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LA_library/matexp.h
jiri b6e5dda896 *** empty log message ***
2006-09-11 23:34:41 +00:00

790 lines
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#ifndef _MATEXP_H_
#define _MATEXP_H_
//general routine for polynomial of a matrix, tuned to minimize the number
//of matrix-matrix multiplications on cost of additions and memory
// the polynom and exp routines will work on any type, for which traits class
// is defined containing definition of an element type, norm and axpy operation
#include "la_traits.h"
#include "laerror.h"
#ifndef NONCBLAS
extern "C" {
#include "cblas.h"
#include "clapack.h"
}
#else
#include "noncblas.h"
#endif
template<class T,class R>
const T polynom0(const T &x, const NRVec<R> &c)
{
int order=c.size()-1;
T z,y;
//trivial reference implementation by horner scheme
if(order==0) {y=x; y=c[0];} //to avoid the problem: we do not know the size of the matrix to contruct a scalar one
else
{
int i;
z=x*c[order];
for(i=order-1; i>=0; i--)
{
if(i<order-1) z=y*x;
y=z+c[i];
}
}
return y;
}
//algorithm which minimazes number of multiplications, at the cost of storage
template<class T,class R>
const T polynom(const T &x, const NRVec<R> &c)
{
int n=c.size()-1;
int i,j,k,m=0,t;
if(n<=4) return polynom0(x,c); //here the horner scheme is optimal
//first find m which minimizes the number of multiplications
j=10*n;
for(i=2;i<=n+1;i++)
{
t=i-2+2*(n/i)-(n%i)?0:1;
if(t<j)
{
j=t;
m=i;
}
}
//allocate array for powers up to m
T *xpows = new T[m];
xpows[0]=x;
for(i=1;i<m;i++) xpows[i]=xpows[i-1]*x;
//run the summation loop
T r,s,f;
k= -1;
for(i=0; i<=n/m;i++)
{
for(j=0;j<m;j++)
{
k++;
if(k>n) break;
if(j==0) {
if(i==0) s=x; /*just to get the dimensions of the matrix*/
s=c[k]; /*create diagonal matrix*/
}
else
LA_traits<T>::axpy(s,xpows[j-1],c[k]); //general s+=xpows[j-1]*c[k]; but more efficient for matrices
}
if(i==0) {r=s; f=xpows[m-1];}
else
{
r+= s*f;
f=f*xpows[m-1];
}
}
delete[] xpows;
return r;
}
//for general objects
template<class T>
const T ncommutator ( const T &x, const T &y, int nest=1, const bool right=1)
{
T z;
if(right) {z=x; while(--nest>=0) z=z*y-y*z;}
else {z=y; while(--nest>=0) z=x*z-z*x;}
return z;
}
template<class T>
const T nanticommutator ( const T &x, const T &y, int nest=1, const bool right=1)
{
T z;
if(right) {z=x; while(--nest>=0) z=z*y+y*z;}
else {z=y; while(--nest>=0) z=x*z+z*x;}
return z;
}
//general BCH expansion (can be written more efficiently in a specialization for matrices)
template<class T>
const T BCHexpansion (const T &h, const T &t, const int n, const bool verbose=0)\
{
T result=h;
double factor=1.;
T z=h;
for(int i=1; i<=n; ++i)
{
factor/=i;
z= z*t-t*z;
if(verbose) cerr << "BCH contribution at order "<<i<<" : "<<z.norm()*factor<<endl;
result+= z*factor;
}
return result;
}
template<class T>
const T ipow( const T &x, int i)
{
if(i<0) laerror("negative exponent in ipow");
if(i==0) {T r=x; r=(typename LA_traits<T>::elementtype)1; return r;}//trick for matrix dimension
if(i==1) return x;
T y,z;
z=x;
while(!(i&1))
{
z = z*z;
i >>= 1;
}
y=z;
while((i >>= 1)/*!=0*/)
{
z = z*z;
if(i&1) y = y*z;
}
return y;
}
inline int nextpow2(const double n)
{
const double log2=log(2.);
if(n<=.75) return 0; //try to keep the taylor expansion short
if(n<=1.) return 1;
return int(ceil(log(n)/log2-log(.75)));
}
template<class T, class C>
NRVec<C> exp_aux(const T &x, int &power,int maxpower= -1, int maxtaylor= -1)
{
//should better be computed by mathematica to have accurate last digits, chebyshev instead, see exp in glibc
static double exptaylor[]={
1.,
1.,
0.5,
0.1666666666666666666666,
0.0416666666666666666666,
0.0083333333333333333333,
0.0013888888888888888888,
0.00019841269841269841253,
2.4801587301587301566e-05,
2.7557319223985892511e-06,
2.7557319223985888276e-07,
2.5052108385441720224e-08,
2.0876756987868100187e-09,
1.6059043836821613341e-10,
1.1470745597729724507e-11,
7.6471637318198164055e-13,
4.7794773323873852534e-14,
2.8114572543455205981e-15,
1.5619206968586225271e-16,
8.2206352466243294955e-18,
4.1103176233121648441e-19,
0.};
double mnorm= x.norm();
power=nextpow2(mnorm);
if(maxpower>=0 && power>maxpower) power=maxpower;
double scale=exp(-log(2.)*power);
//find how long taylor expansion will be necessary
const double precision=1e-14; //decreasing brings nothing
double s,t;
s=mnorm*scale;
int n=0;
t=1.;
do {
n++;
t*=s;
}
while(t*exptaylor[n]>precision);//taylor 0 will terminate in any case
if(maxtaylor>=0 && n>maxtaylor) n=maxtaylor; //useful e.g. if the matrix is nilpotent in order n+1 as the CC T operator for n electrons
int i; //adjust the coefficients in order to avoid scaling the argument
NRVec<C> taylor2(n+1);
for(i=0,t=1.;i<=n;i++)
{
taylor2[i]=exptaylor[i]*t;
t*=scale;
}
return taylor2;
}
//it seems that we do not gain anything by polynom vs polynom0, check the m-optimization!
template<class T>
const T exp(const T &x, bool horner=true, int maxpower= -1, int maxtaylor= -1 )
{
int power;
//prepare the polynom of and effectively scale T
NRVec<typename LA_traits<T>::elementtype> taylor2=exp_aux<T,typename LA_traits<T>::elementtype>(x,power,maxpower,maxtaylor);
T r= horner?polynom0(x,taylor2):polynom(x,taylor2);
//for accuracy summing from the smallest terms up would be better, but this is more efficient for matrices
//power the result back
for(int i=0; i<power; i++) r=r*r;
return r;
}
//this simple implementation seems not to be numerically stable enough
//and probably not efficient either
template<class M, class V>
void exptimesdestructive(const M &mat, V &result, V &rhs, bool transpose=false, const double scale=1., int maxpower= -1, int maxtaylor= -1) //uses just matrix vector multiplication
{
if(mat.nrows()!=mat.ncols()||(unsigned int) mat.nrows() != (unsigned int)rhs.size()) laerror("inappropriate sizes in exptimes");
int power;
//prepare the polynom of and effectively scale the matrix
NRVec<typename LA_traits<V>::elementtype> taylor2=exp_aux<M,typename LA_traits<V>::elementtype>(mat,power,maxpower,maxtaylor);
cerr <<"test power "<<power<<endl;
V tmp;
for(int i=1; i<=(1<<power); ++i) //unfortunatelly, here we have to repeat it many times, unlike if the matrix is stored explicitly
{
if(i>1) rhs=result; //apply again to the result of previous application
else result=rhs;
tmp=rhs; //now rhs can be used as scratch
result*=taylor2[0];
for(int j=1; j<taylor2.size(); ++j)
{
mat.gemv(0.,rhs,transpose?'t':'n',scale,tmp);
tmp=rhs;
result.axpy(taylor2[j],tmp);
}
}
return;
}
template<class M, class V>
const V exptimes(const M &mat, V rhs, bool transpose=false, const double scale=1., int maxpower= -1, int maxtaylor= -1 )
{
V result;
exptimesdestructive(mat,result,rhs,transpose,scale,maxpower,maxtaylor);
return result;
}
#ifdef EXPOKIT
//this comes from dgexpv from expokit, it had to be translated to C to make a template
// dgexpv.f -- translated by f2c (version 20030320).
static double dipow(double x,int i)
{
double y=1.0;
if(x==0.0)
{
if(i>0) return(0.0);
if(i==0) laerror("dipow: 0^0");
return(1./x);
}
if(i) {
if(i<0) {i= -i; x= 1.0/x;}
do
{
if(i&1) y *= x;
x *= x;
}
while((i >>= 1)/*!=0*/);
}
return(y);
}
static int nint(double x)
{
int sgn= x>=0 ? 1 : -1;
return sgn * (int)(abs(x)+.5);
}
#ifdef FORTRAN_
#define FORNAME(x) x##_
#else
#define FORNAME(x) x
#endif
extern "C" void FORNAME(dgpadm)(const int *, int *, double *, double *, int *, double *, int *, int *, int *, int *, int *);
extern "C" void FORNAME(dnchbv)(int *, double *, double *, int *, double *, double *);
//partial template specialization leaving still free room for generic matrix type
template<class M>
extern void dgexpv(const M &mat, NRVec<double> &result, const NRVec<double> &rhs, bool transpose=false, const double t=1., int mxkrylov=0, double tol=1e-14, int mxstep=1000, int mxreject=0, int ideg=6)
{
if(mat.nrows()!= mat.ncols()) laerror("non-square matrix in exptimes");
int n=mat.nrows();
int m;
if(mxkrylov) m=mxkrylov; else m= n>100? 100: (n>1?n-1:1);
if (m > n || m <= 0) laerror("illegal mxkrylov");
if(result.size()!=n || rhs.size()!=n) laerror("inconsistent vector and matrix size in exptimes");
const double *v=rhs;
double *w=result;
double anorm=mat.norm();
int iflag;
/* System generated locals */
int i__1, i__2;
double d__1;
#define pow_dd(x,y) (exp(y*log(x)))
/* Local variables */
int ibrkflag;
double step_min__, step_max__;
int i__, j;
double break_tol__;
int k1;
double p1, p2, p3;
int ih, mh, iv, ns, mx;
double xm;
int j1v;
double hij, sgn, eps, hj1j, sqr1, beta;
double hump;
int ifree, lfree;
double t_old__;
int iexph;
double t_new__;
int nexph;
double t_now__;
int nstep;
double t_out__;
int nmult;
double vnorm;
int nscale;
double rndoff, t_step__, avnorm;
int ireject;
double err_loc__;
int nreject, mbrkdwn;
double tbrkdwn, s_error__, x_error__;
/* -----Purpose----------------------------------------------------------| */
/* --- DGEXPV computes w = exp(t*A)*v - for a General matrix A. */
/* It does not compute the matrix exponential in isolation but */
/* instead, it computes directly the action of the exponential */
/* operator on the operand vector. This way of doing so allows */
/* for addressing large sparse problems. */
/* The method used is based on Krylov subspace projection */
/* techniques and the matrix under consideration interacts only */
/* via the external routine `matvec' performing the matrix-vector */
/* product (matrix-free method). */
/* -----Arguments--------------------------------------------------------| */
/* n : (input) order of the principal matrix A. */
/* m : (input) maximum size for the Krylov basis. */
/* t : (input) time at wich the solution is needed (can be < 0). */
/* v(n) : (input) given operand vector. */
/* w(n) : (output) computed approximation of exp(t*A)*v. */
/* tol : (input/output) the requested accuracy tolerance on w. */
/* If on input tol=0.0d0 or tol is too small (tol.le.eps) */
/* the internal value sqrt(eps) is used, and tol is set to */
/* sqrt(eps) on output (`eps' denotes the machine epsilon). */
/* (`Happy breakdown' is assumed if h(j+1,j) .le. anorm*tol) */
/* anorm : (input) an approximation of some norm of A. */
/* wsp(lwsp): (workspace) lwsp .ge. n*(m+1)+n+(m+2)^2+4*(m+2)^2+ideg+1 */
/* +---------+-------+---------------+ */
/* (actually, ideg=6) V H wsp for PADE */
/* iwsp(liwsp): (workspace) liwsp .ge. m+2 */
/* matvec : external subroutine for matrix-vector multiplication. */
/* synopsis: matvec( x, y ) */
/* double precision x(*), y(*) */
/* computes: y(1:n) <- A*x(1:n) */
/* where A is the principal matrix. */
/* iflag : (output) exit flag. */
/* <0 - bad input arguments */
/* 0 - no problem */
/* 1 - maximum number of steps reached without convergence */
/* 2 - requested tolerance was too high */
/* -----Accounts on the computation--------------------------------------| */
/* Upon exit, an interested user may retrieve accounts on the */
/* computations. They are located in wsp and iwsp as indicated below: */
/* location mnemonic description */
/* -----------------------------------------------------------------| */
/* iwsp(1) = nmult, number of matrix-vector multiplications used */
/* iwsp(2) = nexph, number of Hessenberg matrix exponential evaluated */
/* iwsp(3) = nscale, number of repeated squaring involved in Pade */
/* iwsp(4) = nstep, number of integration steps used up to completion */
/* iwsp(5) = nreject, number of rejected step-sizes */
/* iwsp(6) = ibrkflag, set to 1 if `happy breakdown' and 0 otherwise */
/* iwsp(7) = mbrkdwn, if `happy brkdown', basis-size when it occured */
/* -----------------------------------------------------------------| */
/* wsp(1) = step_min, minimum step-size used during integration */
/* wsp(2) = step_max, maximum step-size used during integration */
/* wsp(3) = dummy */
/* wsp(4) = dummy */
/* wsp(5) = x_error, maximum among all local truncation errors */
/* wsp(6) = s_error, global sum of local truncation errors */
/* wsp(7) = tbrkdwn, if `happy breakdown', time when it occured */
/* wsp(8) = t_now, integration domain successfully covered */
/* wsp(9) = hump, i.e., max||exp(sA)||, s in [0,t] (or [t,0] if t<0) */
/* wsp(10) = ||w||/||v||, scaled norm of the solution w. */
/* -----------------------------------------------------------------| */
/* The `hump' is a measure of the conditioning of the problem. The */
/* matrix exponential is well-conditioned if hump = 1, whereas it is */
/* poorly-conditioned if hump >> 1. However the solution can still be */
/* relatively fairly accurate even when the hump is large (the hump */
/* is an upper bound), especially when the hump and the scaled norm */
/* of w [this is also computed and returned in wsp(10)] are of the */
/* same order of magnitude (further details in reference below). */
/* ----------------------------------------------------------------------| */
/* -----The following parameters may also be adjusted herein-------------| */
/* mxstep = 1000, . mxreject = 0, . ideg = 6, */
/* mxstep : maximum allowable number of integration steps. */
/* The value 0 means an infinite number of steps. */
/* mxreject: maximum allowable number of rejections at each step. */
/* The value 0 means an infinite number of rejections. */
/* ideg : the Pade approximation of type (ideg,ideg) is used as */
/* an approximation to exp(H). The value 0 switches to the */
/* uniform rational Chebyshev approximation of type (14,14) */
/* delta : local truncation error `safety factor' */
/* gamma : stepsize `shrinking factor' */
/* ----------------------------------------------------------------------| */
/* Roger B. Sidje (rbs@maths.uq.edu.au) */
/* EXPOKIT: Software Package for Computing Matrix Exponentials. */
/* ACM - Transactions On Mathematical Software, 24(1):130-156, 1998 */
/* ----------------------------------------------------------------------| */
iflag = 0;
i__1 = m + 2;
int lwsp= n * (m + 2) + i__1 * i__1 * 5 + ideg + 1;
double *wsp = new double[lwsp];
int *iwsp = new int[m+2];
/* Parameter adjustments */
--w;
--v;
--wsp;
--iwsp;
/* --- initialisations ... */
k1 = 2;
mh = m + 2;
iv = 1;
ih = iv + n * (m + 1) + n;
ifree = ih + mh * mh;
lfree = lwsp - ifree + 1;
ibrkflag = 0;
mbrkdwn = m;
nmult = 0;
nreject = 0;
nexph = 0;
nscale = 0;
t_out__ = abs(t);
tbrkdwn = 0.;
step_min__ = t_out__;
step_max__ = 0.;
nstep = 0;
s_error__ = 0.;
x_error__ = 0.;
t_now__ = 0.;
t_new__ = 0.;
p1 = 1.3333333333333333;
L1:
p2 = p1 - 1.;
p3 = p2 + p2 + p2;
eps = (d__1 = p3 - 1., abs(d__1));
if (eps == 0.) {
goto L1;
}
if (tol <= eps) {
tol = sqrt(eps);
}
rndoff = eps * anorm;
break_tol__ = 1e-7;
/* >>> break_tol = tol */
/* >>> break_tol = anorm*tol */
sgn = t>=0?1. : -1.;
cblas_dcopy(n, &v[1], 1, &w[1], 1);
beta = cblas_dnrm2(n, &w[1], 1);
vnorm = beta;
hump = beta;
/* --- obtain the very first stepsize ... */
sqr1 = sqrt(.1);
xm = 1. / (double) (m);
d__1 = (m + 1) / 2.72;
i__1 = m + 1;
p1 = tol * dipow(d__1, i__1) * sqrt((m + 1) * 6.28);
d__1 = p1 / (beta * 4. * anorm);
t_new__ = 1. / anorm * pow_dd(d__1, xm);
d__1 = log10(t_new__) - sqr1;
i__1 = nint(d__1) - 1;
p1 = dipow(10., i__1);
d__1 = t_new__ / p1 + .55;
t_new__ = floor(d__1) * p1;
/* --- step-by-step integration ... */
L100:
if (t_now__ >= t_out__) {
goto L500;
}
++nstep;
/* Computing MIN */
d__1 = t_out__ - t_now__;
t_step__ = min(d__1,t_new__);
p1 = 1. / beta;
i__1 = n;
for (i__ = 1; i__ <= i__1; ++i__) {
wsp[iv + i__ - 1] = p1 * w[i__];
}
i__1 = mh * mh;
for (i__ = 1; i__ <= i__1; ++i__) {
wsp[ih + i__ - 1] = 0.;
}
/* --- Arnoldi loop ... */
j1v = iv + n;
i__1 = m;
for (j = 1; j <= i__1; ++j) {
++nmult;
{
NRVec<double> res(&wsp[j1v],n,true);
NRVec<double> rhs(&wsp[j1v - n],n,true);
mat.gemv(0.,res,transpose?'t':'n',1.,rhs);
}
i__2 = j;
for (i__ = 1; i__ <= i__2; ++i__) {
hij = cblas_ddot(n, &wsp[iv + (i__ - 1) * n], 1, &wsp[j1v], 1);
d__1 = -hij;
cblas_daxpy(n, d__1, &wsp[iv + (i__ - 1) * n], 1, &wsp[j1v], 1);
wsp[ih + (j - 1) * mh + i__ - 1] = hij;
}
hj1j = cblas_dnrm2(n, &wsp[j1v], 1);
/* --- if `happy breakdown' go straightforward at the end ... */
if (hj1j <= break_tol__) {
/* print*,'happy breakdown: mbrkdwn =',j,' h =',hj1j */
k1 = 0;
ibrkflag = 1;
mbrkdwn = j;
tbrkdwn = t_now__;
t_step__ = t_out__ - t_now__;
goto L300;
}
wsp[ih + (j - 1) * mh + j] = hj1j;
d__1 = 1. / hj1j;
cblas_dscal(n, d__1, &wsp[j1v], 1);
j1v += n;
/* L200: */
}
++nmult;
{
NRVec<double> res(&wsp[j1v],n,true);
NRVec<double> rhs(&wsp[j1v - n],n,true);
mat.gemv(0.,res,transpose?'t':'n',1.,rhs);
}
avnorm = cblas_dnrm2(n, &wsp[j1v], 1);
/* --- set 1 for the 2-corrected scheme ... */
L300:
wsp[ih + m * mh + m + 1] = 1.;
/* --- loop while ireject<mxreject until the tolerance is reached ... */
ireject = 0;
L401:
/* --- compute w = beta*V*exp(t_step*H)*e1 ... */
++nexph;
mx = mbrkdwn + k1;
if (ideg != 0) {
/* --- irreducible rational Pade approximation ... */
d__1 = sgn * t_step__;
FORNAME(dgpadm)(&ideg, &mx, &d__1, &wsp[ih], &mh, &wsp[ifree], &lfree, &iwsp[1]
, &iexph, &ns, &iflag);
iexph = ifree + iexph - 1;
nscale += ns;
} else {
/* --- uniform rational Chebyshev approximation ... */
iexph = ifree;
i__1 = mx;
for (i__ = 1; i__ <= i__1; ++i__) {
wsp[iexph + i__ - 1] = 0.;
}
wsp[iexph] = 1.;
d__1 = sgn * t_step__;
FORNAME(dnchbv)(&mx, &d__1, &wsp[ih], &mh, &wsp[iexph], &wsp[ifree + mx]);
}
/* L402: */
/* --- error estimate ... */
if (k1 == 0) {
err_loc__ = tol;
} else {
p1 = (d__1 = wsp[iexph + m], abs(d__1)) * beta;
p2 = (d__1 = wsp[iexph + m + 1], abs(d__1)) * beta * avnorm;
if (p1 > p2 * 10.) {
err_loc__ = p2;
xm = 1. / (double) (m);
} else if (p1 > p2) {
err_loc__ = p1 * p2 / (p1 - p2);
xm = 1. / (double) (m);
} else {
err_loc__ = p1;
xm = 1. / (double) (m - 1);
}
}
/* --- reject the step-size if the error is not acceptable ... */
if (k1 != 0 && err_loc__ > t_step__ * 1.2 * tol && (mxreject == 0 ||
ireject < mxreject)) {
t_old__ = t_step__;
d__1 = t_step__ * tol / err_loc__;
t_step__ = t_step__ * .9 * pow_dd(d__1, xm);
d__1 = log10(t_step__) - sqr1;
i__1 = nint(d__1) - 1;
p1 = dipow(10., i__1);
d__1 = t_step__ / p1 + .55;
t_step__ = floor(d__1) * p1;
#ifdef DEBUG
/* print*,'t_step =',t_old */
/* print*,'err_loc =',err_loc */
/* print*,'err_required =',delta*t_old*tol */
/* print*,'stepsize rejected, stepping down to:',t_step */
#endif
++ireject;
++nreject;
if (mxreject != 0 && ireject > mxreject) {
iflag = 2;
laerror("Failure in DGEXPV: The requested tolerance is too high. Rerun with a smaller value.");
}
goto L401;
}
/* --- now update w = beta*V*exp(t_step*H)*e1 and the hump ... */
/* Computing MAX */
i__1 = 0, i__2 = k1 - 1;
mx = mbrkdwn + max(i__1,i__2);
cblas_dgemv(CblasRowMajor, CblasTrans, n, mx, beta, &wsp[iv], n, &wsp[iexph], 1, 0., &w[1],1);
beta = cblas_dnrm2(n, &w[1], 1);
hump = max(hump,beta);
/* --- suggested value for the next stepsize ... */
d__1 = t_step__ * tol / err_loc__;
t_new__ = t_step__ * .9 * pow_dd(d__1, xm);
d__1 = log10(t_new__) - sqr1;
i__1 = nint(d__1) - 1;
p1 = dipow(10., i__1);
d__1 = t_new__ / p1 + .55;
t_new__ = floor(d__1) * p1;
err_loc__ = max(err_loc__,rndoff);
/* --- update the time covered ... */
t_now__ += t_step__;
/* --- display and keep some information ... */
#ifdef DEBUG
/* print*,'integration',nstep,'---------------------------------' */
/* print*,'scale-square =',ns */
/* print*,'step_size =',t_step */
/* print*,'err_loc =',err_loc */
/* print*,'next_step =',t_new */
#endif
step_min__ = min(step_min__,t_step__);
step_max__ = max(step_max__,t_step__);
s_error__ += err_loc__;
x_error__ = max(x_error__,err_loc__);
if (mxstep == 0 || nstep < mxstep) {
goto L100;
}
iflag = 1;
L500:
/* iwsp(1) = nmult */
/* iwsp(2) = nexph */
/* iwsp(3) = nscale */
/* iwsp(4) = nstep */
/* iwsp(5) = nreject */
/* iwsp(6) = ibrkflag */
/* iwsp(7) = mbrkdwn */
/* wsp(1) = step_min */
/* wsp(2) = step_max */
/* wsp(3) = 0.0d0 */
/* wsp(4) = 0.0d0 */
/* wsp(5) = x_error */
/* wsp(6) = s_error */
/* wsp(7) = tbrkdwn */
/* wsp(8) = sgn*t_now */
/* wsp(9) = hump/vnorm */
/* wsp(10) = beta/vnorm */
if(iflag) laerror("dgexpv error");
delete[] ++wsp; delete[] ++iwsp;
return;
}
#undef FORNAME
#undef pow_dd
#endif
//EXPOKIT
//@@@ power series matrix logarithm?
#endif
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