scribble-math/matrix/matrix.rkt
Jens Axel Søgaard 1ae55396e4 Inital commit
2012-06-20 17:20:30 +02:00

15102 lines
284 KiB
Racket

#reader(lib"read.ss""wxme")WXME0108 ##
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This file uses the GRacket editor format.
Open this file in DrRacket version 5.3.0.6 or later to read it.
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http://racket-lang.org/
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0 0 17 3 3 #";;;"
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0 0 17 3 8 #";;; BUGS"
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0 0 17 3 3 #";;;"
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0 0 22 29 1 #"\n"
0 0 17 3 9 #"; None..."
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0 0 17 3 9 #"; (known)"
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0 0 22 29 1 #"\n"
0 0 17 3 3 #";;;"
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0 0 17 3 8 #";;; TODO"
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0 0 17 3 3 #";;;"
0 0 22 29 1 #"\n"
0 0 22 29 1 #"\n"
0 0 17 3 37 #"; Add tolerance to various functions."
0 0 22 29 1 #"\n"
0 0 17 3 60
#"; If elements are below tolerance they are interpreted as 0."
0 0 22 29 1 #"\n"
0 0 22 29 1 #"\n"
0 0 17 3 47 #"; Partial pivoting is turned of in determinant,"
0 0 22 29 1 #"\n"
0 0 17 3 49 #"; due to the row interchanges in gauss-eliminate!"
0 0 22 29 1 #"\n"
0 0 17 3 54 #"; Find way to communicate the number of row changes to"
0 0 22 29 1 #"\n"
0 0 17 3 42 #"; determinant, so it can correct the sign."
0 0 22 29 1 #"\n"
0 0 22 29 1 #"\n"
0 0 17 3 4 #";;; "
0 0 22 29 1 #"\n"
0 0 17 3 18 #";;; DENSE MATRICES"
0 0 22 29 1 #"\n"
0 0 17 3 3 #";;;"
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0 0 22 29 1 #"\n"
0 0 17 3 59
#"; This module contains an implementation of dense matrices."
0 0 22 29 1 #"\n"
0 0 22 29 1 #"\n"
0 0 17 3 3 #";;;"
0 0 22 29 1 #"\n"
0 0 17 3 17 #";;; Documentation"
0 0 22 29 1 #"\n"
0 0 17 3 3 #";;;"
0 0 22 29 1 #"\n"
0 0 22 29 1 #"\n"
0 0 17 3 19 #"; See link to docs."
0 0 22 29 1 #"\n"
0 0 22 29 1 #"\n"
0 0 17 3 3 #";;;"
0 0 22 29 1 #"\n"
0 0 17 3 17 #";;; Project ideas"
0 0 22 29 1 #"\n"
0 0 17 3 3 #";;;"
0 0 22 29 1 #"\n"
0 0 22 29 1 #"\n"
0 0 17 3 50 #"; 1. Specialized versions of Gaussian elimination."
0 0 22 29 1 #"\n"
0 0 17 3 49 #"; - floats: use Noel's bindings for GSL ?"
0 0 22 29 1 #"\n"
0 0 17 3 17 #"; - integers"
0 0 22 29 1 #"\n"
0 0 17 3 20 #"; - expressions"
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0 0 17 3 65
#"; 2. For integer version see Geddes (two-step integer algorithm)."
0 0 22 29 1 #"\n"
0 0 17 3 67
#"; 3. The integer version can be generealized to gaussian elimantion"
0 0 22 29 1 #"\n"
0 0 17 3 53 #"; of matrices over non-fields (think polynomials)."
0 0 22 29 1 #"\n"
0 0 17 3 55 #"; 4. LUP factorization (easy - modify LU-factorization)"
0 0 22 29 1 #"\n"
0 0 17 3 27 #"; 5. Cholesky factorization"
0 0 22 29 1 #"\n"
0 0 17 3 39 #"; 6. SVD - Singular Value Decomposition"
0 0 22 29 1 #"\n"
0 0 17 3 30 #"; 7. More matrix constructors:"
0 0 22 29 1 #"\n"
0 0 17 3 10 #"; - \""
0 0 17 3 12 #"famous\" ones"
0 0 22 29 1 #"\n"
0 0 17 3 22 #"; - band matrices"
0 0 22 29 1 #"\n"
0 0 17 3 23 #"; - stack matrices"
0 0 22 29 1 #"\n"
0 0 17 3 26 #"; - rotation matrices"
0 0 22 29 1 #"\n"
0 0 17 3 28 #"; - reflection matrices"
0 0 22 29 1 #"\n"
0 0 17 3 28 #"; - permuation matrices"
0 0 22 29 1 #"\n"
0 0 17 3 28 #"; - projection matrices"
0 0 22 29 1 #"\n"
0 0 17 3 26 #"; 8. 2d and 3d-operations:"
0 0 22 29 1 #"\n"
0 0 17 3 19 #"; - hat vector"
0 0 22 29 1 #"\n"
0 0 17 3 22 #"; - cross product"
0 0 22 29 1 #"\n"
0 0 17 3 21 #"; 9. Equation solving"
0 0 22 29 1 #"\n"
0 0 17 3 61
#"; - back and forward substituion for triangular matrices"
0 0 22 29 1 #"\n"
0 0 17 3 16 #"; 10. Predicates"
0 0 22 29 1 #"\n"
0 0 17 3 32 #"; - test for different shapes"
0 0 22 29 1 #"\n"
0 0 17 3 41 #"; - test for different type of entries"
0 0 22 29 1 #"\n"
0 0 17 3 44 #"; 11. Matrix language with convenient syntax"
0 0 22 29 1 #"\n"
0 0 17 3 14 #"; 12. Generics"
0 0 22 29 1 #"\n"
0 0 17 3 48 #"; 13. Rewrite rank and nullity to use QR or SVD."
0 0 22 29 1 #"\n"
0 0 17 3 55 #"; 14. Test algorithms with matrices from Matrix Market."
0 0 22 29 1 #"\n"
0 0 17 3 40 #"; http://math.nist.gov/MatrixMarket/"
0 0 22 29 1 #"\n"
0 0 17 3 35 #"; 15. Implement SVD with ideas from"
0 0 22 29 1 #"\n"
0 0 17 3 42 #"; http://arxiv.org/pdf/0909.4061v2.pdf"
0 0 22 29 1 #"\n"
0 0 22 29 1 #"\n"
0 0 17 3 9 #";;; TESTS"
0 0 22 29 1 #"\n"
0 0 22 29 1 #"\n"
0 0 17 3 38 #"; Tests are situated next to the code."
0 0 22 29 1 #"\n"
0 0 22 29 1 #"\n"
0 0 17 3 36 #"; Run this in DrRacket and evaluate "
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0 0 17 3 8 #"\" test))"
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0 0 17 3 45 #"; in the interaction window to run the tests."
0 0 22 29 1 #"\n"
0 0 17 3 57 #"; Or customize the language in DrRacekt and add the test "
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0 0 17 3 11 #"; module to"
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(
#"; A MATRIX consists of m rows, n columns, and a vector d of m*n elem"
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0 0 17 3 56 #"; for easier to conversion to matrices over expressions."
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0 0 20 3 1 #"6"
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0 0 20 3 1 #"1"
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0 0 20 3 1 #"2"
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0 0 20 3 1 #"6"
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0 0 20 3 1 #"7"
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0 0 14 3 7 #"matrix?"
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0 0 17 3 22 #"; Return copy of M."
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0 0 14 3 1 #"M"
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0 0 14 3 1 #"M"
0 0 22 3 4 #"))))"
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0 0 22 3 1 #" "
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0 0 22 29 1 #"\n"
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0 0 17 3 31 #"; Return the transpose of M."
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0 0 14 3 1 #"x"
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0 0 22 29 1 #"\n"
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0 0 14 3 7 #"matrix?"
0 0 22 3 3 #")])"
0 0 22 29 1 #"\n"
0 0 17 3 22 #"; (matrix-conjugate M)"
0 0 22 29 1 #"\n"
0 0 17 3 31 #"; Return the conjugate of M."
0 0 22 29 1 #"\n"
0 0 17 3 54 #"; (ref (conjugate M) i j) = (conjugate (ref M j i))"
0 0 22 29 1 #"\n"
0 0 22 3 1 #"("
0 0 15 3 6 #"define"
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0 0 14 3 1 #"M"
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0 0 22 3 1 #" "
0 0 14 3 1 #"n"
0 0 22 3 3 #") ("
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0 0 14 3 1 #"M"
0 0 22 3 2 #"))"
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0 0 22 3 3 #" ("
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0 0 22 3 1 #" "
0 0 14 3 1 #"n"
0 0 22 3 3 #" (["
0 0 14 3 1 #"x"
0 0 22 3 2 #" ("
0 0 14 3 9 #"in-matrix"
0 0 22 3 1 #" "
0 0 14 3 1 #"M"
0 0 22 3 5 #")]) ("
0 0 14 3 9 #"conjugate"
0 0 22 3 1 #" "
0 0 14 3 1 #"x"
0 0 22 3 3 #")))"
0 0 22 29 1 #"\n"
0 0 22 3 1 #"("
0 0 14 3 5 #"test="
0 0 22 3 2 #" ("
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0 0 22 3 2 #" ("
0 0 14 3 7 #"matrix:"
0 0 22 3 1 #" "
0 0 20 3 1 #"1"
0 0 22 3 1 #" "
0 0 20 3 1 #"2"
0 0 22 3 2 #" "
0 0 20 3 3 #"3+i"
0 0 22 3 1 #" "
0 0 20 3 3 #"4-i"
0 0 22 3 2 #"))"
0 0 22 29 1 #"\n"
0 0 22 3 8 #" ("
0 0 14 3 7 #"matrix:"
0 0 22 3 1 #" "
0 0 20 3 1 #"1"
0 0 22 3 1 #" "
0 0 20 3 1 #"2"
0 0 22 3 2 #" "
0 0 20 3 3 #"3-i"
0 0 22 3 1 #" "
0 0 20 3 3 #"4+i"
0 0 22 3 2 #"))"
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0 0 22 29 1 #"\n"
0 0 22 29 1 #"\n"
0 0 22 3 1 #"("
0 0 14 3 16 #"provide/contract"
0 0 22 29 1 #"\n"
0 0 22 3 2 #" ["
0 0 14 3 26 #"matrix-hermitian-transpose"
0 0 22 3 2 #" ("
0 0 14 3 2 #"->"
0 0 22 3 1 #" "
0 0 14 3 7 #"matrix?"
0 0 22 3 1 #" "
0 0 14 3 7 #"matrix?"
0 0 22 3 3 #")])"
0 0 22 29 1 #"\n"
0 0 17 3 32 #"; (matrix-hermitian-transpose M)"
0 0 22 29 1 #"\n"
0 0 17 3 31 #"; Return the transpose of M."
0 0 22 29 1 #"\n"
0 0 17 3 54 #"; (ref (transpose M) i j) = (conjugate (ref M j i))"
0 0 22 29 1 #"\n"
0 0 22 3 1 #"("
0 0 15 3 6 #"define"
0 0 22 3 2 #" ("
0 0 14 3 26 #"matrix-hermitian-transpose"
0 0 22 3 1 #" "
0 0 14 3 1 #"M"
0 0 22 3 1 #")"
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0 0 22 3 3 #" ("
0 0 15 3 13 #"define-values"
0 0 22 3 2 #" ("
0 0 14 3 1 #"m"
0 0 22 3 1 #" "
0 0 14 3 1 #"n"
0 0 22 3 3 #") ("
0 0 14 3 11 #"matrix-size"
0 0 22 3 1 #" "
0 0 14 3 1 #"M"
0 0 22 3 2 #"))"
0 0 22 29 1 #"\n"
0 0 22 3 3 #" ("
0 0 15 3 11 #"for*/matrix"
0 0 22 3 1 #" "
0 0 14 3 1 #"n"
0 0 22 3 1 #" "
0 0 14 3 1 #"m"
0 0 22 3 12 #" #:column (["
0 0 14 3 1 #"x"
0 0 22 3 2 #" ("
0 0 14 3 9 #"in-matrix"
0 0 22 3 1 #" "
0 0 14 3 1 #"M"
0 0 22 3 5 #")]) ("
0 0 14 3 9 #"conjugate"
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0 0 22 29 1 #"\n"
0 0 17 3 66
#"; Make the row reducer signal back when no pivot is present."
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0 0 17 3 65
#"; At that moment, the non-existance of a solution is known."
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0 0 17 3 22 #"change-of-basis-matrix"
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0 0 17 3 27 #"; We must \"solve\": X b = y."
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#"; http://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse"
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#"; If A is of full column rank, then the QR method can be used"
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0 0 14 3 1 #"Q"
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0 0 22 3 2 #"))"
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0 0 22 3 2 #" "
0 0 17 3 17 #"; now A* A = R* R"
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0 0 22 3 2 #" "
0 0 17 3 34 #"; A+ = (A* A)^-1 <=> R* R A+ = A*"
0 0 22 29 1 #"\n"
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0 0 14 3 5 #"error"
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0 0 17 3 4 #"200."
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0 0 17 3 16 #"row-echelon-form"
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0 0 17 3 2 #"M)"
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0 0 22 29 1 #"\n"
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