diff options
Diffstat (limited to 'src/algebra/matfuns.spad.pamphlet')
| -rw-r--r-- | src/algebra/matfuns.spad.pamphlet | 48 |
1 files changed, 24 insertions, 24 deletions
diff --git a/src/algebra/matfuns.spad.pamphlet b/src/algebra/matfuns.spad.pamphlet index 59d08748..776895ee 100644 --- a/src/algebra/matfuns.spad.pamphlet +++ b/src/algebra/matfuns.spad.pamphlet @@ -66,7 +66,7 @@ InnerMatrixLinearAlgebraFunctions(R,Row,Col,M):_ -- determines if the ith row of x consists only of zeroes -- internal function: no check on index i for j in minColIndex(x)..maxColIndex(x) repeat - qelt(x,i,j) ^= 0 => return false + qelt(x,i,j) ~= 0 => return false true colAllZeroes?: (M,I) -> Boolean @@ -74,7 +74,7 @@ InnerMatrixLinearAlgebraFunctions(R,Row,Col,M):_ -- determines if the ith column of x consists only of zeroes -- internal function: no check on index j for i in minRowIndex(x)..maxRowIndex(x) repeat - qelt(x,i,j) ^= 0 => return false + qelt(x,i,j) ~= 0 => return false true rowEchelon y == @@ -87,19 +87,19 @@ InnerMatrixLinearAlgebraFunctions(R,Row,Col,M):_ for j in minC..maxC repeat i > maxR => return x n := minR - 1 - -- n = smallest k such that k >= i and x(k,j) ^= 0 + -- n = smallest k such that k >= i and x(k,j) ~= 0 for k in i..maxR repeat - if qelt(x,k,j) ^= 0 then leave (n := k) + if qelt(x,k,j) ~= 0 then leave (n := k) n = minR - 1 => "no non-zeroes" -- put nth row in ith position - if i ^= n then swapRows_!(x,i,n) + if i ~= n then swapRows_!(x,i,n) -- divide ith row by its first non-zero entry b := inv qelt(x,i,j) qsetelt_!(x,i,j,1) for k in (j+1)..maxC repeat qsetelt_!(x,i,k,b * qelt(x,i,k)) -- perform row operations so that jth column has only one 1 for k in minR..maxR repeat - if k ^= i and qelt(x,k,j) ^= 0 then + if k ~= i and qelt(x,k,j) ~= 0 then for k1 in (j+1)..maxC repeat qsetelt_!(x,k,k1,qelt(x,k,k1) - qelt(x,k,j) * qelt(x,i,k1)) qsetelt_!(x,k,j,0) @@ -160,7 +160,7 @@ InnerMatrixLinearAlgebraFunctions(R,Row,Col,M):_ qsetelt_!(w,l,1) basis := cons(w,basis) for k in minC..(j-1) for ll in minR..(l-1) repeat - if qelt(v,k) ^= minR - 1 then + if qelt(v,k) ~= minR - 1 then qsetelt_!(w,ll,-qelt(x,qelt(v,k),j)) qsetelt_!(w,l,1) basis := cons(w,basis) @@ -168,7 +168,7 @@ InnerMatrixLinearAlgebraFunctions(R,Row,Col,M):_ basis determinant y == - (ndim := nrows y) ^= (ncols y) => + (ndim := nrows y) ~= (ncols y) => error "determinant: matrix must be square" -- Gaussian Elimination ndim = 1 => qelt(y,minRowIndex y,minColIndex y) @@ -180,13 +180,13 @@ InnerMatrixLinearAlgebraFunctions(R,Row,Col,M):_ if qelt(x,i,j) = 0 then rown := minR - 1 for k in (i+1)..maxR repeat - qelt(x,k,j) ^= 0 => leave (rown := k) + qelt(x,k,j) ~= 0 => leave (rown := k) if rown = minR - 1 then return 0 swapRows_!(x,i,rown); ans := -ans ans := qelt(x,i,j) * ans; b := -inv qelt(x,i,j) for l in (j+1)..maxC repeat qsetelt_!(x,i,l,b * qelt(x,i,l)) for k in (i+1)..maxR repeat - if (b := qelt(x,k,j)) ^= 0 then + if (b := qelt(x,k,j)) ~= 0 then for l in (j+1)..maxC repeat qsetelt_!(x,k,l,qelt(x,k,l) + b * qelt(x,i,l)) qelt(x,maxR,maxC) * ans @@ -208,7 +208,7 @@ InnerMatrixLinearAlgebraFunctions(R,Row,Col,M):_ map(elt(#1,0),yy)$MATCAT22 inverse x == - (ndim := nrows x) ^= (ncols x) => + (ndim := nrows x) ~= (ncols x) => error "inverse: matrix must be square" ndim = 2 => ans2 : M := zero(ndim, ndim) @@ -457,10 +457,10 @@ MatrixLinearAlgebraFunctions(R,Row,Col,M):Exports == Implementation where ++ of first kind elRow2! : (M,R,I,I) -> M ++ elRow2!(m,a,i,j) adds to row i a*row(m,j) : elementary operation of - ++ second kind. (i ^=j) + ++ second kind. (i ~=j) elColumn2! : (M,R,I,I) -> M ++ elColumn2!(m,a,i,j) adds to column i a*column(m,j) : elementary - ++ operation of second kind. (i ^=j) + ++ operation of second kind. (i ~=j) if R has IntegralDomain then rank: M -> NonNegativeInteger @@ -502,7 +502,7 @@ MatrixLinearAlgebraFunctions(R,Row,Col,M):Exports == Implementation where -- determines if the ith row of x consists only of zeroes -- internal function: no check on index i for j in minColIndex(x)..maxColIndex(x) repeat - qelt(x,i,j) ^= 0 => return false + qelt(x,i,j) ~= 0 => return false true colAllZeroes?: (M,I) -> Boolean @@ -510,7 +510,7 @@ MatrixLinearAlgebraFunctions(R,Row,Col,M):Exports == Implementation where -- determines if the ith column of x consists only of zeroes -- internal function: no check on index j for i in minRowIndex(x)..maxRowIndex(x) repeat - qelt(x,i,j) ^= 0 => return false + qelt(x,i,j) ~= 0 => return false true minorDet:(M,I,List I,I,PrimitiveArray(Union(R,"uncomputed")))-> R @@ -521,7 +521,7 @@ MatrixLinearAlgebraFunctions(R,Row,Col,M):Exports == Implementation where j := first l; l := rest l; pos := true minR := minRowIndex x; minC := minColIndex x; repeat - if qelt(x,j + minR,i + minC) ^= 0 then + if qelt(x,j + minR,i + minC) ~= 0 then ans := md := minorDet(x,m - 2**(j :: NonNegativeInteger),_ concat_!(reverse rl,l),i + 1,v) *_ @@ -534,7 +534,7 @@ MatrixLinearAlgebraFunctions(R,Row,Col,M):Exports == Implementation where pos := not pos; rl := cons(j,rl); j := first l; l := rest l minordet x == - (ndim := nrows x) ^= (ncols x) => + (ndim := nrows x) ~= (ncols x) => error "determinant: matrix must be square" -- minor expansion with (s---loads of) memory n1 : I := ndim - 1 @@ -553,14 +553,14 @@ MatrixLinearAlgebraFunctions(R,Row,Col,M):Exports == Implementation where m -- elementary operation of second kind: add to row i-- - -- a*row j (i^=j) -- + -- a*row j (i~=j) -- elRow2!(m : M,a:R,i:I,j:I) : M == vec:= map(a*#1,row(m,j)) vec:=map("+",row(m,i),vec) setRow!(m,i,vec) m -- elementary operation of second kind: add to column i -- - -- a*column j (i^=j) -- + -- a*column j (i~=j) -- elColumn2!(m : M,a:R,i:I,j:I) : M == vec:= map(a*#1,column(m,j)) vec:=map("+",column(m,i),vec) @@ -579,7 +579,7 @@ MatrixLinearAlgebraFunctions(R,Row,Col,M):Exports == Implementation where if qelt(x,i,j) = 0 then -- candidate for pivot = 0 rown := minR - 1 for k in (i+1)..maxR repeat - if qelt(x,k,j) ^= 0 then + if qelt(x,k,j) ~= 0 then rown := k -- found a pivot leave if rown > minR - 1 then @@ -623,14 +623,14 @@ MatrixLinearAlgebraFunctions(R,Row,Col,M):Exports == Implementation where subMatrix(x,minR,maxR,maxC1,exCol) invertIfCan(y) == - (nr:=nrows y) ^= (ncols y) => + (nr:=nrows y) ~= (ncols y) => error "invertIfCan: matrix must be square" adjRec := adjoint y (den:=recip(adjRec.detMat)) case "failed" => "failed" den::R * adjRec.adjMat adjoint(y) == - (nr:=nrows y) ^= (ncols y) => error "adjoint: matrix must be square" + (nr:=nrows y) ~= (ncols y) => error "adjoint: matrix must be square" maxR := maxRowIndex y maxC := maxColIndex y x := horizConcat(copy y,scalarMatrix(nr,1$R)) @@ -668,7 +668,7 @@ MatrixLinearAlgebraFunctions(R,Row,Col,M):Exports == Implementation where nullSpace m == nullSpace(m)$IMATQF determinant y == - (nrows y) ^= (ncols y) => error "determinant: matrix must be square" + (nrows y) ~= (ncols y) => error "determinant: matrix must be square" fm:=fractionFreeGauss!(copy y) fm(maxRowIndex fm,maxColIndex fm) @@ -736,7 +736,7 @@ MatrixLinearAlgebraFunctions(R,Row,Col,M):Exports == Implementation where un := unitNormal qelt(x,i,j) qsetelt_!(x,i,j,un.canonical) - if un.associate ^= 1 then for jj in (j+1)..maxC repeat + if un.associate ~= 1 then for jj in (j+1)..maxC repeat qsetelt_!(x,i,jj,un.associate * qelt(x,i,jj)) xij := qelt(x,i,j) |