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module linalg; % The Linear Algebra package. %**********************************************************************% % % % Author: Matt Rebbeck, March-July 1994 (at ZIB). % % Modifications by: Walter Tietze. % % % Please report bugs to: Winfried Neun, % % Konrad-Zuse-Zentrum % % fuer Informationstechnik Berlin % % Heilbronner Str. 10 % % 10711 Berlin - Wilmersdorf % % Federal Republic of Germany % % % % (email) neun@sc.ZIB-Berlin.de % % % % % % % % This package provides a selection of useful functions in the field % % of linear algebra: % % % % add_columns add_rows add_to_columns add_to_rows % % augment_columns band_matrix block_matrix char_matrix % % char_poly cholesky coeff_matrix column_dim % % companion copy_into diagonal extend % % get_columns get_rows gram_schmidt hermitian_tp % % hessian hilbert jacobian jordan_block % % lu_decom make_identity matrix_augment matrixp % % matrix_stack minor mult_column mult_row % % pivot pseudo_inverse random_matrix remove_columns % % remove_rows row_dim rows_pivot simplex % % squarep stack_rows sub_matrix svd % % swap_columns swap_entries swap_rows symmetricp % % toeplitz vandermonde kronecker_product % % % % % % % % The package implements the following switches: % % % % imaginary \ % % lower_matrix \ % % not_negative ) for details see the random_matrix comments. % % only_integer / % % upper_matrix / % % % % fast_la (see below). % % % % % % % % For further details about the linear algebra package see the % % linear_algebra.tex file. % % % % % % % % NB: The functions in this package are written to be used from the % % user level. Some of them may well need a bit of fiddling with to get % % them to work from symbolic mode. % % % %**********************************************************************% load_package matrix; create!-package('(linalg lamatrix gramschm ludecom cholesky svd simplex tadjoint), '(contrib linalg)); switch fast_la; % If ON, then the following functions will be faster: % add_columns add_rows augment_columns column_dim % % copy_into make_identity matrix_augment matrix_stack % % minor mult_column mult_row pivot % % remove_columns remove_rows rows_pivot squarep % % stack_rows sub_matrix swap_columns swap_entries % % swap_rows symmetricp % % This is basically done by removing some error checking and doesn't % speed things up too much. You'll need to be making alot of calls to % see the difference. If you get strange error messages with fast_la % ON then thoroughly check your input. symbolic smacro procedure my_reval(n); % % Only revals if it needs to. % if fixp(n) then n else reval(n); symbolic procedure swap_elt(in_list,elt1,elt2); % % Swaps elt elt1 with elt elt2 in in_list. % % NB: destructive. % begin scalar bucket; bucket := nth(in_list,elt1); nth(in_list,elt1) := nth(in_list,elt2); nth(in_list,elt2) := bucket; end; symbolic procedure row_dim(in_mat); % % Finds row dimension of a matrix. % begin if not !*fast_la and not matrixp(in_mat) then rederr "Error in row_dim: input should be a matrix."; return first size_of_matrix(in_mat); end; symbolic procedure column_dim(in_mat); % % Finds column dimension of a matrix. % begin if not !*fast_la and not matrixp(in_mat) then rederr "Error in column_dim: input should be a matrix."; return second size_of_matrix(in_mat); end; flag('(row_dim,column_dim),'opfn); symbolic procedure matrixp(A); % % Tests if input is a matrix (boolean). % if not eqcar(A,'mat) then nil else t; flag('(matrixp),'boolean); flag('(matrixp),'opfn); symbolic procedure size_of_matrix(A); % % Takes matrix and returns list {no. of rows, no. of columns}. % begin integer row_dim,column_dim; row_dim := -1 + length A; column_dim := length cadr A; return {row_dim,column_dim}; end; symbolic procedure companion(poly,x); % % Takes as input a monic univariate polynomial in a variable x. % Returns a companion matrix associated with the polynomial poly(x). % % If C := companion(p,x) and p is a0+a1*x+...+x^n (a univariate monic % polynomial), them C(i,n) = -coeff(p,x,i-1), C(i,i-1) = 1 (i=2..n) % and C(i,j) = 0 for all other i and j. % begin scalar mat1; integer n; n := deg(poly,x); if my_reval coeffn(poly,x,n) neq 1 then msgpri ("Error in companion(first argument): Polynomial", poly, "is not monic.",nil,t); mat1 := mkmatrix(n,n); setmat(mat1,1,n,{'minus,coeffn(poly,x,0)}); for i:=2:n do << setmat(mat1,i,i-1,1); >>; for j:=2:n do << setmat(mat1,j,n,{'minus,coeffn(poly,x,j-1)}); >>; return mat1; end; symbolic procedure find_companion(R,x); % % Given a companion matrix, find_companion will return the associated % polynomial. % begin scalar p; integer rowdim,k; if not matrixp(R) then rederr {"Error in find_companion(first argument): should be a matrix."}; rowdim := row_dim(R); k := 2; while k<=rowdim and getmat(R,k,k-1)=1 do k:=k+1; p := 0; for j:=1:k-1 do << p:={'plus,p,{'times,{'minus,getmat(R,j,k-1)},{'expt,x,j-1}}}; >>; p := {'plus,p,{'expt,x,k-1}}; return p; end; flag('(companion,find_companion),'opfn); symbolic procedure jordan_block(const,mat_dim); % % Takes a constant (const) and an integer (mat_dim) and creates % a jordan block of dimension mat_dim x mat_dim. % begin scalar JB; if not fixp mat_dim then rederr "Error in jordan_block(second argument): should be an integer."; JB := mkmatrix(mat_dim,mat_dim); for i:=1:mat_dim do << for j:=1:mat_dim do << if i=j then << setmat(JB,i,j,const); if i<mat_dim then setmat(JB,i,j+1,1); >>; >>; >>; return JB; end; flag ('(jordan_block),'opfn); symbolic procedure sub_matrix(A,row_list,col_list); % % Removes the sub_matrix from A consisting of the rows in row_list and % the columns in col_list. (Both row_list and col_list can be single % integer values). % begin scalar new_mat; if not !*fast_la and not matrixp(A) then rederr "Error in sub_matrix(first argument): should be a matrix."; new_mat := stack_rows(A,row_list); new_mat := augment_columns(new_mat,col_list); return new_mat; end; % flag('(sub_matrix),'opfn); rtypecar sub_matrix; symbolic procedure copy_into(BB,AA,p,q); % % Copies matrix BB into AA with BB(1,1) at AA(p,q). % begin scalar A,B; integer m,n,r,c; if not !*fast_la then << if not matrixp(BB) then rederr "Error in copy_into(first argument): should be a matrix."; if not matrixp(AA) then rederr "Error in copy_into(second argument): should be a matrix."; if not fixp p then rederr "Error in copy_into(third argument): should be an integer."; if not fixp q then rederr "Error in copy_into(fourth argument): should be an integer."; if p = 0 or q = 0 then << prin2t "***** Error in copy_into: 0 is out of bounds for matrices."; prin2t " The top left element is labelled (1,1) and not (0,0)."; return; >>; >>; m := row_dim(AA); n := column_dim(AA); r := row_dim(BB); c := column_dim(BB); if not !*fast_la and (r+p-1>m or c+q-1>n) then << % Only print offending matrices if they're not too big. if m*n<26 and r*c<26 then << prin2t "***** Error in copy_into: the matrix"; matpri(BB); prin2t " does not fit into"; matpri(AA); prin2 " at position "; prin2 p; prin2 ","; prin2 q; prin2t "."; return; >> else << prin2 "***** Error in copy_into: first matrix does not fit "; prin2 " into second matrix at defined position."; return; >>; >>; A := mkmatrix(m,n); B := mkmatrix(r,c); for i:=1:m do << for j:=1:n do << setmat(A,i,j,getmat(AA,i,j)); >>; >>; for i:=1:r do << for j:=1:c do << setmat(B,i,j,getmat(BB,i,j)); >>; >>; for i:=1:r do << for j:=1:c do << setmat(A,p+i-1,q+j-1,getmat(B,i,j)); >>; >>; return A; end; flag ('(copy_into),'opfn); symbolic procedure copy_mat(u); if pairp u then cons (copy_mat car u, copy_mat cdr u) else u; put('diagonal,'psopfn,'diagonal1); % To allow variable input. symbolic procedure diagonal1(mat_list); % % Can take either a list of arguments or the arguments seperately. % % Takes any number of either scalar entries or square matrices and % creates the diagonal. % begin scalar diag_mat; if pairp mat_list and pairp car mat_list and caar mat_list = 'list then mat_list := cdar mat_list; mat_list := for each elt in mat_list collect reval elt; for each elt in mat_list do << if matrixp(elt) and not squarep(elt) then << % Only print offending matrix if it's not too big. if row_dim(elt)<5 or column_dim(elt)> 5 then << prin2t "***** Error in diagonal: "; matpri(elt); prin2t " is not a square matrix."; rederr ""; >> else rederr "Error in diagonal: input contains non square matrix."; >>; >>; diag_mat := diag({mat_list}); return diag_mat; end; symbolic procedure diag(uu); % % Takes square or scalar matrix entries and creates a matrix with % these matrices on the diagonal. % % What a horrible way to do it! % begin scalar bigA,arg,input,u; integer nargs,n,Aidx,stp,bigsize,smallsize; u := car uu; input := u; bigsize:=0; nargs:=length input; for i:=1:nargs do << arg:=car input; % If scalar entry. if algebraic length(arg) = 1 or eqcar(arg,'quotient) then bigsize:=bigsize+1 else << bigsize:=bigsize+row_dim(arg); >>; input := cdr input; >>; bigA := mkmatrix(bigsize,bigsize); Aidx:=1; input := u; for k:=1:nargs do << arg:=car input; % If scalar entry. if algebraic length(arg) = 1 or eqcar(arg,'quotient) then << setmat(bigA,Aidx,Aidx,arg); Aidx:=Aidx+1; input := cdr input; >> else << smallsize:= row_dim(arg); stp:=smallsize+Aidx-1; for i:=Aidx:stp do << for j:=Aidx:stp do << arg:=car input; % Find (i-Aidx+1)'th row. arg := cdr arg; << n:=1; while n < (i-Aidx+1) do << arg := cdr arg; n:=n+1; >>; >>; arg := car arg; % % Find (j-Aidx+1)'th column elt of i'th row. % << n:=1; while n < (j-Aidx+1) do << arg := cdr arg; n:=n+1; >>; >>; arg := car arg; setmat(bigA,i,j,arg); >>; >>; Aidx := Aidx+smallsize; input := cdr input; >>; >>; return biga; end; symbolic procedure band_matrix(elt_list,sq_size); % % A square band matrix b is created. The elements of the diagonal % are the middle element of elt_list. The elements to the left are % used to fill the required number of subdiagonals and the elements % to the right the superdiagonals. % begin scalar band_matrix; integer i,j,no_elts,middle_pos; if not fixp sq_size then rederr "Error in band_matrix(second argument): should be an integer."; if atom elt_list then elt_list := {elt_list} else if car elt_list = 'list then elt_list := cdr elt_list else rederr "Error in band_matrix(first argument): should be single value or list."; no_elts := length elt_list; if evenp no_elts then rederr "Error in band matrix(first argument): number of elements must be odd."; middle_pos := reval{'quotient,no_elts+1,2}; if my_reval middle_pos > sq_size then rederr "Error in band_matrix: too many elements. Band matrix is overflowing." else band_matrix := mkmatrix(sq_size,sq_size); for i:=1:sq_size do << for j:=1:sq_size do << if middle_pos-i+j > 0 and middle_pos-i+j <= no_elts then setmat(band_matrix,i,j,nth(elt_list,middle_pos-i+j)); >>; >>; return band_matrix; end; flag('(band_matrix),'opfn); symbolic procedure make_identity(sq_size); % % Creates identity matrix. % if not !*fast_la and not fixp sq_size then rederr "Error in make_identity: non integer input." else 'mat. (for i:=1:sq_size collect for j:=1:sq_size collect if i=j then 1 else 0); flag('(make_identity),'opfn); symbolic procedure squarep(in_mat); % % Tests matrix is square. (boolean). % begin scalar tmp; if not !*fast_la and not matrixp(in_mat) then rederr "Error in squarep: non matrix input"; tmp := size_of_matrix(in_mat); if first tmp neq second tmp then return nil else return t; end; flag('(squarep),'boolean); flag('(squarep),'opfn); symbolic procedure swap_rows(in_mat,row1,row2); % % Swaps row1 with rows. % begin scalar new_mat; integer rowdim; if not !*fast_la then << if not matrixp in_mat then rederr "Error in swap_rows(first argument): should be a matrix."; rowdim := row_dim(in_mat); if not fixp row1 then rederr "Error in swap_rows(second argument): should be an integer."; if not fixp row2 then rederr "Error in swap_rows(third argument): should be an integer."; if row1>rowdim or row1=0 then rederr "Error in swap_rows(second argument): out of range for input matrix."; if row2>rowdim or row2=0 then rederr "Error in swap_rows(third argument): out of range for input matrix."; >>; new_mat := copy_mat(in_mat); swap_elt(cdr new_mat,row1,row2); return new_mat; end; symbolic procedure swap_columns(in_mat,col1,col2); % % Swaps col1 with col2. % begin scalar new_mat; integer coldim; if not !*fast_la then << if not matrixp in_mat then rederr "Error in swap_columns(first argument): should be a matrix."; coldim := column_dim(in_mat); if not fixp col1 then rederr "Error in swap_columns(second argument): should be an integer."; if not fixp col2 then rederr "Error in swap_columns(third argument): should be an integer."; if col1>coldim or col1=0 then rederr "Error in swap_columns(second argument): out of range for matrix."; if col2>coldim or col2=0 then rederr "Error in swap_columns(third argument): out of range for input matrix."; >>; new_mat := copy_mat(in_mat); for each row in cdr new_mat do swap_elt(row,col1,col2); return new_mat; end; symbolic procedure swap_entries(in_mat,entry1,entry2); % % Swaps the two entries in in_mat. % % entry1 and entry2 must be lists of the form % {row position,column position}. % begin scalar new_mat; integer rowdim,coldim; if not matrixp(in_mat) then rederr "Error in swap_entries(first argument): should be a matrix."; if atom entry1 or car entry1 neq 'list or length cdr entry1 neq 2 then rederr "Error in swap_entries(second argument): should be list of 2 elements." else entry1 := cdr entry1; if atom entry2 or car entry2 neq 'list or length cdr entry2 neq 2 then rederr "Error in swap_entries(third argument): should be a list of 2 elements." else entry2 := cdr entry2; if not !*fast_la then << rowdim := row_dim(in_mat); coldim := column_dim(in_mat); if not fixp car entry1 then << prin2 "***** Error in swap_entries(second argument): "; prin2t " first element in list must be an integer."; return; >>; if not fixp cadr entry1 then << prin2 "***** Error in swap_entries(second argument): "; prin2t " second element in list must be an integer."; return; >>; if car entry1 > rowdim or car entry1 = 0 then << prin2 "***** Error in swap_entries(second argument): "; prin2t " first element is out of range for input matrix."; return; >>; if cadr entry1 > coldim or cadr entry1 = 0 then << prin2 "***** Error in swap_entries(second argument): "; prin2t " second element is out of range for input matrix."; return; >>; if not fixp car entry2 then << prin2 "***** Error in swap_entries(third argument): "; prin2t " first element in list must be an integer."; return; >>; if not fixp cadr entry2 then << prin2 "***** Error in swap_entries(third argument): "; prin2t " second element in list must be an integer."; return; >>; if car entry2 > rowdim or car entry2 = 0 then << prin2 "***** Error in swap_entries(third argument): "; prin2t " first element is out of range for input matrix."; return; >>; if cadr entry2 > coldim then << prin2 "***** Error in swap_entries(third argument): "; prin2t " second element is out of range for input matrix."; return; >>; >>; new_mat := copy_mat(in_mat); setmat(new_mat,car entry1,cadr entry1, getmat(in_mat,car entry2,cadr entry2)); setmat(new_mat,car entry2,cadr entry2, getmat(in_mat,car entry1,cadr entry1)); return new_mat; end; % flag('(swap_rows,swap_columns,swap_entries),'opfn); rtypecar swap_rows,swap_columns,swap_entries; symbolic procedure get_rows(in_mat,row_list); % % Input is a matrix and either a single row number or a list of row % numbers. % % Extracts either a single row or a number of rows and returns them % in a list of row matrices. % begin integer rowdim,coldim; scalar ans,tmp; if not matrixp(in_mat) then rederr "Error in get_rows(first argument): should be a matrix."; if atom row_list then row_list := {row_list} else if car row_list = 'list then row_list := cdr row_list else << prin2 "***** Error in get_rows(second argument): "; prin2t " should be either an integer or a list of integers."; return; >>; rowdim := row_dim(in_mat); coldim := column_dim(in_mat); for each elt in row_list do << if not fixp elt then rederr "Error in get_rows(second argument): contains non integer."; if elt>rowdim or elt=0 then << prin2 "***** Error in get_rows(second argument): "; rederr "contains row number which is out of range for input matrix."; >>; tmp := 'mat.{nth(cdr in_mat,elt)}; ans := append(ans,{tmp}); >>; return 'list.ans; end; symbolic procedure get_columns(in_mat,col_list); % % Input is a matrix and either a single column number or a list of % column numbers. % % Extracts either a single column or a series of adjacent columns and % returns them in a list of column matrices. % begin integer rowdim,coldim; scalar ans,tmp; if not matrixp(in_mat) then rederr "Error in get_columns(first argument): should be a matrix."; if atom col_list then col_list := {col_list} else if car col_list = 'list then col_list := cdr col_list else << prin2 "***** Error in get_columns(second argument): "; prin2t " should be either an integer or a list of integers."; return; >>; rowdim := row_dim(in_mat); coldim := column_dim(in_mat); for each elt in col_list do << if not fixp elt then rederr "Error in get_columns(second argument): contains non integer."; if elt>coldim or elt=0 then << prin2 "***** Error in get_columns(second argument): "; rederr "contains column number which is out of range for input matrix."; >>; tmp := 'mat.for each row in cdr in_mat collect {nth(row,elt)}; ans := append(ans,{tmp}); >>; return 'list.ans; end; flag('(get_rows,get_columns),'opfn); symbolic procedure stack_rows(in_mat,row_list); % % Stacks all rows pointed to in row_list to form a new matrix. % % row_list can be either an integer or a list of integers. % begin if not !*fast_la and not matrixp in_mat then rederr "Error in stack_rows(first argument): should be a matrix."; if atom row_list then row_list := {row_list} else if car row_list = 'list then row_list := cdr row_list; return 'mat.for each elt in row_list collect nth(cdr in_mat,elt); end; symbolic procedure augment_columns(in_mat,col_list); % % Augments all columns pointed to in col_list to form a new matrix. % % col_list can be either an integer or a list of integers. % begin if not !*fast_la and not matrixp in_mat then rederr "Error in augment_columns(first argument): should be a matrix."; if atom col_list then col_list := {col_list} else if car col_list = 'list then col_list := cdr col_list; return 'mat.for each row in cdr in_mat collect for each elt in col_list collect nth(row,elt); end; % flag('(stack_rows,augment_columns),'opfn); rtypecar stack_rows,augment_columns; symbolic procedure add_rows(in_mat,r1,r2,mult1); % % Replaces row2 (r2) by mult1*r1 + r2. % begin scalar new_mat; integer i,rowdim,coldim; coldim := column_dim(in_mat); if not !*fast_la then << if not matrixp in_mat then rederr "Error in add_rows(first argument): should be a matrix."; rowdim := row_dim(in_mat); if not fixp r1 then rederr "Error in add_rows(second argument): should be an integer."; if not fixp r2 then rederr "Error in add_rows(third argument): should be an integer."; if r1>rowdim or r1=0 then rederr "Error in add_rows(second argument): out of range for input matrix."; if r2>rowdim or r2=0 then rederr "Error in add_rows(third argument): out of range for input matrix."; >>; new_mat := copy_mat(in_mat); % Efficiency. if (my_reval mult1) = 0 then return new_mat; for i:=1:coldim do setmat(new_mat,r2,i,reval {'plus,{'times,mult1, getmat(new_mat,r1,i)},getmat(in_mat,r2,i)}); return new_mat; end; symbolic procedure add_columns(in_mat,c1,c2,mult1); % % Replaces column2 (c2) by mult1*c1 + c2. % begin scalar new_mat; integer i,rowdim,coldim; rowdim := row_dim(in_mat); if not !*fast_la then << if not matrixp in_mat then rederr "Error in add_columns(first argument): should be a matrix."; coldim := column_dim(in_mat); if not fixp c1 then rederr "Error in add_columns(second argument): should be an integer."; if not fixp c2 then rederr "Error in add_columns(third argument): should be an integer."; if c1>coldim or c1=0 then rederr "Error in add_columns(second argument): out of range for input matrix."; if c2>rowdim or c2=0 then rederr "Error in add_columns(third argument): out of range for input matrix."; >>; new_mat := copy_mat(in_mat); % Why not be efficient. if (my_reval mult1) = 0 then return new_mat; for i:=1:rowdim do setmat(new_mat,i,c2,{'plus,{'times,mult1,getmat(new_mat,i,c1)}, getmat(in_mat,i,c2)}); return new_mat; end; % flag('(add_rows,add_columns),'opfn); rtypecar add_rows,add_columns; symbolic procedure add_to_rows(in_mat,row_list,value); % % Adds value to each element in each row in row_list. % % row_list can be either an integer or a list of integers. % begin scalar new_mat; integer i,rowdim,coldim; if not matrixp in_mat then rederr "Error in add_to_row(first argument): should be a matrix."; if atom row_list then row_list := {row_list} else if car row_list = 'list then row_list := cdr row_list else << prin2 "***** Error in add_to_rows(second argument): "; prin2t " should be either integer or a list of integers."; return; >>; rowdim := row_dim(in_mat); coldim := column_dim(in_mat); new_mat := copy_mat(in_mat); for each row in row_list do << if not fixp row then rederr "Error in add_to_row(second argument): should be an integer."; if row>rowdim or row=0 then << prin2 "***** Error in add_to_rows(second argument): "; rederr "contains row which is out of range for input matrix."; >>; for i:=1:coldim do setmat(new_mat,row,i,{'plus,getmat(new_mat,row,i),value}); >>; return new_mat; end; symbolic procedure add_to_columns(in_mat,col_list,value); % % Adds value to each element in each column in col_list. % % col_list can be either an integer or a list of integers. % begin scalar new_mat; integer i,rowdim,coldim; if not matrixp in_mat then rederr "Error in add_to_columns(first argument): should be a matrix."; if atom col_list then col_list := {col_list} else if car col_list = 'list then col_list := cdr col_list else << prin2 "***** Error in add_to_columns(second argument): "; prin2t " should be either integer or list of integers."; return; >>; rowdim := row_dim(in_mat); coldim := column_dim(in_mat); new_mat := copy_mat(in_mat); for each col in col_list do << if not fixp col then rederr "Error in add_to_columns(second argument): should be an integer."; if col>coldim or col=0 then << prin2 "***** Error in add_to_columns(second argument): "; rederr "contains column which is out of range for input matrix."; >>; for i:=1:rowdim do setmat(new_mat,i,col,{'plus,getmat(new_mat,i,col),value}); >>; return new_mat; end; % flag('(add_to_rows,add_to_columns),'opfn); rtypecar add_to_rows,add_to_columns; symbolic procedure mult_rows(in_mat,row_list,mult1); % % Replaces rows specified in row_list by row * mult1. % begin scalar new_mat; integer i,rowdim,coldim; if not !*fast_la and not matrixp(in_mat) then rederr "Error in mult_rows(first argument): should be a matrix."; if atom row_list then row_list := {row_list} else if car row_list = 'list then row_list := cdr row_list; rowdim := row_dim(in_mat); coldim := column_dim(in_mat); new_mat := copy_mat(in_mat); for each row in row_list do << if not !*fast_la and not fixp row then rederr "Error in mult_rows(second argument): contains non integer."; if not !*fast_la and (row>rowdim or row=0) then << prin2 "***** Error in mult_rows(second argument): "; rederr "contains row that is out of range for input matrix."; >>; for i:=1:coldim do << setmat(new_mat,row,i,reval {'times,mult1,getmat(in_mat,row,i)}); >>; >>; return new_mat; end; symbolic procedure mult_columns(in_mat,column_list,mult1); % % Replaces columns specified in column_list by column * mult1. % begin scalar new_mat; integer i,rowdim,coldim; if not !*fast_la and not matrixp(in_mat) then rederr "Error in mult_columns(first argument): should be a matrix."; if atom column_list then column_list := {column_list} else if car column_list = 'list then column_list := cdr column_list; rowdim := row_dim(in_mat); coldim := column_dim(in_mat); new_mat := copy_mat(in_mat); for each column in column_list do << if not !*fast_la and not fixp column then rederr "Error in mult_columns(second argument): contains non integer."; if not !*fast_la and (column>coldim or column=0) then << prin2 "***** Error in mult_columns(second argument): "; rederr "contains column that is out of range for input matrix."; >>; for i:=1:rowdim do << setmat(new_mat,i,column, reval {'times,mult1,getmat(in_mat,i,column)}); >>; >>; return new_mat; end; % flag('(mult_rows,mult_columns),'opfn); rtypecar mult_rows,mult_columns; %%%%%%%%%%%%%%%%%%%%% matrix_augment/matrix_stack %%%%%%%%%%%%%%%%%%%%%% put('matrix_augment,'psopfn,'matrix_augment1); symbolic procedure matrix_augment1(matrices); % % Takes any number of matrices and joins them horizontally. % % Can take either a list of matrices or the matrices as seperate % arguments. % begin scalar mat_list,new_list,new_row; if pairp matrices and pairp car matrices and caar matrices = 'list then matrices := cdar matrices; if not !*fast_la then << mat_list := for each elt in matrices collect reval elt; for each elt in mat_list do if not matrixp(elt) then rederr "Error in matrix_augment: non matrix in input."; >>; const_rows_test(mat_list); for i:=1:row_dim(first mat_list) do << new_row := {}; for each mat1 in mat_list do new_row := append(new_row,nth(cdr mat1,i)); new_list := append(new_list,{new_row}); >>; return 'mat.new_list; end; put('matrix_stack,'psopfn,'matrix_stack1); symbolic procedure matrix_stack1(matrices); % % Takes any number of matrices and joins them vertically. % % Can take either a list of matrices or the matrices as seperate % arguments. % begin scalar mat_list,new_list; if pairp matrices and pairp car matrices and caar matrices = 'list then matrices := cdar matrices; if not !*fast_la then << mat_list := for each elt in matrices collect reval elt; for each elt in mat_list do if not matrixp(elt) then rederr "Error in matrix_stack: non matrix in input."; >>; const_columns_test(mat_list); for each mat1 in mat_list do new_list := append(new_list,cdr mat1); return 'mat.new_list; end; symbolic procedure no_rows(mat_list); % % Takes list of matrices and sums the no. of rows. % for each mat1 in mat_list sum row_dim(mat1); symbolic procedure no_cols(mat_list); % % Takes list of matrices and sums the no. of columns. % for each mat1 in mat_list sum column_dim(mat1); symbolic procedure const_rows_test(mat_list); % % Tests that each matrix in mat_list has the same number of rows % (otherwise augmentation not possible). % begin integer i,listlen,rowdim; listlen := length(mat_list); rowdim := row_dim(car mat_list); i := 1; while i<listlen and row_dim(car mat_list) = row_dim(cadr mat_list) do << i := i+1; mat_list := cdr mat_list; >>; if i=listlen then return rowdim else << prin2 "***** Error in matrix_augment: "; rederr "all input matrices must have the same row dimension."; >>; end; symbolic procedure const_columns_test(mat_list); % % Tests that each matrix in mat_list has the same number of columns % (otherwise stacking not possible). % begin integer i,listlen,coldim; listlen := length(mat_list); coldim := column_dim(car mat_list); i := 1; while i<listlen and column_dim(car mat_list) = column_dim(cadr mat_list) do << i := i+1; mat_list := cdr mat_list; >>; if i=listlen then return coldim else << prin2 "***** Error in matrix_stack: "; rederr "all input matrices must have the same column dimension."; return; >>; end; %%%%%%%%%%%%%%%%%%%% end matrix_augment/matrix_stack %%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%% block_matrix %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% symbolic procedure block_matrix(rows,cols,mat_list); % % Creates a matrix consisting of rows*cols matrices which are taken % sequentially from the mat_list. % begin scalar block_mat,row_list; integer rowdim,coldim,start_row,start_col,i,j; if not fixp rows then rederr "Error in block_matrix(first argument): should be an integer."; if rows=0 then << prin2 "***** Error in block_matrix(first argument): "; prin2t " should be an integer greater than 0."; return; >>; if not fixp cols then rederr "Error in block_matrix(second argument): should be an integer."; if cols=0 then << prin2 "***** Error in block_matrix(second argument): "; prin2t " should be an integer greater than 0."; return; >>; if matrixp mat_list then mat_list := {mat_list} else if pairp mat_list and car mat_list = 'list then mat_list := cdr mat_list else << prin2 "***** Error in block_matrix(third argument): "; prin2t " should be either a single matrix or a list of matrices."; return; >>; if rows*cols neq length mat_list then rederr "Error in block_matrix(third argument): Incorrect number of matrices."; row_list := create_row_list(rows,cols,mat_list); rowdim := check_rows(row_list); coldim := check_cols(row_list); block_mat := mkmatrix(rowdim,coldim); start_row := 1; start_col := 1; for i:=1:length row_list do << for j:=1:cols do << block_mat := copy_into(nth(nth(row_list,i),j),block_mat, start_row,start_col); start_col := start_col + column_dim(nth(nth(row_list,i),j)); >>; start_col := 1; start_row := start_row + row_dim(nth(nth(row_list,i),1)); >>; return block_mat; end; flag('(block_matrix),'opfn); symbolic procedure create_row_list(rows,cols,mat_list); % % Takes mat_list and creates a list of rows elements each of which is % a list containing cols elements (ordering left to right). % eg: create_row_list(3,2,{a,b,c,d,e,f}) will return % {{a,b},{c,d},{e,f}}. % begin scalar row_list,tmp_list; integer i,j,increment; increment := 1; for i:=1:rows do << tmp_list := {}; for j:=1:cols do << tmp_list := append(tmp_list,{nth(mat_list,increment)}); increment := increment + 1; >>; row_list := append(row_list,{tmp_list}); >>; return row_list; end; symbolic procedure check_cols(row_list); % % Checks each element in row_list has same number of columns. % Returns this number. % begin integer i,listlen; i := 1; listlen := length(row_list); while i<listlen and no_cols(nth(row_list,i)) = no_cols(nth(row_list,i+1)) do i:=i+1; if i=listlen then return no_cols(nth(row_list,i)) else << prin2 "***** Error in block_matrix: column dimensions of matrices "; prin2t " into block_matrix are not compatible"; return; >>; end; symbolic procedure check_rows(row_list); % % Checks all matrices in each element in row_list contains same % amount of rows. % Returns the sum of all of these row numbers (ie: number of rows % required in the block matrix). % begin integer i,listlen,rowdim,eltlen,j; i := 1; listlen := length(row_list); while i<=listlen do << eltlen := length nth(row_list,i); j := 1; while j<eltlen do << if row_dim(nth(nth(row_list,i),j)) = row_dim(nth(nth(row_list,i),j+1)) then j := j+1 else << prin2 "***** Error in block_matrix: row dimensions of "; rederr "matrices into block_matrix are not compatible"; >>; >>; rowdim := rowdim + row_dim(nth(nth(row_list,i),j)); i := i+1; >>; return rowdim; end; %%%%%%%%%%%%%%%%%%%%%%%%%%%% end block_matrix %%%%%%%%%%%%%%%%%%%%%%%%%% put('vandermonde,'psopfn,'vandermonde1); symbolic procedure vandermonde1(variables); % % Input can be either a list or individual arguments. % % Creates the Vandermonde matrix. % ie: the square matrix in which the (i,j)'th entry is % nth(variables,i)^(j-1). % begin scalar vand,in_list; integer i,j,sq_size; if pairp variables and pairp car variables and caar variables = 'list then variables := cdar variables; in_list := for each elt in variables collect my_reval elt; sq_size := length in_list; vand := mkmatrix(sq_size,sq_size); for i:=1:sq_size do << for j:=1:sq_size do << setmat(vand,i,j, reval{'expt,nth(in_list,i),{'plus,j,{'minus,1}}}); >>; >>; return vand; end; put('toeplitz,'psopfn,'toeplitz1); symbolic procedure toeplitz1(variables); % % Input can be either a list or individual arguments. % % Creates the Toeplitz matrix. % ie: the square matrix in which the first element is placed on the % diagonal and the nth(variables,i) element is placed on the (i-1) % sub and super diagonals. % begin scalar toep,in_list; integer i,j,sq_size; if pairp variables and pairp car variables and caar variables = 'list then variables := cdar variables; in_list := for each elt in variables collect my_reval elt; sq_size := length in_list; toep := mkmatrix(sq_size,sq_size); for i:=1:sq_size do << for j:=0:i-1 do << setmat(toep,i,i-j,nth(in_list,j+1)); setmat(toep,i-j,i,nth(in_list,j+1)); >>; >>; return toep; end; %%%%%%%%%%%%%%%%%%%%%%%%% kronecker_product %%%%%%%%%%%%%%%%%%%%%%%%%%%% symbolic procedure kronecker_product(AA,BB); % % Copies matrix BB into AA with BB(1,1) at AA(p,q). % begin scalar A,B; integer m,n,r,c; if not !*fast_la then << if not matrixp(aa) then rederr "Error in kronecker_product (first argument): should be a matrix."; if not matrixp(bb) then rederr "Error in kronecker_product (second argument): should be a matrix."; >>; m := row_dim(AA); n := column_dim(AA); r := row_dim(BB); c := column_dim(BB); A := mkmatrix(m*r,n*c); for i:=1:m do for j:=1:n do << B := getmat(AA,i,j); for ii:=1:c do for jj := 1 : r do setmat(A,(i-1)*r+jj,(j-1)*c+ii, reval list('times,b, getmat(bb,jj,ii))); >>; return A; end; % flag('(kronecker_product),'opfn); rtypecar kronecker_product; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% minor %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% symbolic procedure minor(in_mat,row,col); % % Removes row (row) and column (col) from in_mat. % begin scalar min; if not !*fast_la then << if not matrixp(in_mat) then rederr "Error in minor(first argument): should be a matrix."; if not fixp row then rederr "Error in minor(second argument): should be an integer."; if row>row_dim(in_mat) or row=0 then rederr "Error in minor(second argument): out of range for input matrix."; if not fixp col then rederr "Error in minor(third argument): should be an integer."; if col>column_dim(in_mat) or col=0 then rederr "Error in minor(second argument): out of range for input matrix."; >>; min := remove_rows(in_mat,row); min := remove_columns(min,col); return min; end; symbolic procedure remove_rows(in_mat,row_list); % % Removes each row in row_list from in_mat. % % row_list can be either an integer or a list of integers. % begin scalar unique_row_list,new_list; integer rowdim,row; if not !*fast_la and not matrixp(in_mat) then rederr "Error in remove_rows(first argument): non matrix input."; if atom row_list then row_list := {row_list} else if car row_list = 'list then row_list := cdr row_list else << prin2 "***** Error in remove_rows(second argument): "; prin2t " should be either an integer or a list of integers."; return; >>; % Remove any repititions in row_list (I'm assuming here that if the % user has inputted the same row more than once then the meaning % is to only remove that row once). unique_row_list := {}; for each row in row_list do << if not intersection({row},unique_row_list) then unique_row_list := append(unique_row_list,{row}); >>; rowdim := row_dim(in_mat); if not !*fast_la then << for each row in unique_row_list do if not fixp row then rederr "Error in remove_rows(second argument): contains a non integer."; % rowdim := row_dim(in_mat); % coldim := column_dim(in_mat); for each row in unique_row_list do if row>rowdim or row=0 then rederr "Error in remove_rows(second argument): out of range for input matrix."; if length unique_row_list = rowdim then << prin2 "***** Warning in remove_rows:"; prin2t " all the rows have been removed. Returning nil."; return nil; >>; >>; for row:=1:rowdim do if not intersection({row},unique_row_list) then new_list := append(new_list,{nth(cdr in_mat,row)}); return 'mat.new_list; end; symbolic procedure remove_columns(in_mat,col_list); % % Removes each column in col_list from in_mat. % % col_list can be either an integer or a list of integers. % begin scalar unique_col_list,new_list,row_list; integer coldim,row,col; if not !*fast_la and not matrixp(in_mat) then rederr "Error in remove_columns(first argument): non matrix input."; if atom col_list then col_list := {col_list} else if car col_list = 'list then col_list := cdr col_list else << prin2 "***** Error in remove_columns(second argument): "; prin2t " should be either an integer or a list of integers."; return; >>; % Remove any repititions in col_list (I'm assuming here that if the % user has inputted the same column more than once then the meaning % is to only remove that column once). unique_col_list := {}; for each col in col_list do << if not intersection({col},unique_col_list) then unique_col_list := append(unique_col_list,{col}); >>; coldim := column_dim(in_mat); if not !*fast_la then << for each col in unique_col_list do if not fixp col then rederr "Error in remove_columns(second argument): contains a non integer."; for each col in unique_col_list do if col>coldim or col=0 then rederr "Error in remove_columns(second argument): out of range for matrix."; if length unique_col_list = coldim then << prin2 "***** Warning in remove_columns: "; prin2t " all the columns have been removed. Returning nil."; return nil; >>; >>; for each row in cdr in_mat do << row_list := {}; for col:=1:coldim do << if not intersection({col},unique_col_list) then row_list := append(row_list,{nth(row,col)}); >> ; new_list := append(new_list,{row_list}); >>; return 'mat.new_list; end; flag('(minor,remove_rows,remove_columns),'opfn); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end minor %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% begin random_matrix/im_random_matrix %%%%%%%%%%%%%%%%% switch imaginary; % If ON, then random_matrix creates a random % matrix with imaginary entries. switch not_negative; % If ON, then the random matrix functions create % matrices with only positive entries. In the % imaginary case, each entry x+iy will have x and % y both > 0. Not that this really means a great % deal mathematically apart from each guy sitting % right up there in the top right hand corner of % the argand plane but oh well. switch only_integer; % If ON, then the random matrix functions will % create matrices with only integer entries. In % the imaginary case, each entry x+iy will have % x and y as integers. switch symmetric; % If ON, random matrix is symmetric. switch upper_matrix; % If ON, then the random matrix is an upper % triagonal matrix. switch lower_matrix; % If ON, then the random matrix is a lower % triagonal matrix. symbolic procedure random_minus(limit); % % Creates random number in the range -limit < number < limit. % begin scalar r; r := random(limit); if evenp random(1000) then r := {'minus,r}; return r end; symbolic procedure random_make_minus(u); % % Randomly makes u negative. % if evenp random(1000) then {'minus,u} else u; symbolic procedure random_matrix(rowdim,coldim,limit); % % Creates an rowdim by coldim matrix with random entries in the bound % -limit < entry < limit. % begin scalar randmat,random_decimal; integer i,j,start,current_precision; if !*lower_matrix and !*upper_matrix then << prin2 "***** Error in random matrix: "; prin2t " both upper_matrix and lower_matrix switches are on."; return; >>; if !*upper_matrix and !*symmetric then << prin2 "***** Error in random_matriix: "; prin2t " both upper_matrix and symmetric switches are on."; return; >>; if !*lower_matrix and !*symmetric then << prin2 "***** Error in random_matrix: "; prin2t " both lower_matrix and symmetric switches are on."; return; >>; if not fixp limit then limit := algebraic floor(abs(limit)); if not fixp rowdim then rederr "Error in random_matrix(first argument): should be an integer."; if rowdim=0 then rederr "Error in random_matrix(first argument): should be integer > than 0."; if not fixp coldim then rederr "Error in random_matrix(second argument): should be an integer."; if coldim=0 then << prin2 "***** Error in random_matrix(second argument): "; prin2t " should be an integer greater than 0."; return; >>; current_precision := precision 0; if !*imaginary then randmat := im_random_matrix(rowdim,coldim,limit) else << start := 1; randmat := mkmatrix(rowdim,coldim); for i:=1:rowdim do << if !*symmetric or !*lower_matrix then coldim := i else if !*upper_matrix then start := i; for j:=start:coldim do begin scalar r1, r2; r1 := random(limit); r2 := random(10^current_precision); random_decimal := {'plus,r1,{'quotient, r2, 10^current_precision}}; if !*only_integer and !*not_negative then setmat(randmat,i,j,random(limit)) else if !*only_integer then setmat(randmat,i,j,random_minus(limit)) else if !*not_negative then setmat(randmat,i,j,random_decimal) else setmat(randmat,i,j,random_make_minus(random_decimal)); if !*symmetric then setmat(randmat,j,i,getmat(randmat,i,j)); end; >>; >>; return randmat; end; flag('(random_matrix),'opfn); symbolic procedure im_random_matrix(rowdim,coldim,limit); % % Creates an rowdim by coldim matrix with random imaginary entries. % The entrirs are of the form x+iy where x and y are in the bound % -limit < x,y < limit. % begin scalar randmat,random_decimal,im_random_decimal; integer i,j,start,current_precision; start := 1; current_precision := precision 0; randmat := mkmatrix(rowdim,coldim); for i:=1:rowdim do << if !*symmetric or !*lower_matrix then coldim := i else if !*upper_matrix then start := i; for j:=start:coldim do begin scalar r1, r2; r1 := random(limit); r2 := random(10^current_precision); random_decimal := {'plus,1,{'quotient, r2, 10^current_precision}}; r1 := random(limit); r2 := random(10^current_precision); im_random_decimal := {'plus,r1,{'quotient, r2, 10^current_precision}}; if !*only_integer and !*not_negative then << r1 := random(limit); r2 := random(limit); setmat(randmat,i,j,{'plus,r1, {'times,'i,r2}}) >> else if !*only_integer then << r1 := random_minus(limit); r2 := random_minus(limit); setmat(randmat,i,j,{'plus,r1, {'times,'i,r2}}) >> else if !*not_negative then setmat(randmat,i,j,{'plus,random_decimal, {'times,'i,im_random_decimal}}) else << r1 := random_make_minus(random_decimal); r2 := random_make_minus(im_random_decimal); setmat(randmat,i,j,{'plus,r1, {'times,'i,r2}}) >>; if !*symmetric then setmat(randmat,j,i,getmat(randmat,i,j)); end; >>; return randmat; end; % flag('(im_random_matrix),'opfn); rtypecar im_random_matrix; %%%%%%%%%%%%%%%%%% end random_matrix/im_random_matrix %%%%%%%%%%%%%%%%%% symbolic procedure extend(in_mat,rows,cols,entry); % % Extends in_mat by rows rows (!) and cols columns. New entries are % initialised to entry. % begin scalar ex_mat; integer rowdim,coldim,i,j; if not matrixp(in_mat) then rederr "Error in extend(first argument): should be a matrix."; if not fixp rows then rederr "Error in extend(second argument): should be an integer."; if not fixp cols then rederr "Error in extend(third argument): should be an integer."; rowdim := row_dim(in_mat); coldim := column_dim(in_mat); ex_mat := mkmatrix(rowdim+rows,coldim+cols); ex_mat := copy_into(in_mat,ex_mat,1,1); for i:=1:rowdim+rows do << for j:=1:coldim+cols do << if i<=rowdim and j<=coldim then <<>> else setmat(ex_mat,i,j,entry); >>; >>; return ex_mat; end; flag('(extend),'opfn); rtypecar extend; %%%%%%%%%%%%%%%%%%%%% begin char_matrix/char_poly %%%%%%%%%%%%%%%%%%%%%% symbolic procedure char_matrix(in_mat,lmbda); % % Create characteristic matrix. ie: C := lmbda*I - in_mat. % in_ mat must be square. % begin scalar carmat; integer rowdim; if not matrixp(in_mat) then rederr "Error in char_matrix(first argument): should be a matrix."; if not squarep(in_mat) then rederr "Error in char_matrix(first argument): must be a square matrix."; rowdim := row_dim(in_mat); carmat := {'plus,{'times,lmbda,make_identity(rowdim)}, {'minus,in_mat}}; return carmat; end; symbolic procedure char_poly(in_mat,lmbda); % % Finds characteristic polynomial of matrix in_mat. % ie: det(lmbda*I - in_mat). % begin scalar chpoly,carmat; if not matrixp(in_mat) then rederr "Error in char_poly(first argument): should be a matrix."; carmat := char_matrix(in_mat,lmbda); chpoly := algebraic det(carmat); return chpoly; end; flag('(char_matrix char_poly),'opfn); %%%%%%%%%%%%%%%%%%%%%%%% end char_matrix/char_poly %%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%% begin pivot/rows_pivot %%%%%%%%%%%%%%%%%%%%%% symbolic procedure pivot(in_mat,pivot_row,pivot_col); % % Converts all elements in pivot column (apart from the one in pivot % row) to 0. % begin scalar piv_mat,ratio; integer i,j,rowdim,coldim; if not !*fast_la and not matrixp(in_mat) then rederr "Error in pivot(first argument): should be a matrix."; rowdim := row_dim(in_mat); coldim := column_dim(in_mat); if not !*fast_la then << if not fixp pivot_row then rederr "Error in pivot(second argument): should be an integer."; if pivot_row>rowdim or pivot_row=0 then rederr "Error in pivot(second argument): out of range for input matrix."; if not fixp pivot_col then rederr "Error in pivot(third argument): should be an integer."; if pivot_col>coldim or pivot_col=0 then rederr "Error in pivot(third argument): out of range for input matrix."; if getmat(in_mat,pivot_row,pivot_col) = 0 then rederr "Error in pivot: cannot pivot on a zero entry."; >>; piv_mat := copy_mat(in_mat); piv_mat := copy_mat(in_mat); for i:=1:rowdim do << for j:=1:coldim do << if i = pivot_row then <<>> else << ratio := {'quotient,getmat(in_mat,i,pivot_col), getmat(in_mat,pivot_row,pivot_col)}; setmat(piv_mat,i,j,{'plus,getmat(in_mat,i,j),{'minus, {'times,ratio,getmat(in_mat,pivot_row,j)}}}); >>; >>; >>; return piv_mat; end; symbolic procedure rows_pivot(in_mat,pivot_row,pivot_col,row_list); % % Same as pivot but only rows a .. to .. b, where row_list = {a,b}, % are changed. % % rows_pivot will work if row_list is just an integer. % begin scalar piv_mat,ratio; integer j,rowdim,coldim; rowdim := row_dim(in_mat); coldim := column_dim(in_mat); if not !*fast_la then << if not matrixp(in_mat) then rederr "Error in rows_pivot(first argument): should be a matrix."; rowdim := row_dim(in_mat); coldim := column_dim(in_mat); if not fixp pivot_row then rederr "Error in pivot(second argument): should be an integer."; if pivot_row>rowdim or pivot_row=0 then rederr "Error in rows_pivot(second argument): out of range for input matrix."; if not fixp pivot_col then rederr "Error in pivot(third argument): should be an integer."; if pivot_col>coldim or pivot_col=0 then rederr "Error in rows_pivot(third argument): out of range for input matrix."; >>; if atom row_list then row_list := {row_list} else if pairp row_list and car row_list = 'list then row_list := cdr row_list else << prin2 "***** Error in rows_pivot(fourth argument): "; prin2t " should be either an integer or a list of integers."; return; >>; if getmat(in_mat,pivot_row,pivot_col) = 0 then rederr "Error in rows_pivot: cannot pivot on a zero entry."; piv_mat := copy_mat(in_mat); for each elt in row_list do << if not !*fast_la then << if not fixp elt then rederr "Error in rows_pivot: fourth argument contains a non integer."; if elt>rowdim or elt=0 then << prin2 "***** Error in rows_pivot(fourth argument): "; rederr "contains row which is out of range for input matrix."; >>; >>; for j:=1:coldim do << if elt = pivot_row then <<>> else << ratio := {'quotient,getmat(in_mat,elt,pivot_col), getmat(in_mat,pivot_row,pivot_col)}; setmat(piv_mat,elt,j,{'plus,getmat(in_mat,elt,j),{'minus, {'times,ratio,getmat(in_mat,pivot_row,j)}}}); >>; >>; >>; return piv_mat; end; flag('(pivot,rows_pivot),'opfn); %%%%%%%%%%%%%%%%%%%%%%%%%%%%% end pivot/rows_pivot %%%%%%%%%%%%%%%%%%%%% symbolic procedure jacobian(exp_list,var_list); % % jacobian(exp,var) computes the Jacobian matrix of exp w.r.t. var. % The (i,j)'th entry is diff(nth(exp,i),nth(var,j)). % begin scalar jac,exp1,var1; integer i,j,rowdim,coldim; if atom exp_list then exp_list := {exp_list} else if car exp_list neq 'list then rederr "Error in jacobian(first argument): expressions must be in a list." else exp_list := cdr exp_list; if atom var_list then var_list := {var_list} else if car var_list neq 'list then rederr "Error in jacobian(second argument): variables must be in a list." else var_list := cdr var_list; rowdim := length exp_list; coldim := length var_list; jac := mkmatrix(rowdim,coldim); for i:=1:rowdim do << for j:=1:coldim do << exp1 := nth(exp_list,i); var1 := nth(var_list,j); setmat(jac,i,j,algebraic df(exp1,var1)); >>; >>; return jac; end; flag('(jacobian),'opfn); symbolic procedure hessian(poly,variables); % % variables can be either a list or a single variable. % % A Hessian matrix is a matrix whose (i,j)'th entry is % df(df(poly,nth(var,i)),nth(var,j)) % % where df is the derivative. % begin scalar hess_mat,part1,part2,elt; integer row,col,sq_size; if atom variables then variables := {variables} else if car variables = 'list then variables := cdr variables else << prin2 "***** Error in hessian(second argument): "; prin2t " should be either a single variable or a list of variables."; return; >>; sq_size := length variables; hess_mat := mkmatrix(sq_size,sq_size); for row:=1:sq_size do << for col:=1:sq_size do << part1 := nth(variables,row); part2 := nth(variables,col); elt := algebraic df(df(poly,part1),part2); setmat(hess_mat,row,col,elt); >>; >>; return hess_mat; end; flag('(hessian),'opfn); symbolic procedure hermitian_tp(in_mat); % % Computes the Hermitian transpose (HT say) of in_mat. % % The (i,j)'th element of HT = conjugate of the (j,i)'th element of % in__mat. % begin scalar h_tp,element; integer row,col; if not matrixp(in_mat) then rederr "Error in hermitian_tp: non matrix input."; h_tp := algebraic tp(in_mat); for row:=1:row_dim(h_tp) do << for col:=1:column_dim(h_tp) do << element := getmat(h_tp,row,col); setmat(h_tp,row,col, algebraic (repart(element) - i*impart(element))); >>; >>; return h_tp; end; flag('(hermitian_tp),'opfn); symbolic procedure hilbert(sq_size,value); % % The Hilbert matrix is symmetric and the (i,j)'th entry in % 1/(i+j-x). % begin scalar hil_mat,denom; integer row,col; if not fixp sq_size or sq_size<1 then rederr "Error in hilbert(first argument): must be a positive integer."; hil_mat := mkmatrix(sq_size,sq_size); for row:=1:sq_size do << for col:=1:sq_size do << if (denom := reval{'plus,row,col,{'minus,value}}) = 0 then rederr "Error in hilbert: division by zero." else setmat(hil_mat,row,col,{'quotient,1,denom}); >>; >>; return hil_mat; end; flag('(hilbert),'opfn); %%%%%%%%%%%%%%%%%%%%%%%% begin coeff_matrix %%%%%%%%%%%%%%%%%%%%%%%%%%% put('coeff_matrix,'psopfn,'coeff_matrix1); % To allow variable input. symbolic procedure coeff_matrix1(equation_list); % % Given the system of linear equations, coeff_matrix returns {A,X,b} % s.t. AX = b. % % Input can be either a list of linear equations or the linear % equations as individual arguments. % begin scalar variable_list,A,X,b; if pairp car equation_list and caar equation_list = 'list then equation_list := cdar equation_list; equation_list := remove_equals(equation_list); variable_list := get_variable_list(equation_list); if variable_list = nil then rederr "Error in coeff_matrix: no variables in input."; check_linearity(equation_list,variable_list); A := get_A(equation_list,variable_list); X := get_X(variable_list); b := get_b(equation_list,variable_list); return {'list,A,X,b}; end; symbolic procedure remove_equals(equation_list); % % If any of the equations are equalities the equalities are removed % to leave a list of polynomials. % begin equation_list := for each equation in equation_list collect if pairp equation and car equation = 'equal then reval{'plus,cadr equation,{'minus,caddr equation}} else equation; return equation_list; end; symbolic procedure get_variable_list(equation_list); % % Gets hold of all variables from the equations in equation_list. % begin scalar variable_list; for each equation in equation_list do variable_list := union(get_coeffs(equation),variable_list); return reverse variable_list; end; symbolic procedure check_linearity(equation_list,variable_list); % % Checks that we really are dealing with a system of linear equations. % for each equation in equation_list do << for each variable in variable_list do << if deg(equation,variable) > 1 then rederr "Error in coeff_matrix: the equations are not linear."; >>; >>; symbolic procedure get_A(equation_list,variable_list); begin scalar A,element,var_elt; integer row,col,length_equation_list,length_variable_list; length_equation_list := length equation_list; length_variable_list := length variable_list; A := mkmatrix(length equation_list,length variable_list); for row:=1:length_equation_list do << for col:=1:length_variable_list do << element := nth(equation_list,row); var_elt := nth(variable_list,col); setmat(A,row,col,algebraic coeffn(element,var_elt,1)); >>; >>; return A; end; symbolic procedure get_b(equation_list,variable_list); % % Puts the integer parts of all the equations into a column matrix. % begin scalar substitution_list,integer_list,b; integer length_integer_list,row; substitution_list := 'list.for each variable in variable_list collect {'equal,variable,0}; integer_list := for each equation in equation_list collect algebraic sub(substitution_list,equation); length_integer_list := length integer_list; b := mkmatrix(length_integer_list,1); for row:=1:length_integer_list do setmat(b,row,1,-nth(integer_list,row)); return b; end; symbolic procedure get_X(variable_list); begin scalar X; integer row,length_variable_list; length_variable_list := length variable_list; X := mkmatrix(length_variable_list,1); for row := 1:length variable_list do setmat(X,row,1,nth(variable_list,row)); return X; end; symbolic procedure get_coeffs(poly); % % Gets all kernels in a poly. % begin scalar ker_list_num,ker_list_den; ker_list_num := kernels !*q2f simp reval num poly; ker_list_den := kernels !*q2f simp reval den poly; ker_list_num := union(ker_list_num,ker_list_den); return ker_list_num; end; %%%%%%%%%%%%%%%%%%%%%%%%%% end coeff_matrix %%%%%%%%%%%%%%%%%%%%%%%%%%% % Smacro used in other modules. symbolic smacro procedure my_revlis(u); % % As my_reval but for lists. % for each j in u collect my_reval(j); endmodule; %linear algebra. end;