns. All elements
201 have the value value.
202 integer nrows
204 Number of rows
205 integer ncols
207 Number of columns
208 double value
210 Uniform value to be used (default: 0.0)
211 ::math::linearalgebra::getrow matrix row ?imin? ?imax?
214 Returns a single row of a matrix as a list
215 list matrix
217 Matrix in question
218 integer row
220 Index of the row to return
221 integer imin
223 Minimum index of the column (default: 0)
224 integer imax
226 Maximum index of the column (default: ncols-1)
227 ::math::linearalgebra::setrow matrix row newvalues ?imin? ?imax?
230 Set a single row of a matrix to new values (this list must have
231 the same number of elements as the number of columns in the
232 matrix)
233 list matrix
235 name of the matrix in question
236 integer row
238 Index of the row to update
239 list newvalues
241 List of new values for the row
242 integer imin
244 Minimum index of the column (default: 0)
245 integer imax
247 Maximum index of the column (default: ncols-1)
248 ::math::linearalgebra::getcol matrix col ?imin? ?imax?
251 Returns a single column of a matrix as a list
252 list matrix
254 Matrix in question
255 integer col
257 Index of the column to return
258 integer imin
260 Minimum index of the row (default: 0)
261 integer imax
263 Maximum index of the row (default: nrows-1)
264 ::math::linearalgebra::setcol matrix col newvalues ?imin? ?imax?
267 Set a single column of a matrix to new values (this list must
268 have the same number of elements as the number of rows in the
269 matrix)
270 list matrix
272 name of the matrix in question
273 integer col
275 Index of the column to update
276 list newvalues
278 List of new values for the column
279 integer imin
281 Minimum index of the row (default: 0)
282 integer imax
284 Maximum index of the row (default: nrows-1)
285 ::math::linearalgebra::getelem matrix row col
288 Returns a single element of a matrix/vector
289 list matrix
291 Matrix or vector in question
292 integer row
294 Row of the element
295 integer col
297 Column of the element (not present for vectors)
298 ::math::linearalgebra::setelem matrix row ?col? newvalue
301 Set a single element of a matrix (or vector) to a new value
302 list matrix
304 name of the matrix in question
305 integer row
307 Row of the element
308 integer col
310 Column of the element (not present for vectors)
311 ::math::linearalgebra::swaprows matrix irow1 irow2 ?imin? ?imax?
314 Swap two rows in a matrix completely or only a selected part
315 list matrix
317 name of the matrix in question
318 integer irow1
320 Index of first row
321 integer irow2
323 Index of second row
324 integer imin
326 Minimum column index (default: 0)
327 integer imin
329 Maximum column index (default: ncols-1)
330 ::math::linearalgebra::swapcols matrix icol1 icol2 ?imin? ?imax?
333 Swap two columns in a matrix completely or only a selected part
334 list matrix
336 name of the matrix in question
337 integer irow1
339 Index of first column
340 integer irow2
342 Index of second column
343 integer imin
345 Minimum row index (default: 0)
346 integer imin
348 Maximum row index (default: nrows-1)
349 Querying matrices and vectors
351 ::math::linearalgebra::show obj ?format? ?rowsep? ?colsep?
353 Return a string representing the vector or matrix, for easy
354 printing. (There is currently no way to print fixed sets of
355 columns)
356 list obj
358 Matrix or vector in question
359 string format
361 Format for printing the numbers (default: %6.4f)
362 string rowsep
364 String to use for separating rows (default: newline)
365 string colsep
367 String to use for separating columns (default: space)
368 ::math::linearalgebra::dim obj
371 Returns the number of dimensions for the object (either 0 for a
372 scalar, 1 for a vector and 2 for a matrix)
373 any obj
375 Scalar, vector, or matrix
376 ::math::linearalgebra::shape obj
379 Returns the number of elements in each dimension for the object
380 (either an empty list for a scalar, a single number for a vector
381 and a list of the number of rows and columns for a matrix)
382 any obj
384 Scalar, vector, or matrix
385 ::math::linearalgebra::conforming type obj1 obj2
388 Checks if two objects (vector or matrix) have conforming shapes,
389 that is if they can be applied in an operation like addition or
390 matrix multiplication.
391 string type
393 Type of check:
394 · "shape" - the two objects have the same shape (for
396 all element-wise operations)
397 · "rows" - the two objects have the same number of
399 rows (for use as A and b in a system of linear
400 equations Ax = b
401 · "matmul" - the first object has the same number of
403 columns as the number of rows of the second
404 object. Useful for matrix-matrix or matrix-vector
405 multiplication.
406 list obj1
408 First vector or matrix (left operand)
409 list obj2
411 Second vector or matrix (right operand)
412 ::math::linearalgebra::symmetric matrix ?eps?
415 Checks if the given (square) matrix is symmetric. The argument
416 eps is the tolerance.
417 list matrix
419 Matrix to be inspected
420 float eps
422 Tolerance for determining approximate equality (defaults
423 to 1.0e-8)
424 Basic operations
426 ::math::linearalgebra::norm vector type
428 Returns the norm of the given vector. The type argument can be:
429 1, 2, inf or max, respectively the sum of absolute values, the
430 ordinary Euclidean norm or the max norm.
431 list vector
433 Vector, list of coefficients
434 string type
436 Type of norm (default: 2, the Euclidean norm)
437 ::math::linearalgebra::norm_one vector
439 Returns the L1 norm of the given vector, the sum of absolute
440 values
441 list vector
443 Vector, list of coefficients
444 ::math::linearalgebra::norm_two vector
446 Returns the L2 norm of the given vector, the ordinary Euclidean
447 norm
448 list vector
450 Vector, list of coefficients
451 ::math::linearalgebra::norm_max vector ?index?
453 Returns the Linf norm of the given vector, the maximum absolute
454 coefficient
455 list vector
457 Vector, list of coefficients
458 integer index
460 (optional) if non zero, returns a list made of the maxi‐
461 mum value and the index where that maximum was found. if
462 zero, returns the maximum value.
463 ::math::linearalgebra::normMatrix matrix type
466 Returns the norm of the given matrix. The type argument can be:
467 1, 2, inf or max, respectively the sum of absolute values, the
468 ordinary Euclidean norm or the max norm.
469 list matrix
471 Matrix, list of row vectors
472 string type
474 Type of norm (default: 2, the Euclidean norm)
475 ::math::linearalgebra::dotproduct vect1 vect2
478 Determine the inproduct or dot product of two vectors. These
479 must have the same shape (number of dimensions)
480 list vect1
482 First vector, list of coefficients
483 list vect2
485 Second vector, list of coefficients
486 ::math::linearalgebra::unitLengthVector vector
489 Return a vector in the same direction with length 1.
490 list vector
492 Vector to be normalized
493 ::math::linearalgebra::normalizeStat mv
496 Normalize the matrix or vector in a statistical sense: the mean
497 of the elements of the columns of the result is zero and the
498 standard deviation is 1.
499 list mv
501 Vector or matrix to be normalized in the above sense
502 ::math::linearalgebra::axpy scale mv1 mv2
505 Return a vector or matrix that results from a "daxpy" operation,
506 that is: compute a*x+y (a a scalar and x and y both vectors or
507 matrices of the same shape) and return the result.
508 Specialised variants are: axpy_vect and axpy_mat (slightly
510 faster, but no check on the arguments)
511 double scale
513 The scale factor for the first vector/matrix (a)
514 list mv1
516 First vector or matrix (x)
517 list mv2
519 Second vector or matrix (y)
520 ::math::linearalgebra::add mv1 mv2
523 Return a vector or matrix that is the sum of the two arguments
524 (x+y)
525 Specialised variants are: add_vect and add_mat (slightly faster,
527 but no check on the arguments)
528 list mv1
530 First vector or matrix (x)
531 list mv2
533 Second vector or matrix (y)
534 ::math::linearalgebra::sub mv1 mv2
537 Return a vector or matrix that is the difference of the two
538 arguments (x-y)
539 Specialised variants are: sub_vect and sub_mat (slightly faster,
541 but no check on the arguments)
542 list mv1
544 First vector or matrix (x)
545 list mv2
547 Second vector or matrix (y)
548 ::math::linearalgebra::scale scale mv
551 Scale a vector or matrix and return the result, that is: compute
552 a*x.
553 Specialised variants are: scale_vect and scale_mat (slightly
555 faster, but no check on the arguments)
556 double scale
558 The scale factor for the vector/matrix (a)
559 list mv
561 Vector or matrix (x)
562 ::math::linearalgebra::rotate c s vect1 vect2
565 Apply a planar rotation to two vectors and return the result as
566 a list of two vectors: c*x-s*y and s*x+c*y. In algorithms you
567 can often easily determine the cosine and sine of the angle, so
568 it is more efficient to pass that information directly.
569 double c
571 The cosine of the angle
572 double s
574 The sine of the angle
575 list vect1
577 First vector (x)
578 list vect2
580 Seocnd vector (x)
581 ::math::linearalgebra::transpose matrix
584 Transpose a matrix
585 list matrix
587 Matrix to be transposed
588 ::math::linearalgebra::matmul mv1 mv2
591 Multiply a vector/matrix with another vector/matrix. The result
592 is a matrix, if both x and y are matrices or both are vectors,
593 in which case the "outer product" is computed. If one is a vec‐
594 tor and the other is a matrix, then the result is a vector.
595 list mv1
597 First vector/matrix (x)
598 list mv2
600 Second vector/matrix (y)
601 ::math::linearalgebra::angle vect1 vect2
604 Compute the angle between two vectors (in radians)
605 list vect1
607 First vector
608 list vect2
610 Second vector
611 ::math::linearalgebra::crossproduct vect1 vect2
614 Compute the cross product of two (three-dimensional) vectors
615 list vect1
617 First vector
618 list vect2
620 Second vector
621 ::math::linearalgebra::matmul mv1 mv2
624 Multiply a vector/matrix with another vector/matrix. The result
625 is a matrix, if both x and y are matrices or both are vectors,
626 in which case the "outer product" is computed. If one is a vec‐
627 tor and the other is a matrix, then the result is a vector.
628 list mv1
630 First vector/matrix (x)
631 list mv2
633 Second vector/matrix (y)
634 Common matrices and test matrices
636 ::math::linearalgebra::mkIdentity size
638 Create an identity matrix of dimension size.
639 integer size
641 Dimension of the matrix
642 ::math::linearalgebra::mkDiagonal diag
645 Create a diagonal matrix whose diagonal elements are the ele‐
646 ments of the vector diag.
647 list diag
649 Vector whose elements are used for the diagonal
650 ::math::linearalgebra::mkRandom size
653 Create a square matrix whose elements are uniformly distributed
654 random numbers between 0 and 1 of dimension size.
655 integer size
657 Dimension of the matrix
658 ::math::linearalgebra::mkTriangular size ?uplo? ?value?
661 Create a triangular matrix with non-zero elements in the upper
662 or lower part, depending on argument uplo.
663 integer size
665 Dimension of the matrix
666 string uplo
668 Fill the upper (U) or lower part (L)
669 double value
671 Value to fill the matrix with
672 ::math::linearalgebra::mkHilbert size
675 Create a Hilbert matrix of dimension size. Hilbert matrices are
676 very ill-conditioned with respect to eigenvalue/eigenvector
677 problems. Therefore they are good candidates for testing the
678 accuracy of algorithms and implementations.
679 integer size
681 Dimension of the matrix
682 ::math::linearalgebra::mkDingdong size
685 Create a "dingdong" matrix of dimension size. Dingdong matrices
686 are imprecisely represented, but have the property of being very
687 stable in such algorithms as Gauss elimination.
688 integer size
690 Dimension of the matrix
691 ::math::linearalgebra::mkOnes size
694 Create a square matrix of dimension size whose entries are all
695 1.
696 integer size
698 Dimension of the matrix
699 ::math::linearalgebra::mkMoler size
702 Create a Moler matrix of size size. (Moler matrices have a very
703 simple Choleski decomposition. It has one small eigenvalue and
704 it can easily upset elimination methods for systems of linear
705 equations.)
706 integer size
708 Dimension of the matrix
709 ::math::linearalgebra::mkFrank size
712 Create a Frank matrix of size size. (Frank matrices are fairly
713 well-behaved matrices)
714 integer size
716 Dimension of the matrix
717 ::math::linearalgebra::mkBorder size
720 Create a bordered matrix of size size. (Bordered matrices have a
721 very low rank and can upset certain specialised algorithms.)
722 integer size
724 Dimension of the matrix
725 ::math::linearalgebra::mkWilkinsonW+ size
728 Create a Wilkinson W+ of size size. This kind of matrix has
729 pairs of eigenvalues that are very close together. Usually the
730 order (size) is odd.
731 integer size
733 Dimension of the matrix
734 ::math::linearalgebra::mkWilkinsonW- size
737 Create a Wilkinson W- of size size. This kind of matrix has
738 pairs of eigenvalues with opposite signs, when the order (size)
739 is odd.
740 integer size
742 Dimension of the matrix
743 Common algorithms
745 ::math::linearalgebra::solveGauss matrix bvect
747 Solve a system of linear equations (Ax=b) using Gauss elimina‐
748 tion. Returns the solution (x) as a vector or matrix of the
749 same shape as bvect.
750 list matrix
752 Square matrix (matrix A)
753 list bvect
755 Vector or matrix whose columns are the individual b-vec‐
756 tors
757 ::math::linearalgebra::solvePGauss matrix bvect
759 Solve a system of linear equations (Ax=b) using Gauss elimina‐
760 tion with partial pivoting. Returns the solution (x) as a vector
761 or matrix of the same shape as bvect.
762 list matrix
764 Square matrix (matrix A)
765 list bvect
767 Vector or matrix whose columns are the individual b-vec‐
768 tors
769 ::math::linearalgebra::solveTriangular matrix bvect ?uplo?
772 Solve a system of linear equations (Ax=b) by backward substitu‐
773 tion. The matrix is supposed to be upper-triangular.
774 list matrix
776 Lower or upper-triangular matrix (matrix A)
777 list bvect
779 Vector or matrix whose columns are the individual b-vec‐
780 tors
781 string uplo
783 Indicates whether the matrix is lower-triangular (L) or
784 upper-triangular (U). Defaults to "U".
785 ::math::linearalgebra::solveGaussBand matrix bvect
787 Solve a system of linear equations (Ax=b) using Gauss elimina‐
788 tion, where the matrix is stored as a band matrix (cf. STORAGE).
789 Returns the solution (x) as a vector or matrix of the same shape
790 as bvect.
791 list matrix
793 Square matrix (matrix A; in band form)
794 list bvect
796 Vector or matrix whose columns are the individual b-vec‐
797 tors
798 ::math::linearalgebra::solveTriangularBand matrix bvect
801 Solve a system of linear equations (Ax=b) by backward substitu‐
802 tion. The matrix is supposed to be upper-triangular and stored
803 in band form.
804 list matrix
806 Upper-triangular matrix (matrix A)
807 list bvect
809 Vector or matrix whose columns are the individual b-vec‐
810 tors
811 ::math::linearalgebra::determineSVD A eps
814 Determines the Singular Value Decomposition of a matrix: A = U S
815 Vtrans. Returns a list with the matrix U, the vector of singu‐
816 lar values S and the matrix V.
817 list A Matrix to be decomposed
819 float eps
821 Tolerance (defaults to 2.3e-16)
822 ::math::linearalgebra::eigenvectorsSVD A eps
825 Determines the eigenvectors and eigenvalues of a real symmetric
826 matrix, using SVD. Returns a list with the matrix of normalized
827 eigenvectors and their eigenvalues.
828 list A Matrix whose eigenvalues must be determined
830 float eps
832 Tolerance (defaults to 2.3e-16)
833 ::math::linearalgebra::leastSquaresSVD A y qmin eps
836 Determines the solution to a least-sqaures problem Ax ~ y via
837 singular value decomposition. The result is the vector x.
838 Note that if you add a column of 1s to the matrix, then this
840 column will represent a constant like in: y = a*x1 + b*x2 + c.
841 To force the intercept to be zero, simply leave it out.
842 list A Matrix of independent variables
844 list y List of observed values
846 float qmin
848 Minimum singular value to be considered (defaults to 0.0)
849 float eps
851 Tolerance (defaults to 2.3e-16)
852 ::math::linearalgebra::choleski matrix
855 Determine the Choleski decomposition of a symmetric positive
856 semidefinite matrix (this condition is not checked!). The result
857 is the lower-triangular matrix L such that L Lt = matrix.
858 list matrix
860 Matrix to be decomposed
861 ::math::linearalgebra::orthonormalizeColumns matrix
864 Use the modified Gram-Schmidt method to orthogonalize and nor‐
865 malize the columns of the given matrix and return the result.
866 list matrix
868 Matrix whose columns must be orthonormalized
869 ::math::linearalgebra::orthonormalizeRows matrix
872 Use the modified Gram-Schmidt method to orthogonalize and nor‐
873 malize the rows of the given matrix and return the result.
874 list matrix
876 Matrix whose rows must be orthonormalized
877 ::math::linearalgebra::dger matrix alpha x y ?scope?
880 Perform the rank 1 operation A + alpha*x*y' inline (that is: the
881 matrix A is adjusted). For convenience the new matrix is also
882 returned as the result.
883 list matrix
885 Matrix whose rows must be adjusted
886 double alpha
888 Scale factor
889 list x A column vector
891 list y A column vector
893 list scope
895 If not provided, the operation is performed on all
896 rows/columns of A if provided, it is expected to be the
897 list {imin imax jmin jmax} where:
898 · imin Minimum row index
900 · imax Maximum row index
902 · jmin Minimum column index
904 · jmax Maximum column index
906 ::math::linearalgebra::dgetrf matrix
909 Computes an LU factorization of a general matrix, using partial,
910 pivoting with row interchanges. Returns the permutation vector.
911 The factorization has the form
913 P * A = L * U
916 where P is a permutation matrix, L is lower triangular with unit
919 diagonal elements, and U is upper triangular. Returns the per‐
920 mutation vector, as a list of length n-1. The last entry of the
921 permutation is not stored, since it is implicitely known, with
922 value n (the last row is not swapped with any other row). At
923 index #i of the permutation is stored the index of the row #j
924 which is swapped with row #i at step #i. That means that each
925 index of the permutation gives the permutation at each step, not
926 the cumulated permutation matrix, which is the product of permu‐
927 tations.
928 list matrix
930 On entry, the matrix to be factored. On exit, the fac‐
931 tors L and U from the factorization P*A = L*U; the unit
932 diagonal elements of L are not stored.
933 ::math::linearalgebra::det matrix
936 Returns the determinant of the given matrix, based on PA=LU
937 decomposition, i.e. Gauss partial pivotal.
938 list matrix
940 Square matrix (matrix A)
941 list ipiv
943 The pivots (optionnal). If the pivots are not provided,
944 a PA=LU decomposition is performed. If the pivots are
945 provided, we assume that it contains the pivots and that
946 the matrix A contains the L and U factors, as provided by
947 dgterf. b-vectors
948 ::math::linearalgebra::largesteigen matrix tolerance maxiter
951 Returns a list made of the largest eigenvalue (in magnitude) and
952 associated eigenvector. Uses iterative Power Method as provided
953 as algorithm #7.3.3 of Golub & Van Loan. This algorithm is used
954 here for a dense matrix (but is usually used for sparse matri‐
955 ces).
956 list matrix
958 Square matrix (matrix A)
959 double tolerance
961 The relative tolerance of the eigenvalue (default:1.e-8).
962 integer maxiter
964 The maximum number of iterations (default:10).
965 Compability with the LA package Two procedures are provided for compat‐
967 ibility with Hume's LA package:
968 ::math::linearalgebra::to_LA mv
970 Transforms a vector or matrix into the format used by the origi‐
971 nal LA package.
972 list mv
974 Matrix or vector
975 ::math::linearalgebra::from_LA mv
977 Transforms a vector or matrix from the format used by the origi‐
978 nal LA package into the format used by the present implementa‐
979 tion.
980 list mv
982 Matrix or vector as used by the LA package
983
985 While most procedures assume that the matrices are given in full form,
986 the procedures solveGaussBand and solveTriangularBand assume that the
987 matrices are stored as band matrices. This common type of "sparse"
988 matrices is related to ordinary matrices as follows:
989 · "A" is a full-size matrix with N rows and M columns.
991 · "B" is a band matrix, with m upper and lower diagonals and n
993 rows.
994 · "B" can be stored in an ordinary matrix of (2m+1) columns (one
996 for each off-diagonal and the main diagonal) and n rows.
997 · Element i,j (i = -m,...,m; j =1,...,n) of "B" corresponds to
999 element k,j of "A" where k = M+i-1 and M is at least (!) n, the
1000 number of rows in "B".
1001 · To set element (i,j) of matrix "B" use:
1003 setelem B $j [expr {$N+$i-1}] $value
1006 (There is no convenience procedure for this yet)
1009
1011 There is a difference between the original LA package by Hume and the
1012 current implementation. Whereas the LA package uses a linear list, the
1013 current package uses lists of lists to represent matrices. It turns out
1014 that with this representation, the algorithms are faster and easier to
1015 implement.
1016 The LA package was used as a model and in fact the implementation of,
1018 for instance, the SVD algorithm was taken from that package. The set of
1019 procedures was expanded using ideas from the well-known BLAS library
1020 and some algorithms were updated from the second edition of J.C. Nash's
1021 book, Compact Numerical Methods for Computers, (Adam Hilger, 1990) that
1022 inspired the LA package.
1023 Two procedures are provided to make the transition between the two
1025 implementations easier: to_LA and from_LA. They are described above.
1026
1028 Odds and ends: the following algorithms have not been implemented yet:
1029 · determineQR
1031 · certainlyPositive, diagonallyDominant
1033
1035 If you load this package in a Tk-enabled shell like wish, then the com‐
1036 mand
1037 namespace import ::math::linearalgebra
1039 results in an error message about "scale". This is due to the fact that
1040 Tk defines all its commands in the global namespace. The solution is to
1041 import the linear algebra commands in a namespace that is not the
1042 global one:
1043 package require math::linearalgebra
1046 namespace eval compute {
1047 namespace import ::math::linearalgebra::*
1048 ... use the linear algebra version of scale ...
1049 }
1050 To use Tk's scale command in that same namespace you can rename it:
1052 namespace eval compute {
1055 rename ::scale scaleTk
1056 scaleTk .scale ...
1057 }
1058
1061 This document, and the package it describes, will undoubtedly contain
1062 bugs and other problems. Please report such in the category math ::
1063 linearalgebra of the Tcllib Trackers
1064 [http://core.tcl.tk/tcllib/reportlist]. Please also report any ideas
1065 for enhancements you may have for either package and/or documentation.
1066 When proposing code changes, please provide unified diffs, i.e the out‐
1068 put of diff -u.
1069 Note further that attachments are strongly preferred over inlined
1071 patches. Attachments can be made by going to the Edit form of the
1072 ticket immediately after its creation, and then using the left-most
1073 button in the secondary navigation bar.
1074
1076 least squares, linear algebra, linear equations, math, matrices,
1077 matrix, vectors
1078
1080 Mathematics
1081
1083 Copyright (c) 2004-2008 Arjen Markus <arjenmarkus@users.sourceforge.net>
1084 Copyright (c) 2004 Ed Hume <http://www.hume.com/contact.us.htm>
1085 Copyright (c) 2008 Michael Buadin <relaxkmike@users.sourceforge.net>
1086tcllib 1.1.5 math::linearalgebra(n)