### genmat.m

The matrix package again, but 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/rational>) 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.