import Miranda 2.066 from upstream

1 files changed, 84 insertions, 0 deletions

diff --git a/miralib/ex/genmat.m b/miralib/ex/genmat.m
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+||The matrix package again, but this time parameterised over an arbitrary
+||element type, with a zero, a unit and four functions of arithmetic.
+||Example - to instantiate this package with numbers as the element type,
+||in another script, say:-
+||    %include <ex/genmat> { elem==num; zero=0; unit=1; 
+||                           plus=+; minus=-; times=*; divide=/; }
+
+||However another possibility would be to use the package to do matrix
+||calculations over rationals (as defined in <ex/rat>) thus:-
+||    %include <ex/genmat>
+||             { elem==rational; zero=mkrat 0; unit=mkrat 1; 
+||               plus=rplus; minus=rminus; times=rtimes; divide=rdiv; }
+
+%export matrix idmat matadd matsub matmult prescalmult postscalmult
+	mkrow mkcol det adjoint inv
+
+%free { elem::type; zero,unit::elem; 
+	plus,minus,times,divide::elem->elem->elem;
+      }
+
+matrix == [[elem]]
+
+idmat :: num->matrix ||identity matrix of given size
+idmat n = [[delta i j|j<-[1..n]]|i<-[1..n]]
+          where
+	  delta i j = unit, if i=j
+		    = zero, otherwise
+
+matadd :: matrix->matrix->matrix
+matadd x y = map2 vadd x y
+             where
+	     vadd x y = map2 plus x y
+
+matsub :: matrix->matrix->matrix
+matsub x y = map2 vsub x y
+             where
+	     vsub = map2 minus
+
+matmult :: matrix->matrix->matrix
+matmult x y = outer inner x (transpose y)  ||*
+inner x y = summate (map2 times x y)
+outer f x y = [[f a b|b<-y]|a<-x]
+
+||*note that transpose is already defined in the standard environment
+
+summate = foldl plus zero
+
+prescalmult :: elem->matrix->matrix ||premultiply a matrix by a scalar
+prescalmult n x = map (map (times n)) x
+
+postscalmult :: elem->matrix->matrix ||postmultiply a matrix by a scalar
+postscalmult n x = map (map ($times n)) x
+
+||we need both the above because element multiplication may not be
+||commutative
+
+mkrow :: [elem]->matrix ||make vector into matrix with a single row
+mkrow x = [x]
+
+mkcol :: [elem]->matrix ||make vector into matrix with a single column
+mkcol x = map (:[]) x
+
+det :: matrix->elem ||determinant, of square matrix
+det [[a]] = a
+det xs = summate [(xs!0!i) $times cofactor 0 i xs|i<-index xs], if #xs=#xs!0
+       = error "det of nonsquare matrix", otherwise
+cofactor i j xs = parity (i+j) $times det (minor i j xs)
+minor i j xs = [omit j x | x<-omit i xs]
+omit i x = take i x ++ drop (i+1) x
+
+parity::num->elem
+parity i = unit, if i mod 2 = 0
+         = zero $minus unit, otherwise
+
+adjoint :: matrix->matrix ||adjoint, of square matrix
+adjoint xs = transpose[[cofactor i j xs | j<-index xs] | i <- index xs]
+
+inv :: matrix->matrix ||inverse, of non-singular square matrix
+inv xs = transpose[[cofactor i j xs $divide h | j<-index xs] | i <- index xs]
+	 where
+	 h = det xs
+||The above is a literal transcription of the mathematical definition of
+||matrix inverse.  A less naive version of the package would rewrite 
+||this to use Gaussian elimination.