float support in doc

CLAUDE.md

@@ -18,7 +18,7 @@ decisions and plans agreed with the author, so that they survive a fresh clone.
   Quick test program: `g++ -std=gnu++11 -I<repo> -O0 -DDEBUG -DNO_STRASSEN -DFORTRAN_ -DNONCLAPACK X.cc <repo>/.libs/libla.a -lcblas -lblas -llapack`
   (add `-fsanitize=address` for memory checks).
 - Regression test of the optimizers: `make check` builds `testoptimize`; `./testoptimize` must report
-  "123 passed, 0 failed" (exit code = number of failures). New optimizer tests go to testoptimize.cc;
+  "137 passed, 0 failed" (exit code = number of failures). New optimizer tests go to testoptimize.cc;
   test problems live in testoptimize.h.
 - `t` (t.cc) is a collection of if(0)/if(1) test blocks, most read n from stdin (`echo 10 | ./t`).
 - NONCBLAS check: copy the repo with `cp -a` (keeps timestamps, otherwise automake reruns and fails),
@@ -90,10 +90,15 @@ decisions and plans agreed with the author, so that they survive a fresh clone.
   its NEA oscillator strength f=dE_min/dE is a placeholder for the true one.
 
 ## Planned (only when the author asks)
-- float version: float and complex<float> must ALWAYS use BLAS/LAPACK (s*/c* routines, noncblas wrappers)
+- float version (IN PROGRESS since 2026-11-05): float and complex<float> must ALWAYS use BLAS/LAPACK (s*/c* routines,
-  like double/complex<double>, never generic loops. INSTANTIZE(float) in optimize.cc currently needs float
+  noncblas wrappers) like double/complex<double>, never generic loops. Done: noncblas, vec, mat, smat, nonclass (+ complexify
-  versions of diagonalize, linear_solve, singular_decomposition. BLAS-based NRVec members are meant for
+  BLAS for NRSMat/NRMat), sparsemat, sparsesmat, tensor, fourindex, polynomial, header templates (matexp,
-  floating types only, not integers.
+  conjgrad, gmres, davidson, lanczos, diis: no changes needed beyond the defaults). Float twins are generated from the double specializations (placed right after them). Precision-
+  dependent defaults live in LA_precision<T> (la_traits.h, only double and float; double values unchanged). optimize done
+  2026-11-05: own trait opt_precision<T> in optimize.h, INSTANTIZE(float), float tests in testoptimize.cc (test_float);
+  float LM on badly scaled problems needs my_scaling=true. reg/regsurf, csrmat, strassen, CUDA stay double-only.
+  Open: Tensor Tucker bugs (complex roundtrip, inverseorder=true abort, also double; bugtests/tensor_tucker_bug.cc).
+  BLAS-based NRVec members are meant for floating types only, not integers.
 - NRSMat: possible bool member `hermitian` toggling hermitian vs complex symmetric behaviour
   (Tensor(const NRSMat&) must then respect it).
 - SparseSMat: possible member `symmetry_type` (0 none, 1 symmetric, 2 hermitian) for NRSMat conversion,

doc/concepts.tex

@@ -153,12 +153,36 @@ the mixed real--complex functions such as \code{gemv} of a complex vector with a
 The functions \code{LA::real(c)} and \code{LA::imag(c)} return references to the real and imaginary part of a
 \code{std::complex}, which can be used as l-values.
 
+Default thresholds of the numerical methods depend on the floating point precision and are collected in
+\code{LA_precision<T>} (\header{la_traits.h}), which is defined for \code{T = double} and \code{float} only (any other type
+gives a compile-time error); functions with an argument of type \code{T} use \code{LA_precision<typename LA_traits<T>::normtype>},
+so that complex types take the values of their real component. The double values are those used by the library before.
+\begin{center}
+\begin{tabular}{llll}
+\toprule
+member & double & float & used by \\
+\midrule
+\code{svd_threshold()} & $10^{-12}$ & $10^{-5}$ & \code{svd_solve}, \code{Tensor::Tucker} \\
+\code{taylor_precision()} & $10^{-14}$ & $10^{-7}$ & \code{exp}, \code{sincos}, \code{exptimes} (\header{matexp.h}) \\
+\code{conjgrad_tol()} & $10^{-8}$ & $10^{-5}$ & \code{conjgrad} \\
+\code{gmres_eps()} & $10^{-7}$ & $10^{-5}$ & \code{gmres} \\
+\code{eigensolver_eps()} & $10^{-6}$ & $10^{-4}$ & \code{davidson}, \code{lanczos} \\
+\code{newton_threshold()} & $10^{-14}$ & $10^{-6}$ & \code{Polynomial::newton} \\
+\code{det_tiny()} & $10^{-300}$ & $10^{-37}$ & underflow guard of the determinant in \code{linear_solve} \\
+\code{sparse_epsilon()} & $10^{-15}$ & $10^{-7}$ & zero elements of \code{SparseMat}, \code{SparseSMat} (\code{SPARSEEPSILON}) \\
+\code{cholesky_epsilon()} & $10^{-16}$ & $10^{-8}$ & pivots skipped by \code{SparseSMat::cholesky} (\code{CHOLESKYEPSILON}) \\
+\bottomrule
+\end{tabular}
+\end{center}
+
 \subsection{Element types and instantiation}
 
 The class templates are explicitly instantiated in the library for \code{double}, \code{std::complex<double>}
 and the integer types \code{char}, \code{short}, \code{int}, \code{long}, \code{long long} and their unsigned
-variants. The BLAS and LAPACK backed functions are implemented for \code{double} and \code{std::complex<double>}
+variants; \code{NRVec}, \code{NRMat}, \code{NRSMat}, \code{SparseMat}, \code{SparseSMat}, \code{Tensor}, \code{fourindex} (with \code{unsigned char} indices) and \code{Polynomial} also for \code{float} and
-(some also for mixed real--complex arguments); calling them for other types results in a run-time
+\code{std::complex<float>} (single precision support, 2026, in progress for the other classes). The BLAS and LAPACK backed
+functions are implemented for \code{double} and \code{std::complex<double>} (some also for mixed real--complex arguments)
+and for their single precision counterparts, which always use the single precision BLAS/LAPACK routines (\code{s}/\code{c}); calling them for other types results in a run-time
 \code{laerror} (``not implemented for this type'') or a link error. Functions implemented in the headers work
 for any type with the necessary arithmetic operations.
 

doc/fourindex.tex

@@ -24,9 +24,9 @@ Indices are opaque to the sparse classes (they are never compared with the dimen
 1-based indexing are possible there; the dense classes use indices from 1.
 
 Only \code{fourindex<unsigned char,double>}, \code{fourindex_ext<unsigned char,double>} and
-\code{fourindex_dense<nosymmetry,double,unsigned char>} are explicitly instantiated in fourindex.cc;
+\code{fourindex_dense<nosymmetry,double,unsigned char>} and the same with \code{float} are explicitly instantiated in fourindex.cc;
 everything else is implemented in the header and instantiated on use, except \code{fscanf()} and
-\code{fprintf()} which exist only for \code{fourindex<unsigned char,double>}.
+\code{fprintf()} which exist only for \code{fourindex<unsigned char,double>} and \code{fourindex<unsigned char,float>}.
 
 \section{Symmetry types}\label{sec:fourindex:symmetry}
 
@@ -131,7 +131,7 @@ The symmetry must be set (not \code{undefined_symmetry}) for \code{piterator} an
 \lafunc{unsigned long get(int fd, bool withattr=true)}{read the same format until end of file, \emph{adding} the elements to the list; returns their number.}
 \lafunc{template <class I, class T> std::ostream& operator<<(std::ostream &s, const fourindex<I,T> &x)}{text output: the dimension, lines \code{i j k l value}, and a final line with four terminators and 0.}
 \lafunc{template <class I, class T> std::istream& operator>>(std::istream &s, fourindex<I,T> &x)}{read the text format; \code{x} is resized to the read dimension (the symmetry is not set) and reading stops at the first line containing the terminator of \code{x} as an index.}
-\lafunc{void fscanf(FILE *f); void fprintf(FILE *f, char *format) const}{C-style text I/O, only for \code{fourindex<unsigned char,double>}; line format \code{value i j k l}. \code{fscanf} adds elements until a line in which any index equals the terminator (compared after conversion to the index type, so $-1$ in the file matches the default terminator 255); \code{fprintf} applies \code{format} (which must consume a \code{double} and four \code{int}s) to each element and ends with a line \code{0 t t t t} with the terminator $t$. With terminator 0 this is the FCIDUMP-like convention (blocks separated by zero lines, read by consecutive \code{fscanf} calls; a one-electron line $i\,j\,0\,0$ also ends a block).}
+\lafunc{void fscanf(FILE *f); void fprintf(FILE *f, char *format) const}{C-style text I/O, only for \code{fourindex<unsigned char,double>} and \code{<unsigned char,float>}; line format \code{value i j k l}. \code{fscanf} adds elements until a line in which any index equals the terminator (compared after conversion to the index type, so $-1$ in the file matches the default terminator 255); \code{fprintf} applies \code{format} (which must consume a \code{double} and four \code{int}s) to each element and ends with a line \code{0 t t t t} with the terminator $t$. With terminator 0 this is the FCIDUMP-like convention (blocks separated by zero lines, read by consecutive \code{fscanf} calls; a one-electron line $i\,j\,0\,0$ also ends a block).}
 \end{lareference}
 
 \section{Disc storage: \texttt{fourindex\_ext<I,T>}}\label{sec:fourindex:ext}

doc/iterative.tex

@@ -14,6 +14,9 @@ or of the normal equations $A\Tr Ax=A\Tr b$ for rectangular $A$ (least squares);
 \item the class \code{DIIS} (\header{diis.h}): Pulay's DIIS convergence acceleration of a general fixed-point iteration;
 \item \code{bisection_find()} and \code{interpolation_find()} (\header{bisection.h}): search in a sorted array.
 \end{itemize}
+All of them work for \code{T = float} and \code{std::complex<float>} as well (where the double versions support the
+corresponding real or complex type); their default thresholds then come from \code{LA_precision<float>}
+(section~\ref{sec:concepts:traits}).
 Dense (LAPACK based) diagonalization, linear solvers and matrix functions are in chapter~\ref{ch:nonclass}; the small
 subspace problems of \code{davidson} and \code{lanczos} are solved by the routines described there
 (\code{diagonalize}, \code{gdiagonalize}, \code{diagonalize3}, \code{linear_solve}). The quasi-Newton GDIIS optimizer built on
@@ -96,7 +99,7 @@ problem.
 \laheader{davidson.h}
 
 \begin{lareference}
-\lafunc{template <typename T, typename Matrix> void davidson(const Matrix &bigmat, NRVec<T> &eivals, NRVec<T> *eivecs, const char *eivecsfile, int nroots=1, const bool verbose=0, const double eps=1e-6, const bool incore=1, int maxit=100, const int maxkrylov = 500, void (*initguess)(NRVec<T> &)=NULL, const typename LA_traits<T>::normtype *target=NULL)}{
+\lafunc{template <typename T, typename Matrix> void davidson(const Matrix &bigmat, NRVec<T> &eivals, NRVec<T> *eivecs, const char *eivecsfile, int nroots=1, const bool verbose=0, const double eps=LA_precision<normtype>::eigensolver_eps(), const bool incore=1, int maxit=100, const int maxkrylov = 500, void (*initguess)(NRVec<T> &)=NULL, const typename LA_traits<T>::normtype *target=NULL)}{
 Davidson's method \cite{iterative:davidson1975} for the \code{nroots} lowest eigenvalues of \code{bigmat}.
 \begin{itemize}
 \item \code{eivals}: on return the eigenvalues in its first \code{nroots} elements (ascending, or by increasing distance from
@@ -167,7 +170,7 @@ implicitly given one, built column by column by \code{gemv} on unit vectors (for
 \laheader{lanczos.h}
 
 \begin{lareference}
-\lafunc{template <typename T, typename Matrix> void lanczos(const Matrix &bigmat, NRVec<T> &eivals, NRVec<T> *eivecs, const char *eivecsfile, int nroots=1, const bool verbose=0, const double eps=1e-6, const bool incore=1, int maxit=100, const int maxkrylov = 1000, void (*initguess)(NRVec<T> &)=NULL, const typename LA_traits<T>::normtype *target=NULL)}{
+\lafunc{template <typename T, typename Matrix> void lanczos(const Matrix &bigmat, NRVec<T> &eivals, NRVec<T> *eivecs, const char *eivecsfile, int nroots=1, const bool verbose=0, const double eps=LA_precision<normtype>::eigensolver_eps(), const bool incore=1, int maxit=100, const int maxkrylov = 1000, void (*initguess)(NRVec<T> &)=NULL, const typename LA_traits<T>::normtype *target=NULL)}{
 Lanczos method for the \code{nroots} lowest (or closest to \code{*target}) eigenvalues of a hermitian matrix. The parameters
 have the same meaning as for \code{davidson}, with these differences: \code{bigmat.issymmetric()} must be \code{true}
 (\code{laerror} otherwise); the start vector is normalized by the function; \code{incore=false} keeps only the Lanczos vectors
@@ -194,7 +197,7 @@ the method fails with \code{laerror} when more than \code{maxit} iterations are
 \laheader{conjgrad.h}
 
 \begin{lareference}
-\lafunc{template<typename T, typename Matrix> bool conjgrad(const Matrix &bigmat, const NRVec<T> &b, NRVec<T> &x, const bool doguess=true, const double tol=1e-8, const int itmax=1000, const bool verbose=true, bool issquare=1, const bool precondition=1)}{
+\lafunc{template<typename T, typename Matrix> bool conjgrad(const Matrix &bigmat, const NRVec<T> &b, NRVec<T> &x, const bool doguess=true, const double tol=LA_precision<normtype>::conjgrad_tol(), const int itmax=1000, const bool verbose=true, bool issquare=1, const bool precondition=1)}{
 Solves $Ax=b$ by the (Jacobi preconditioned) conjugate gradient method (the source refers to the book of Stoer and Bulirsch).
 Returns \code{true} if converged, \code{false} after \code{itmax}+1 iterations without convergence; \code{x} then holds the
 last iterate.
@@ -227,7 +230,7 @@ $p\cdot Ap$ (or $|Ap|^2$) or the residual product $r\cdot z$ vanishes. Only real
 \laheader{gmres.h}
 
 \begin{lareference}
-\lafunc{template<typename T, typename Matrix> bool gmres(const Matrix &bigmat, const NRVec<T> &b, NRVec<T> &x, const bool doguess=1, const double eps=1e-7, const int MAXIT=50, const bool verbose=1, bool square=1, const bool precondition=1, int neustart=0, const int incore=1)}{
+\lafunc{template<typename T, typename Matrix> bool gmres(const Matrix &bigmat, const NRVec<T> &b, NRVec<T> &x, const bool doguess=1, const double eps=LA_precision<normtype>::gmres_eps(), const int MAXIT=50, const bool verbose=1, bool square=1, const bool precondition=1, int neustart=0, const int incore=1)}{
 Solves $Ax=b$ by restarted GMRES with Givens rotations (after H.~R.~Schwarz, original implementation by M.~Warken). Returns
 \code{true} if converged; otherwise \code{false}, with \code{x} holding the last iterate.
 \begin{itemize}

doc/nonclass.tex

@@ -119,7 +119,7 @@ scaled). Returns \code{info}.}
 \lafunc{const NRMat<T> calcsvdinverse(NRMat<T> a, double thr=0, bool do_absadjoint=false)}{Inverse via SVD, $V\,\mathrm{diag}(1/s_i)\,U\Hc$,
 singular values $s_i<$\code{thr} are treated as zero (pseudoinverse). With \code{do_absadjoint} returns
 $V\,\mathrm{diag}(\prod_{j\ne i}s_j)\,U\Hc$, the adjugate matrix up to the phase $\det U\det V$. Square matrices only.}
-\lafunc{NRVec<T> svd_solve(NRMat<T> &a, const NRVec<T> &b, double thres=1e-12, int *nullity=NULL); NRMat<T> svd_solve(NRMat<T> &a, const NRMat<T> &b, double thres=1e-12, int *nullity=NULL)}{Least-squares
+\lafunc{NRVec<T> svd_solve(NRMat<T> &a, const NRVec<T> &b, normtype thres=LA_precision<normtype>::svd_threshold(), int *nullity=NULL); NRMat<T> svd_solve(NRMat<T> &a, const NRMat<T> &b, normtype thres=LA_precision<normtype>::svd_threshold(), int *nullity=NULL)}{Least-squares
 solution $x=V\,\mathrm{diag}(1/s_i)\,U\Hc b$ discarding singular values $\le$\code{thres} (no contamination by the null space);
 the number of discarded ones is stored in \code{*nullity}. Here (unlike \code{linear_solve}) the right-hand sides are the
 \emph{columns} of the matrix \code{b} ($m\times n_{\mathrm{rhs}}$); \code{a} ($m\times n$, $m\ge n$) is destroyed.}
@@ -223,8 +223,8 @@ so they can be used with implicitly defined (``direct'') matrices as in chapter~
 needs \code{size()}, \code{operator*=}, \code{axpy} and \code{resize()}.
 
 The exponential uses scaling and squaring: the power $p$ is chosen so that $\|A\|/2^p\lesssim 0.75$
-(\code{mynextpow2}), the Taylor series of the scaled argument is truncated when the next term would be below $10^{-14}$
+(\code{mynextpow2}), the Taylor series of the scaled argument is truncated when the next term would be below
-relative to the norm, and the result is squared $p$ times. The scaling is put into the Taylor coefficients,
+\code{LA_precision<normtype>::taylor_precision()} ($10^{-14}$ for double, $10^{-7}$ for float) relative to the norm, and the result is squared $p$ times. The scaling is put into the Taylor coefficients,
 so the argument itself is not copied. \code{maxpower} (if ${}\ge0$) limits $p$, \code{maxtaylor} (if ${}\ge0$) limits
 the order of the Taylor polynomial, which is useful and exact for nilpotent matrices (e.g.\ coupled-cluster excitation
 operators).

doc/optimize.tex

@@ -23,12 +23,13 @@ Methods are written on top of a small infrastructure (evaluation counter, iterat
 common drivers (line search, first order, trust region), so that a new method usually consists of a short class supplying the
 search direction, the update rule or the trust-region step (section~\ref{sec:optimize:drivers}).
 
-All templates are parameterized by the real type \code{T}, but optimize.cc instantiates them for \code{T = double} only.
+All templates are parameterized by the real type \code{T}; optimize.cc instantiates them for \code{T = double} and
-The capability interfaces, adapters and the hyperparameter classes are header-only templates; the optimizers, the adapters with
+\code{float}. The capability interfaces, adapters and the hyperparameter classes are header-only templates; the optimizers, the
-member functions in optimize.cc (\code{FrozenOptProblem}, \code{ParameterShift}, \code{FiniteDifference},
+adapters with member functions in optimize.cc (\code{FrozenOptProblem}, \code{ParameterShift}, \code{FiniteDifference},
-\code{OptLeastSquares}) and the auxiliary functions exist in the library only for \code{double}.
+\code{OptLeastSquares}) and the auxiliary functions exist in the library for \code{double} and \code{float}.
 The common drivers \code{linesearch_minimize}, \code{firstorder_minimize} and \code{trustregion_minimize}, the extension
-points for user-written methods, are explicitly instantiated for \code{double} as well.
+points for user-written methods, are explicitly instantiated for both types as well. The defaults depending on the precision
+are taken from \code{opt_precision<T>} (section~\ref{sec:optimize:hyper}).
 
 The data members of all classes of this chapter are small, except vectors held by \code{FrozenOptProblem} and
 \code{optevaluator}; these are ordinary \code{NRVec} objects following the reference counting and copy-on-demand scheme of
@@ -254,6 +255,29 @@ hyperparameters                    maxstep(200) maxfunc(0) f_thr(0) x_thr(0) sad
  +- Cobyla_hyperparameters         rhobeg(0.5) rhoend(1e-6) flexible_radius(true); maxstep=100000
 \end{lstlisting}
 
+The defaults given above and below are those for \code{double}. The precision-dependent ones are taken from the trait
+\code{opt_precision<T>} (\header{optimize.h}, analogous to \code{LA_precision<T>} of section~\ref{sec:concepts:traits}), defined
+for \code{T = double} and \code{float} only; the double values are those used by the library before. In float the gradient norm
+reachable by a line search is limited by the resolution of $f$ (about $\epsilon|f|$ divided by the step), hence the larger
+\code{g_thr}.
+\begin{center}
+\begin{tabular}{llll}
+\toprule
+member & double & float & default of \\
+\midrule
+\code{fd_central_step()} & $10^{-6}$ & $5\cdot10^{-3}$ & \code{hh} of \code{FiniteDifference}, \code{ParameterShift} \\
+\code{fd_forward_step()} & $10^{-7}$ & $3\cdot10^{-4}$ & \code{h} of \code{OptLeastSquares}, \code{my_fd_step} (LM) \\
+\code{g_thr()} & $10^{-6}$ & $10^{-3}$ & \code{my_g_thr} \\
+\code{sy_thr()} & $10^{-12}$ & $10^{-6}$ & \code{my_sy_thr} (quasi-Newton) \\
+\code{sr1_thr()} & $10^{-8}$ & $10^{-4}$ & \code{my_sr1_thr} (SR1) \\
+\code{update_thr()} & $10^{-8}$ & $10^{-4}$ & \code{my_update_thr} (trust region) \\
+\code{nm_f_thr()}, \code{nm_x_thr()} & $10^{-8}$, $10^{-6}$ & $10^{-5}$, $10^{-4}$ & \code{my_f_thr}, \code{my_x_thr} (Nelder-Mead) \\
+\code{powell_f_thr()}, \code{powell_x_thr()} & $10^{-10}$, $10^{-6}$ & $10^{-6}$, $10^{-4}$ & \code{my_f_thr}, \code{my_x_thr} (Powell) \\
+\code{rhoend()} & $10^{-6}$ & $10^{-4}$ & \code{my_rhoend} (COBYLA) \\
+\bottomrule
+\end{tabular}
+\end{center}
+
 \paragraph{Base classes.}
 \begin{lareference}
 \lafunc{template<typename T> class hyperparameters}{Parameters common to all methods.}
@@ -671,16 +695,19 @@ saddle order.}
 
 For least-squares problems providing \code{OptResiduals} (typically derived from \code{OptLeastSquares}): the step $h$ solves
 $(J\Tr J+\mu D)h=-J\Tr r$ (\code{linear_solve}), $D=I$ or $D=\mathrm{diag}(J\Tr J)$ (Marquardt scaling); the damping starts at
-$\mu=\tau\max_i(J\Tr J)_{ii}$ and is updated by the gain ratio $\varrho$ \cite{optimize:madsen2004} (alg.~3.16): for $\varrho>0$
+$\mu=\tau\max_i(J\Tr J)_{ii}$ (with scaling $\mu=\tau$, the same relative to $D$) and is updated by the gain ratio $\varrho$ \cite{optimize:madsen2004} (alg.~3.16): for $\varrho>0$
 the step is accepted and $\mu\leftarrow\mu\max(1/3,1-(2\varrho-1)^3)$, $\nu=2$, otherwise $\mu\leftarrow\mu\nu$, $\nu\leftarrow2\nu$.
 The Jacobian comes from \code{jacobian()} of the problem or from forward differences (relative step \code{my_fd_step}, $n$
 residual evaluations counted in \code{my_maxfunc}). Convergence: $|J\Tr r|<$ \code{my_g_thr}, the general $f$/$x$ criterion, or
-when a rejected step becomes negligible at machine precision ($|h|\le10\epsilon(|x|+\epsilon)$). The function value returned is
+when a rejected step becomes negligible at machine precision ($h_i^2\le(10\epsilon)^2(x_i^2+\epsilon^2)$ for all $i$,
+tested per component to be scale invariant, cf.\ \cite{optimize:madsen2004} eq.~3.16). The function value returned is
 $f=\frac12|r|^2$. The problem's \code{funcgrad()} is not used.
 \begin{lareference}
 \lafunc{template<typename T> class LM_hyperparameters : public gradient_hyperparameters<T>}{}
 \lafunc{T my_tau}{Initial damping factor $\tau$ (\code{1e-3}).}
-\lafunc{bool my_scaling}{Marquardt scaling $D=\mathrm{diag}(J\Tr J)$ (default \code{false}, $D=I$).}
+\lafunc{bool my_scaling}{Marquardt scaling $D=\mathrm{diag}(J\Tr J)$ (default \code{false}, $D=I$). Recommended for
+badly scaled parameters, in float usually necessary (otherwise the steps in the weakly curved directions fall below the
+resolution of $f$).}
 \lafunc{T my_fd_step}{Relative step of the forward-difference Jacobian (\code{1e-7}).}
 \lafunc{template<typename T> class LM_Optimizer : public Optimizer<T>}{\code{laerror} if the problem does not provide
 \code{OptResiduals}. Verbose name Levenberg-Marquardt.}

doc/polynomial.tex

@@ -10,7 +10,7 @@ This chapter describes the polynomial class \code{Polynomial<T>} and the exact r
 \code{template <typename T> class Polynomial : public NRVec<T>} represents $p(x)=\sum_{i=0}^{d}c_ix^i$; the coefficient $c_i$ is the vector element
 \code{p[i]} (indexed from 0, i.e.\ by the power) and the vector has $d+1$ elements. The member \code{size()} is made private to prevent confusion; use
 \code{degree()}. The leading coefficient may be zero (e.g.\ after a subtraction); \code{simplify()} removes such terms. The class is explicitly
-instantiated for \code{int}, \code{unsigned int}, \code{double}, \code{std::complex<double>}; root finding and SVD-based gcd require a floating point type.
+instantiated for \code{int}, \code{unsigned int}, \code{double}, \code{std::complex<double>}, \code{float}, \code{std::complex<float>}; root finding and SVD-based gcd require a floating point type.
 
 All inherited \code{NRVec} functionality (norms, \code{randomize}, \code{clear}, I/O operators printing the number of coefficients followed by the
 coefficients, \ldots) is available. \code{Polynomial} follows the copy-on-demand scheme of section~\ref{sec:concepts:refcount}: its modifying members call
@@ -80,7 +80,7 @@ vector, hence also a \code{Polynomial} (section~\ref{sec:nonclass:matexp}).
 \lafunc{NRVec<typename LA_traits<T>::complextype> roots() const}{all roots as eigenvalues of the companion matrix (\code{gdiagonalize}, chapter~\ref{ch:nonclass}); for \code{double}
 	and \code{std::complex<double>} only (laerror for integer types).}
 \lafunc{NRVec<T> realroots(const typename LA_traits<T>::normtype thr) const}{roots with $|\mathrm{Im}|<$ \code{thr}, real parts sorted ascending (\code{T = double}).}
-\lafunc{T newton(const T x0, const typename LA_traits<T>::normtype thr=1e-14, const int maxit=1000) const}{Newton iteration from \code{x0} until $|p(x)|<$ \code{thr}; no convergence report.}
+\lafunc{T newton(const T x0, const typename LA_traits<T>::normtype thr=LA_precision<normtype>::newton_threshold(), const int maxit=1000) const}{Newton iteration from \code{x0} until $|p(x)|<$ \code{thr}; no convergence report.}
 \lafunc{template <typename T> Polynomial<T> polyfromroots(const NRVec<T> &roots, const int skip= -1)}{monic $\prod_{i\ne\mathrm{skip}}(x-r_i)$.}
 \lafunc{template <typename T> Polynomial<T> lagrange_interpolation(const NRVec<T> &x, const NRVec<T> &y)}{interpolating polynomial of degree $n-1$ through the $n$ points; laerror for repeated $x$.}
 \lafunc{template <typename T> NRMat<T> Sylvester(const Polynomial<T> &p, const Polynomial<T> &q)}{Sylvester matrix of dimension $\deg p+\deg q$ (its determinant is the resultant).}

doc/sparse.tex

@@ -23,7 +23,7 @@
 Common definitions in \code{sparsemat.h}:
 \begin{lareference}
 \lafunc{typedef unsigned int SPMatindex; typedef int SPMatindexdiff}{Index type and index difference type. All row and column indices count from 0.}
-\lafunc{const double SPARSEEPSILON=1e-15}{Threshold: elements with absolute value below (or equal to) it are dropped by \code{simplify()} and by most constructors and accumulating operations.}
+\lafunc{const double SPARSEEPSILON=1e-15}{Kept for user code. The library itself uses the precision-dependent threshold \code{LA_precision<normtype>::sparse_epsilon()} (section~\ref{sec:concepts:traits}; $10^{-15}$ for double, i.e.\ $\epsilon_{\rm sp}$, and $10^{-7}$ for float): elements with absolute value below (or equal to) it are dropped by \code{simplify()} and by most constructors and accumulating operations; it is denoted $\epsilon_{\rm sp}$ below.}
 \lafunc{template<typename T> struct matel {T elem; SPMatindex row; SPMatindex col; matel *next;}}{List element of \code{SparseMat}; also the value type returned by the iterators of both classes.}
 \end{lareference}
 Both classes are explicitly instantiated for \code{double} and \code{std::complex<double>}.
@@ -43,7 +43,7 @@ which no longer exists, is not included by any file and is not installed; it is
 The nonzero elements form a singly linked list of \code{matel<T>} (pointer \code{list}); new elements are
 prepended. The list is in no particular order and may contain several entries with the same
 $(i,j)$, which represent their sum; \code{simplify()} merges such duplicates, removes elements with
-$|a_{ij}|\le$ \code{SPARSEEPSILON} and leaves the list sorted by rows. Operations needing an ordering
+$|a_{ij}|\le$ $\epsilon_{\rm sp}$ and leaves the list sorted by rows. Operations needing an ordering
 (\code{simplify}, sparse products, \code{dot}) build private arrays of pointers sorted by rows or by columns
 (member \code{sort()}); these arrays and the cached element count \code{nonzero} are kept until the next
 operation which calls the private \code{unsort()}: \code{add()} (and everything built on it), \code{simplify},
@@ -86,7 +86,7 @@ Most modifying members call \code{copyonwrite()}: \code{simplify}, \code{operato
 \lafunc{SparseMat()}{Undefined empty matrix ($0\times0$, no reference count).}
 \lafunc{SparseMat(const SPMatindex n, const SPMatindex m)}{Defined empty (all zero) $n\times m$ matrix, not symmetric.}
 \lafunc{SparseMat(const SparseMat &rhs)}{Shallow copy; an empty matrix gets its own reference count.}
-\lafunc{explicit SparseMat(const NRMat<T> &rhs)}{From a dense matrix, elements with $|a_{ij}|>$ \code{SPARSEEPSILON}.}
+\lafunc{explicit SparseMat(const NRMat<T> &rhs)}{From a dense matrix, elements with $|a_{ij}|>$ $\epsilon_{\rm sp}$.}
 \lafunc{explicit SparseMat(const NRSMat<T> &rhs)}{From a packed symmetric matrix; symmetric flag set, lower triangle stored.}
 \lafunc{~SparseMat()}{Destructor.}
 \lafunc{SparseMat & operator=(const SparseMat &rhs)}{Shallow copy (also copies the symmetric flag).}
@@ -107,8 +107,8 @@ duplicates summed), \code{explicit NRSMat<T>(const SparseMat<T>&)} (requires the
 \subsection{Element insertion and access}
 \begin{lareference}
 \lafunc{inline void add(const SPMatindex n, const SPMatindex m, const T elem)}{Prepend element $(n,m)$; it is summed with existing entries at the same position. No checks, no \code{copyonwrite()}, see section~\ref{sec:sparse:refcount}. The matrix must have been dimensioned (constructor with dimensions or \code{resize}).}
-\lafunc{void addsafe(const SPMatindex n, const SPMatindex m, const T elem)}{As \code{add}, but ignores $|$\code{elem}$|\le$ \code{SPARSEEPSILON}, calls \code{copyonwrite()} and creates the reference count of an undefined matrix (range check only with the lowercase \code{debug} macro).}
+\lafunc{void addsafe(const SPMatindex n, const SPMatindex m, const T elem)}{As \code{add}, but ignores $|$\code{elem}$|\le\epsilon_{\rm sp}$, calls \code{copyonwrite()} and creates the reference count of an undefined matrix (range check only with the lowercase \code{debug} macro).}
-\lafunc{void simplify(const double sparseepsilon=SPARSEEPSILON)}{Sum duplicates, remove elements with $|a_{ij}|\le$ \code{sparseepsilon}, sort by rows; symmetric matrices are first moved to the upper triangle ($i\le j$).}
+\lafunc{void simplify(const typename LA_traits<T>::normtype sparseepsilon=LA_precision<normtype>::sparse_epsilon())}{Sum duplicates, remove elements with $|a_{ij}|\le$ \code{sparseepsilon}, sort by rows; symmetric matrices are first moved to the upper triangle ($i\le j$).}
 \lafunc{unsigned int length() const}{Number of list elements (cached until the next modification, see section~\ref{sec:sparse:storage}).}
 \lafunc{unsigned int sort(int type) const}{Build the internal array sorted by rows (\code{type=0}) or columns (\code{type=1}) and return the number of elements. Logically const, but unfolds a symmetric matrix (\code{setunsymmetric}). Normally called internally only.}
 \lafunc{inline matel<T> *getlist() const; void setlist(matel<T> *l)}{Raw access to the list head (no reference count handling; \code{setlist} does not free the old list).}
@@ -245,7 +245,7 @@ copies).
 \lafunc{SparseSMat()}{Undefined empty matrix.}
 \lafunc{explicit SparseSMat(const SPMatindex n); explicit SparseSMat(const SPMatindex n, const SPMatindex m)}{Empty (zero) $n\times n$ or $n\times m$ matrix.}
 \lafunc{SparseSMat(const SparseSMat &rhs)}{Shallow copy.}
-\lafunc{explicit SparseSMat(const NRSMat<T> &rhs)}{From a packed symmetric (complex: hermitian) matrix (both triangles inserted, the upper one conjugated, $|a_{ij}|>$ \code{SPARSEEPSILON}).}
+\lafunc{explicit SparseSMat(const NRSMat<T> &rhs)}{From a packed symmetric (complex: hermitian) matrix (both triangles inserted, the upper one conjugated, $|a_{ij}|>$ $\epsilon_{\rm sp}$).}
 \lafunc{explicit SparseSMat(const NRMat<T> &rhs)}{From a dense $n\times m$ matrix, nonzero elements copied as they are (no symmetrization).}
 \lafunc{explicit SparseSMat(const SparseMat<T> &rhs)}{From a \code{SparseMat} (same dimensions); duplicates are summed, a symmetric \code{SparseMat} is unfolded without conjugation (complex symmetric, as \code{SparseMat} defines it).}
 \lafunc{explicit SparseSMat(const CSRMat<T> &rhs)}{Declared but not implemented (\code{CSRMat} is not supported).}
@@ -266,7 +266,7 @@ Dense from sparse: \code{explicit NRMat<T>(const SparseSMat<T>&)} copies all sto
 \lafunc{inline void add(const SPMatindex n, const SPMatindex m, const T elem, const bool both=true)}{Add \code{elem} to element $(n,m)$ and, if \code{both} and $n\ne m$, its complex conjugate to $(m,n)$. Creates the row map if needed; no \code{copyonwrite()}; range check in DEBUG only. Resulting zeros remain until \code{simplify()}.}
 \lafunc{inline const std::map<SPMatindex,T> *line(const SPMatindex n) const}{Read-only access to row $n$ (\code{NULL} if empty).}
 \lafunc{inline ROWTYPE & operator[](const SPMatindex i)}{Row $i$ as a reference to the pointer \code{std::map<SPMatindex,T>*} (\code{typedef ROWTYPE}), possibly \code{NULL}. Non-const only.}
-\lafunc{unsigned long long simplify()}{Remove elements with $|a_{ij}|<$ \code{SPARSEEPSILON} and empty rows; returns the number of remaining elements.}
+\lafunc{unsigned long long simplify()}{Remove elements with $|a_{ij}|<$ $\epsilon_{\rm sp}$ and empty rows; returns the number of remaining elements.}
 \lafunc{inline unsigned long long length()}{\code{simplify()} (non-const).}
 \lafunc{int nrows() const; int ncols() const; inline bool issymmetric() const}{Dimensions; \code{issymmetric()} is always true.}
 \lafunc{iterator begin() const; iterator end() const}{Iterator over all stored elements, row by row in increasing column order. \code{*it} is a \code{matel<T>} \emph{copy} held in the iterator (writing to it does not change the matrix). It is meant for comparison with \code{end()} or via \code{it.notend()}; \code{iterator(const SparseSMat&)} starts at the first element.}
@@ -288,7 +288,7 @@ Dense from sparse: \code{explicit NRMat<T>(const SparseSMat<T>&)} copies all sto
 \lafunc{const T* diagonalof(NRVec<T> &r, const bool divide=0, bool cache=false) const}{As for \code{SparseMat} (square matrices only).}
 \lafunc{SparseSMat transpose(bool conj=false) const}{Transposed (optionally conjugated) copy.}
 \lafunc{void transposeme() const}{Raises an error (not implemented).}
-\lafunc{SparseSMat<T> cholesky(void) const}{Pivoted Cholesky decomposition of a positive semidefinite symmetric/hermitian matrix (not checked): returns $R$ with $R\Hc R = A$ (\code{result.transpose(true)*result} reproduces the input). The pivoting is done a priori by sorting the diagonal; pivots with real part below \code{CHOLESKYEPSILON}$=10^{-16}$ are skipped (rank-deficient case), elements below \code{SPARSEEPSILON} are dropped. $R$ is upper triangular only in the permuted basis.}
+\lafunc{SparseSMat<T> cholesky(void) const}{Pivoted Cholesky decomposition of a positive semidefinite symmetric/hermitian matrix (not checked): returns $R$ with $R\Hc R = A$ (\code{result.transpose(true)*result} reproduces the input). The pivoting is done a priori by sorting the diagonal; pivots with real part below \code{LA_precision<normtype>::cholesky_epsilon()} ($10^{-16}$ for double, the old \code{CHOLESKYEPSILON}, $10^{-8}$ for float) are skipped (rank-deficient case), elements below $\epsilon_{\rm sp}$ are dropped. $R$ is upper triangular only in the permuted basis.}
 \lafunc{SparseSMat submatrix(const int fromrow, const int torow, const int fromcol, const int tocol) const}{Block (inclusive bounds).}
 \lafunc{void storesubmatrix(const int fromrow, const int fromcol, const SparseSMat &rhs)}{Store \code{rhs} into the block starting at $(\mathtt{fromrow},\mathtt{fromcol})$: the old elements of the block are removed first (as \code{NRMat::storesubmatrix}); calls \code{copyonwrite()}.}
 \end{lareference}
@@ -302,7 +302,7 @@ Dense from sparse: \code{explicit NRMat<T>(const SparseSMat<T>&)} copies all sto
 \lafunc{SparseSMat<T> otimes_sparse(const NRVec<T> &lhs, const NRVec<T> &rhs, const bool conjugate=false, const T &scale=1)}{\emph{Symmetrized} outer product, symmetric (complex: hermitian) regardless of the input vectors (unlike \code{SparseMat}):
 with $M=\mathtt{scale}\cdot l\,r\Tr$ (or $l\,r\Hc$ if \code{conjugate}) the result is $S=M+M\Hc$ ($=l\,r\Tr+r\,l\Tr$ for real types),
 including the doubled diagonal $S_{ii}=M_{ii}+M_{ii}^*$. Note that \code{otimes_sparse(x,x,true)} is $2\,x x\Hc$; use \code{scale=0.5} for $x x\Hc$.
-Elements with $|S_{ij}|\le$ \code{SPARSEEPSILON} are omitted. Both vectors must have the same length (\code{laerror} otherwise).}
+Elements with $|S_{ij}|\le$ $\epsilon_{\rm sp}$ are omitted. Both vectors must have the same length (\code{laerror} otherwise).}
 \lafunc{SparseSMat<T> NRMat<T>::operator*(const SparseSMat<T> &rhs) const}{Declared in mat.h: dense times sparse, general product $AS$ with a sparse result (row $i$ accumulated as $\sum_k A_{ik}\,(\text{row }k\text{ of }S)$, zero $A_{ik}$ skipped; simplified). Instantiated for \code{double} and \code{std::complex<double>}.}
 \end{lareference}
 In \ref{ch:nonclass}: \code{complexmatrix()}, \code{realmatrix()} and \code{imagmatrix()} build a

doc/tensor.tex

@@ -18,8 +18,8 @@ Typical uses are many-body quantities (amplitudes, integrals, density matrices)
 in groups of indices should be exploited for storage.
 Limitations stated in the source:
 \begin{itemize}
-\item the implementation is ``rudimentary''; only \code{Tensor<double>} and \code{Tensor<std::complex<double> >}
+\item the implementation is ``rudimentary''; only \code{Tensor<double>}, \code{Tensor<std::complex<double> >} and their single precision
-	are instantiated (tensor.cc);
+	counterparts \code{Tensor<float>}, \code{Tensor<std::complex<float> >} are instantiated (tensor.cc);
 \item index types \code{LA_index} and \code{LA_largeindex} are \code{int}, so the total size is limited
 	to $2^{31}$ elements (\code{NRVec} limitation);
 \item symmetry \emph{between} index groups (e.g.\ $(ij|kl)=(kl|ij)$) is not supported; one would have to
@@ -291,7 +291,7 @@ Names are permuted together with the indices unless stated otherwise.
 \begin{lareference}
 \lafunc{void apply_permutation_algebra(const Tensor &rhs, const PermutationAlgebra<int,T> &pa, bool inverse=false, T alpha=1, T beta=0, bool conjugate_by_parity=false)}{$\mathtt{*this} \leftarrow \beta\,\mathtt{*this}$, then for all stored elements $I$ of \code{*this}: $\mathtt{this}(I) \mathrel{+}= \alpha \sum_P c_P\, \mathtt{rhs}(P(I))$, with $P$ acting on the flat index list (inverse permutations if \code{inverse}); with \code{conjugate_by_parity} odd permutations take the complex conjugate. Can represent general (anti)symmetrizers, e.g.\ \code{general_antisymmetrizer} (chapter~\ref{ch:permutation}). \code{*this} may have a symmetry which \code{rhs} lacks. The names of \code{*this} are taken from (or checked against) the names of \code{rhs} permuted by the first permutation. Element-by-element, not optimally efficient.}
 \lafunc{void apply_permutation_algebra(const NRVec<Tensor> &rhsvec, const PermutationAlgebra<int,T> &pa, bool inverse=false, T alpha=1, T beta=0)}{the same with \code{rhs} being the outer product of the tensors in \code{rhsvec}, which is not formed explicitly (e.g.\ Grassmann products building a density matrix from cumulants); rather inefficient.}
-\lafunc{NRVec<NRMat<T> > Tucker(typename LA_traits<T>::normtype thr=1e-12, bool inverseorder=false)}{HOSVD/Tucker decomposition: \code{*this} is replaced by the flat core tensor and the factor matrices (one per index, columns orthonormal) are returned; singular values below \code{thr} are dropped, which reduces the corresponding range of the core.}
+\lafunc{NRVec<NRMat<T> > Tucker(typename LA_traits<T>::normtype thr=LA_precision<normtype>::svd_threshold(), bool inverseorder=false)}{HOSVD/Tucker decomposition: \code{*this} is replaced by the flat core tensor and the factor matrices (one per index, columns orthonormal) are returned; singular values below \code{thr} are dropped, which reduces the corresponding range of the core.}
 \lafunc{Tensor inverseTucker(const NRVec<NRMat<T> > &x, bool inverseorder=false) const}{rebuild the tensor from the core (\code{*this}) and the matrices returned by \code{Tucker} with the same \code{inverseorder}.}
 \lafunc{Tensor linear_transform(const NRVec<NRMat<T> > &x) const}{transform every index of group $g$ by the matrix \code{x[g]} (\code{x[g].ncols()} must equal the range; the new range is \code{x[g].nrows()}), preserving the group symmetries: $t'_{\dots a \dots} = \sum_i x_{ai}\, t_{\dots i\dots}$.}
 \lafunc{Tensor linear_transform(const int g, const Tensor<T> &x) const; Tensor linear_transform(const int g, const NRMat<T> &x) const}{transform only group \code{g}, keeping its position and symmetry; \code{x} must be a flat rank-2 tensor (useful for raising/lowering indices) or a matrix.}