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/* modified for doe macsyma with define_variable */
/* this is the file eigen > dsk:share;.
this is the source code for the new eigen package and it is
macsyma batchable, i.e. batch(eigen,>,dsk,share);. if you do not want
to waste time (and/or paper...) the fastloadable version is on the file
eigen fasl dsk:share;. you can load the latter using macsyma's
loadfile command, i.e. loadfile(eigen,fasl,dsk,share);. the functions
are described in the file eigen usage dsk:share;, and the demo file in
which the functions are demonstrated is eigen demo dsk:share;. */
/* commented out of doe macsyma
eval_when(translate_file,
modedeclare([hermitianmatrix,nondiagonalizable,knowneigvals,
knowneigvects],boolean,
[index1,index2,index3,index4,dimnsn,count,%rnum],fixnum),
declare([hermitianmatrix,nondiagonalizable,knowneigvals,
knowneigvects,listeigvects,listeigvals,%rnum,listarith,programmode,
algebraic,realonly,multiplicities,solveexplicit],special))$ */
eval_when([translate,batch,demo,load,loadfile],
mi(var)::=buildq([var],mode_identity(fixnum,var)),
dv(var)::=buildq([var],define_variable(var,false,boolean)))$
/* commented out of doe macsyma
sstatus(feature,eigen)$
hermitianmatrix:false$
nondiagonalizable:false$
knowneigvals:false$
knowneigvects:false$
listeigvects:[]$
listeigvals:[]$
*/
dv(hermitianmatrix)$ dv(nondiagonalizable)$ dv(knowneigvals)$
dv(knowneigvects)$
define_variable(listeigvects,[],list)$
define_variable(listeigvals,[],list)$
define_variable(rightmatrix,[],any)$
define_variable(leftmatrix,[],any)$
/* conjugate(x):=sublis('[%i=-%i],x)$ */
innerproduct(x,y):=block([listarith],listarith:true,ratsimp(conjugate(x).y))$
/*
unitvector(x):=block([listarith,intrn],listarith:true,intrn:innerproduct(x,x),
intrn:sqrt(intrn),x/intrn)$
*/
unitvector(x):=block([listarith],listarith:true,x/sqrt(innerproduct(x,x)))$
columnvector(x):=transpose(matrix(x))$
gramschmidt(x%,[myinnerproduct]):=
block([listarith,dimnsn,listall,intern,count,denom,unit,index1,
index2],
if myinnerproduct=[] then myinnerproduct:innerproduct
else myinnerproduct:first(myinnerproduct),
mode_declare([dimnsn,count,index1,index2],fixnum,
[listall],list,[intern,denom,unit],any),
listarith:true,dimnsn:mi(length(x%)),listall:[part(x%,1)],
count:1,if dimnsn=1 then return(x%)
else (for index1:2 thru dimnsn do
(unit:part(x%,index1),for index2 thru count do
(intern:part(listall,index2),denom : apply (myinnerproduct, [intern, intern]),
unit:factor(ratsimp(unit - apply (myinnerproduct, [intern, unit])*intern/denom
))),
count:count+1,listall:endcons(unit,listall)),
return(listall)))$
eigenvalues(mat):=
block([dimnsn,listall,solution,multiplicities,solveexplicit,
dummy:?gensym(),index2],
mode_declare([dimnsn,index2],fixnum,[listall,solution],list,
[dummy],any),
listall:[],
solveexplicit:true,
dimnsn:mi(length(mat)),
solution:block([programmode:true],
solve(charpoly(mat,dummy),dummy)),
if solution=[] then
(print(" "),print("eigenvalues: solve is unable to find the roots of the characteristic polynomial."),
return(listall))
else if apply("+",multiplicities) < dimnsn then
(print(" "),print("eigenvalues: solve is unable to find some of the roots of the characteristic polynomial."),
dimnsn:apply("+",multiplicities)),
for index2 thru dimnsn do
(dimnsn:mi(dimnsn-part(multiplicities,index2)+1),
listall:endcons(rhs(part(solution,index2)),listall)),
listall:endcons(multiplicities,[listall]),
return(listall))$
/* EIGENVECTORS - Compute eigenvectors of the matrix mat.
The return value is a list of two elements. The first is a list of the
eigenvalues of mat and a list of the multiplicities of the eigenvalues,
found by calling eigenvalues(). The second is a list of lists of
eigenvectors. There is one list of eigenvectors for each eigenvalue.
There may be one or more eigenvectors in each list.
The following flags are used:
nondiagonalizable is set to true when the matrix is nondiagonalizable
after eigenvectors returns, or false otherwise.
hermitianmatrix when true, causes the degenerate eigenvectors of the
Hermitian matrix to be orthogonalized using the Gram-Schmidt algorithm.
knowneigvals when true causes the eigen package to assume the eigenvalues
of the matrix are known to the user and stored under the global name
listeigvals. listeigvals should be set to a list similar to the output
eigenvalues.
*/
eigenvectors(mat):=
block([eigvecs,equations,unknowns,solution,eigvals,dimnsn,
count,vectr,index3,index4,index2,index1,matrx,mmatrx,notknwn,
unit,multiplicities,%rnum,realonly,algebraic,interm,intern],
local(solution, mmatrx),
mode_declare([equations,unknowns,solution,eigvals,
unit,interm],list,
[dimnsn,count,index3,index4,index2,index1],fixnum,
[vectr,matrx,mmatrx,intern,notknwn],any),
unknowns:[],dimnsn:mi(length(mat)),
count:mi(dimnsn),notknwn:?gensym(),
mat:rationalize(mat),
if knowneigvals then eigvals:listeigvals
else eigvals:eigenvalues(mat),
if eigvals=[] then (nondiagonalizable:true,return(eigvals))
else (multiplicities:part(eigvals,2),
count:length(multiplicities),
for index1 thru dimnsn do
unknowns:endcons(concat(notknwn,index1),unknowns),
vectr:columnvector(unknowns),matrx:mat.vectr,
nondiagonalizable:false,
eigvecs:[],realonly:false,algebraic:true,
for index1 thru count do
(mmatrx:matrx-part(eigvals,1,index1)*vectr,
equations:[],
for index2 thru dimnsn do
equations:cons(mmatrx[index2,1],equations),%rnum:0,
solution:algsys(equations,unknowns),
interm:map('rhs,solution[1]),
unit:[],if %rnum#part(multiplicities,index1)
then nondiagonalizable:true,
for index3 thru %rnum do
(intern:substvectk(%rnum_list,index3,interm),
unit:append(unit,[intern])),
if unit=[] then
(print(" "),print("eigenvectors: the eigenvector(s) for the",
index1,"th eigenvalue will be missing.")),
if hermitianmatrix and %rnum>1 then unit:gramschmidt(unit),
eigvecs:append(eigvecs,[unit])),
return([eigvals, eigvecs])))$
/* the first arg is of the form [r1,r2,r3].
we want to construct [r1=0,r2=1,r3=0] for example. */
substvectk(l,n,exp):=(mode_declare(l,list,n,fixnum,exp,any),
block([sub_list:[],j:0],mode_declare(sub_list,list,j,fixnum),
for var in l do (mode_declare(var,any),
j:j+1,sub_list:cons(var = if j=n then 1 else 0,sub_list)),
sublis(sub_list,exp)))$
uniteigenvectors (M):= block ([uv], local(uv),
mode_declare (uv, list, [i, j], fixnum),
if knowneigvects
then uv : copy (listeigvects)
else uv : copy (eigenvectors (M)),
if uv = []
then []
else
(for i thru length (uv[2])
do for j thru length (uv[2][i])
/* modification in-place (OK due to copy above) */
do uv[2][i][j] : ratsimp (unitvector (uv[2][i][j])),
uv));
similaritytransform(mat):=
block([listvec,listuevec], local(listuevec),
mode_declare([listvec,listuevec],list),
listuevec:uniteigenvectors(mat),
if nondiagonalizable then return(listuevec)
else (listvec: apply (append, listuevec[2]),
rightmatrix:transpose(apply('matrix,listvec)),
if hermitianmatrix then
leftmatrix:conjugate(transpose(rightmatrix))
else leftmatrix:rightmatrix^^-1,
return(listuevec)))$
conj(x):=conjugate(x)$
inprod(x,y):=innerproduct(x,y)$
uvect(x):=unitvector(x)$
covect(x):=columnvector(x)$
gschmit(x%, [f]):= if f = [] then gramschmidt(x%) else gramschmidt (x%, first(f))$
eivals(mat):=eigenvalues(mat)$
eivects(mat):=eigenvectors(mat)$
ueivects(mat):=uniteigenvectors(mat)$
simtran(mat):=similaritytransform(mat)$
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