Documentation can be found at http://tcllib.sourceforge.net/doc/math.html * http://tcllib.sourceforge.net/doc/bigfloat.html * http://tcllib.sourceforge.net/doc/bignum.html * http://tcllib.sourceforge.net/doc/calculus.html * http://tcllib.sourceforge.net/doc/combinatorics.html * http://tcllib.sourceforge.net/doc/constants.html * http://tcllib.sourceforge.net/doc/fourier.html * http://tcllib.sourceforge.net/doc/fuzzy.html * http://tcllib.sourceforge.net/doc/geometry.html * http://tcllib.sourceforge.net/doc/interpolate.html * http://tcllib.sourceforge.net/doc/linalg.html * http://tcllib.sourceforge.net/doc/optimize.html * http://tcllib.sourceforge.net/doc/polynomials.html * http://tcllib.sourceforge.net/doc/qcomplex.html * http://tcllib.sourceforge.net/doc/rational_funcs.html * http://tcllib.sourceforge.net/doc/roman.html * http://tcllib.sourceforge.net/doc/romberg.html * http://tcllib.sourceforge.net/doc/special.html This package is a part of the [Tcllib] distribution. It contains: * ::math::cov -- coefficient of variation of 3 or more arguments * ::math::fibonacci -- Return the nth Fibonacci number * ::math::integrate -- calculate the area under a curve defined by an argument of a list of 5 or more x,y data pairs * ::math::max -- Return the maximum of two or more arguments * ::math::mean -- Return the arithmetic mean of two or more arguments * ::math::min -- Return the minimum of two or more arguments * ::math::product -- Return the product of two or more arguments * ::math::random -- Return a random number, value chosen based on an argument selecting one of these ranges: (null) choose a number between 0 and 1 val choose a number between 0 and val val1 val2 choose a number between val1 and val2 * ::[math::roman] -- Handling of [Roman Numerals]. * ::math::sigma -- population standard deviation value (from 3 or more arguments) * ::math::stats -- calculate mean, standard deviation, and coefficient of variation from 3 or more arguments. * ::math::sum -- calculate arithmetic sum of two or more arguments Things to keep in mind: * http://www.speech.kth.se/~beskow/tcl/ has commands for [matrix] and [vector] operations. * Also note [tcllib]/[struct]/[matrix], * [nap] (and its overview [http://tcl-nap.sourceforge.net/overview.html]), * [VKIT], * and the [BLT] [vector] functionality. * [Additional math functions] is the corresponding Wiki sandbox. If you pick tcllib up from its CVS repository, you will be able to get access to even more functionality: * ::[math::bigfloat]::abs, acos, add, addInt2Float, asin, atan, ceil, ... * ::[math::bignum]::cmp, tostr, add, div, fromstr, max, min, mul, abs, iszero, setsign, ... * ::[math::calculus]::eulerStep, heunStep, rungeKuttaStep, ... * ::[math::combinatorics] Beta, factorialList, pascal, InitializeFactorial, InitializePascal, choose, ln_Gamma, factorial * ::math::complexnumbers::complex, +, -, *, / mod, pow, real, sin, sqrt, ... * ::math::constants::constants::pi e, ln10 onethird eps print-constants, find_huge, find_tiny * ::math::fourier::dft, ... * ::math::geometry::calculateDistanceToLine, findClosestPointOnLine, angle, areaPolygon, bbox, calculateDistanceToLine, ... * ::math::interpolate::interp-linear, neville, ... * ::math::linearalgebra::angle, ... * ::math::cov, expectDouble, expectInteger, fibonacci, integrate, max, mean, min, product, random, sigma, stats, sum, ... * ::math::optimize::minimum, maximum, min_bound_1d, ... * ::math::polynomials::polynomial, addPolyn, subPolyn, ... * ::math::rationalnumbers::rationalFunction, ratioCmd, evalRatio, ... * ::math::special::elliptic_K, ... * ::math::special::exponential_Ei, ... * ::math::special::Gamma, ... * ::math::special::I_n, J-1/2, J0, J1, J1/2, Jn ... * ::math::special::legendre, ... * ::math::statistics::BasicStats, histogram, corr, ... * ::math::statistics::filter, map, samplescount, ... * ::math::statistics::Inverse-cdf-exponential, cdf-normal, ... * ::math::statistics::pdf-normal, pdf-uniform, ... * ::math::statistics::plot-scale, plot-xydata, ... * ::math::statistics::random-exponential, random-normal, ... ---- [FF] - 2007-06-11 - I implemented some missing features of '''math::linearalgebra'''. (If they are going to be included in tcllib -I hope for that- you can delete this comment.) [LV] - 2007 June 11 To get code included in tcllib, visit http://tcllib.sf.net/ and submit a '''feature request''' from the sf.net project page. ---- package require math::linearalgebra namespace eval ::math::linearalgebra { namespace export joinMatrix cancelrow cancelcol namespace export cofactor cofactorMatrix namespace export adjointMatrix namespace export determinant invert rank namespace export mkRandom namespace export round } # joinMatrix -- # Join two matrices along the specified dimension # Arguments: # dim Dimension to join [1,2] # mv1,mv2 Matrices to be treated # # Result: # Matrix result of the join # proc ::math::linearalgebra::joinMatrix { dim mv1 mv2 } { set result $mv1 switch -exact -- $dim { 1 { foreach r $mv2 {lappend $result $r} } 2 { set l [llength $mv1] for {set i 0} {$i < $l} {incr i} { lset result $i [concat [lindex $mv1 $i] [lindex $mv2 $i]] } } default { return -code error "do we work on ${dim}D matrices?" } } return $result } # cancelrow -- # Return a matrix minus the specified row # Arguments: # matrix Matrix to work on # row to delete # # Result: # Matrix result of the operation # proc ::math::linearalgebra::cancelrow { matrix row } { return [concat [lrange $matrix 0 $row-1] [lrange $matrix $row+1 end]] } # cancelrow -- # Return a matrix minus the specified column # Arguments: # matrix Matrix to work on # col to delete # # Result: # Matrix result of the operation # proc ::math::linearalgebra::cancelcol { matrix col } { set result {} foreach r $matrix { lappend result [concat [lrange $r 0 $col-1] [lrange $r $col+1 end]] } return $result } # cofactor -- # Compute the cofactor of the specified element # Arguments: # matrix Matrix to work on # row # col # # Result: # The cofactor of element at a{row,col} # proc ::math::linearalgebra::cofactor { matrix row col } { set submatrix [cancelcol [cancelrow $matrix $row] $col] return [determinant $submatrix] } # cofactorMatrix -- # Compute the cofactor of the specified matrix # Arguments: # matrix Matrix to work on # # Result: # The cofactor of specified matrix # proc ::math::linearalgebra::cofactorMatrix { matrix } { set result {} lassign [shape $matrix] rows cols for {set i 0} {$i < $rows} {incr i} { set newrow {} for {set j 0} {$j < $cols} {incr j} { lappend newrow [expr ((($i+$j)%2)==0?1:-1)*[cofactor $matrix $i $j]] } lappend result $newrow } return $result } # adjointMatrix -- # The adjoint matrix is the transpose of the cofactor matrix # Arguments: # matrix # # Result: # The adjoint matrix # proc ::math::linearalgebra::adjointMatrix {matrix} { return [transpose [cofactorMatrix $matrix]] } # determinant -- # Compute the cofactor of the specified element # Arguments: # matrix Matrix to work on # row # col # # Result: # The cofactor of element at a{row,col} # proc ::math::linearalgebra::determinant { matrix } { set shape [shape $matrix] if { [lindex $shape 0] != [lindex $shape 1] } { return -code error "determinant only defined for a square matrix" } switch -exact -- [join $shape x] { 1x1 { return [lindex [lindex $matrix 0] 0] } 2x2 { lassign [getrow $matrix 0] a b lassign [getrow $matrix 1] c d return [expr {($a*$d)-($c*$b)}] } default { set det 0 set sign 0 set row_no 0 foreach row $matrix { set sign [expr {($sign==1)?(-1):(1)}] set det [expr { $det+$sign*[lindex $row 0]*[cofactor $matrix $row_no 0] }] incr row_no } return $det } } } # invert -- # Perform the matrix inversion using the adjoint method # Note: probably the Gauss-Jordan elimination over an # augmented matrix is *much* faster. # Arguments: # matrix Matrix to work on # # Result: # The inverted matrix # proc ::math::linearalgebra::invert { matrix } { set det [determinant $matrix] if { $det == 0 } { return -code error "cannot invert a singular matrix" } return [scale_mat [expr 1./$det] [adjointMatrix $matrix]] } Useful for testing: proc ::math::linearalgebra::mkRandom { rows {cols {}} args} { if {$cols == {}} {set cols $rows} set min 0 set max 1 if {[llength $args] == 1} { set max $args } elseif {[llength $args] == 2} { lassign $args min max } set result {} for {set i 0} {$i < $rows} {incr i} { set row {} for {set j 0} {$j < $cols} {incr j} { lappend row [expr {$min+$max*rand()}] } lappend result $row } return $result } proc ::math::linearalgebra::round { matrix } { set result {} foreach row $matrix { set newrow {} foreach el $row { lappend newrow [expr {round($el)}] } lappend result $newrow } return $result } Also I believe these should go in '''math::combinatorics''': proc ::math::combinations {n k} { set l [list] for {set i 0} {$i < $n} {incr i} {lappend l $i} return [lcombinations $l $k] } proc ::math::lcombinations {list size} { if {$size == 0} {return [list [list]]} set retval {} for {set i 0} {($i + $size) <= [llength $list]} {incr i} { set firstElement [lindex $list $i] set remainingElements [lrange $list [expr {$i+1}] end] foreach subset [lcombinations $remainingElements [expr {$size-1}]] { lappend retval [linsert $subset 0 $firstElement] } } return $retval } Here's a possible implementation of rank: proc ::math::linearalgebra::rank { matrix {checkR {}} } { lassign [shape $matrix] rows cols if {$checkR == {}} {set checkR [expr ($cols>$rows)?$rows:$cols]} if {$checkR == 1} {return 1} set col_idx [combinations $cols $checkR] set row_idx [combinations $rows $checkR] foreach coll $col_idx { foreach rowl $row_idx { set m {} for {set i 0} {$i < [llength $rowl]} {incr i} { set row {} for {set j 0} {$j < [llength $coll]} {incr j} { lappend row [getelem $matrix [lindex $rowl $i] [lindex $coll $j]] } lappend m $row } set det [determinant $m] if {$det != 0} {return $checkR} } } return [rank $matrix [incr checkR -1]] } [Lars H]: That's horribly inefficient (exponential time), unfortunately. The normal way of computing the rank of a matrix is rather to do some kind of factorisation that reveals the rank (e.g. LU factorisation = Gauss elimination, QR factorisation, etc.). With the available facilities, I would suggest doing a SVD decomposition and count nonzero elements of the S vector. ---- [Category Package], subset [Tcllib], [Category Mathematics]