Mercurial > octave
view liboctave/numeric/oct-norm.cc @ 30898:51a3d3a69193
maint: Use "fcn" as preferred abbreviation for "function" in liboctave/.
* DAEFunc.h, DASPK.cc, DASSL.cc, LSODE.cc, ODEFunc.h, eigs-base.cc,
eigs-base.h, oct-norm.cc, mx-inlines.cc:
Replace "func", "fun" in documentation and variable names with "fcn".
author | Rik <rik@octave.org> |
---|---|
date | Tue, 05 Apr 2022 13:20:48 -0700 |
parents | 796f54d4ddbf |
children | a96f68a48e9e e88a07dec498 |
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//////////////////////////////////////////////////////////////////////// // // Copyright (C) 2008-2022 The Octave Project Developers // // See the file COPYRIGHT.md in the top-level directory of this // distribution or <https://octave.org/copyright/>. // // This file is part of Octave. // // Octave is free software: you can redistribute it and/or modify it // under the terms of the GNU General Public License as published by // the Free Software Foundation, either version 3 of the License, or // (at your option) any later version. // // Octave is distributed in the hope that it will be useful, but // WITHOUT ANY WARRANTY; without even the implied warranty of // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the // GNU General Public License for more details. // // You should have received a copy of the GNU General Public License // along with Octave; see the file COPYING. If not, see // <https://www.gnu.org/licenses/>. // //////////////////////////////////////////////////////////////////////// #if defined (HAVE_CONFIG_H) # include "config.h" #endif #include <cmath> #include <algorithm> #include <limits> #include <vector> #include "Array.h" #include "CColVector.h" #include "CMatrix.h" #include "CRowVector.h" #include "CSparse.h" #include "MArray.h" #include "dColVector.h" #include "dDiagMatrix.h" #include "dMatrix.h" #include "dRowVector.h" #include "dSparse.h" #include "fCColVector.h" #include "fCMatrix.h" #include "fCRowVector.h" #include "fColVector.h" #include "fDiagMatrix.h" #include "fMatrix.h" #include "fRowVector.h" #include "lo-error.h" #include "lo-ieee.h" #include "lo-mappers.h" #include "mx-cm-s.h" #include "mx-fcm-fs.h" #include "mx-fs-fcm.h" #include "mx-s-cm.h" #include "oct-cmplx.h" #include "oct-norm.h" #include "quit.h" #include "svd.h" namespace octave { // Theory: norm accumulator is an object that has an accum method able // to handle both real and complex element, and a cast operator // returning the intermediate norm. Reference: Higham, N. "Estimating // the Matrix p-Norm." Numer. Math. 62, 539-555, 1992. // norm accumulator for the p-norm template <typename R> class norm_accumulator_p { public: norm_accumulator_p () { } // we need this one for Array norm_accumulator_p (R pp) : m_p(pp), m_scl(0), m_sum(1) { } template <typename U> void accum (U val) { octave_quit (); R t = std::abs (val); if (m_scl == t) // we need this to handle Infs properly m_sum += 1; else if (m_scl < t) { m_sum *= std::pow (m_scl/t, m_p); m_sum += 1; m_scl = t; } else if (t != 0) m_sum += std::pow (t/m_scl, m_p); } operator R () { return m_scl * std::pow (m_sum, 1/m_p); } private: R m_p, m_scl, m_sum; }; // norm accumulator for the minus p-pseudonorm template <typename R> class norm_accumulator_mp { public: norm_accumulator_mp () { } // we need this one for Array norm_accumulator_mp (R pp) : m_p(pp), m_scl(0), m_sum(1) { } template <typename U> void accum (U val) { octave_quit (); R t = 1 / std::abs (val); if (m_scl == t) m_sum += 1; else if (m_scl < t) { m_sum *= std::pow (m_scl/t, m_p); m_sum += 1; m_scl = t; } else if (t != 0) m_sum += std::pow (t/m_scl, m_p); } operator R () { return m_scl * std::pow (m_sum, -1/m_p); } private: R m_p, m_scl, m_sum; }; // norm accumulator for the 2-norm (euclidean) template <typename R> class norm_accumulator_2 { public: norm_accumulator_2 () : m_scl(0), m_sum(1) { } void accum (R val) { R t = std::abs (val); if (m_scl == t) m_sum += 1; else if (m_scl < t) { m_sum *= pow2 (m_scl/t); m_sum += 1; m_scl = t; } else if (t != 0) m_sum += pow2 (t/m_scl); } void accum (std::complex<R> val) { accum (val.real ()); accum (val.imag ()); } operator R () { return m_scl * std::sqrt (m_sum); } private: static inline R pow2 (R x) { return x*x; } //-------- R m_scl, m_sum; }; // norm accumulator for the 1-norm (city metric) template <typename R> class norm_accumulator_1 { public: norm_accumulator_1 () : m_sum (0) { } template <typename U> void accum (U val) { m_sum += std::abs (val); } operator R () { return m_sum; } private: R m_sum; }; // norm accumulator for the inf-norm (max metric) template <typename R> class norm_accumulator_inf { public: norm_accumulator_inf () : m_max (0) { } template <typename U> void accum (U val) { if (math::isnan (val)) m_max = numeric_limits<R>::NaN (); else m_max = std::max (m_max, std::abs (val)); } operator R () { return m_max; } private: R m_max; }; // norm accumulator for the -inf pseudonorm (min abs value) template <typename R> class norm_accumulator_minf { public: norm_accumulator_minf () : m_min (numeric_limits<R>::Inf ()) { } template <typename U> void accum (U val) { if (math::isnan (val)) m_min = numeric_limits<R>::NaN (); else m_min = std::min (m_min, std::abs (val)); } operator R () { return m_min; } private: R m_min; }; // norm accumulator for the 0-pseudonorm (hamming distance) template <typename R> class norm_accumulator_0 { public: norm_accumulator_0 () : m_num (0) { } template <typename U> void accum (U val) { if (val != static_cast<U> (0)) ++m_num; } operator R () { return m_num; } private: unsigned int m_num; }; // OK, we're armed :) Now let's go for the fun template <typename T, typename R, typename ACC> inline void vector_norm (const Array<T>& v, R& res, ACC acc) { for (octave_idx_type i = 0; i < v.numel (); i++) acc.accum (v(i)); res = acc; } // dense versions template <typename T, typename R, typename ACC> void column_norms (const MArray<T>& m, MArray<R>& res, ACC acc) { res = MArray<R> (dim_vector (1, m.columns ())); for (octave_idx_type j = 0; j < m.columns (); j++) { ACC accj = acc; for (octave_idx_type i = 0; i < m.rows (); i++) accj.accum (m(i, j)); res.xelem (j) = accj; } } template <typename T, typename R, typename ACC> void row_norms (const MArray<T>& m, MArray<R>& res, ACC acc) { res = MArray<R> (dim_vector (m.rows (), 1)); std::vector<ACC> acci (m.rows (), acc); for (octave_idx_type j = 0; j < m.columns (); j++) { for (octave_idx_type i = 0; i < m.rows (); i++) acci[i].accum (m(i, j)); } for (octave_idx_type i = 0; i < m.rows (); i++) res.xelem (i) = acci[i]; } // sparse versions template <typename T, typename R, typename ACC> void column_norms (const MSparse<T>& m, MArray<R>& res, ACC acc) { res = MArray<R> (dim_vector (1, m.columns ())); for (octave_idx_type j = 0; j < m.columns (); j++) { ACC accj = acc; for (octave_idx_type k = m.cidx (j); k < m.cidx (j+1); k++) accj.accum (m.data (k)); res.xelem (j) = accj; } } template <typename T, typename R, typename ACC> void row_norms (const MSparse<T>& m, MArray<R>& res, ACC acc) { res = MArray<R> (dim_vector (m.rows (), 1)); std::vector<ACC> acci (m.rows (), acc); for (octave_idx_type j = 0; j < m.columns (); j++) { for (octave_idx_type k = m.cidx (j); k < m.cidx (j+1); k++) acci[m.ridx (k)].accum (m.data (k)); } for (octave_idx_type i = 0; i < m.rows (); i++) res.xelem (i) = acci[i]; } // now the dispatchers #define DEFINE_DISPATCHER(FCN_NAME, ARG_TYPE, RES_TYPE) \ template <typename T, typename R> \ RES_TYPE FCN_NAME (const ARG_TYPE& v, R p) \ { \ RES_TYPE res; \ if (p == 2) \ FCN_NAME (v, res, norm_accumulator_2<R> ()); \ else if (p == 1) \ FCN_NAME (v, res, norm_accumulator_1<R> ()); \ else if (lo_ieee_isinf (p)) \ { \ if (p > 0) \ FCN_NAME (v, res, norm_accumulator_inf<R> ()); \ else \ FCN_NAME (v, res, norm_accumulator_minf<R> ()); \ } \ else if (p == 0) \ FCN_NAME (v, res, norm_accumulator_0<R> ()); \ else if (p > 0) \ FCN_NAME (v, res, norm_accumulator_p<R> (p)); \ else \ FCN_NAME (v, res, norm_accumulator_mp<R> (p)); \ return res; \ } DEFINE_DISPATCHER (vector_norm, MArray<T>, R) DEFINE_DISPATCHER (column_norms, MArray<T>, MArray<R>) DEFINE_DISPATCHER (row_norms, MArray<T>, MArray<R>) DEFINE_DISPATCHER (column_norms, MSparse<T>, MArray<R>) DEFINE_DISPATCHER (row_norms, MSparse<T>, MArray<R>) // The approximate subproblem in Higham's method. Find lambda and mu such // that norm ([lambda, mu], p) == 1 and norm (y*lambda + col*mu, p) is // maximized. // Real version. As in Higham's paper. template <typename ColVectorT, typename R> static void higham_subp (const ColVectorT& y, const ColVectorT& col, octave_idx_type nsamp, R p, R& lambda, R& mu) { R nrm = 0; for (octave_idx_type i = 0; i < nsamp; i++) { octave_quit (); R fi = i * static_cast<R> (M_PI) / nsamp; R lambda1 = cos (fi); R mu1 = sin (fi); R lmnr = std::pow (std::pow (std::abs (lambda1), p) + std::pow (std::abs (mu1), p), 1/p); lambda1 /= lmnr; mu1 /= lmnr; R nrm1 = vector_norm (lambda1 * y + mu1 * col, p); if (nrm1 > nrm) { lambda = lambda1; mu = mu1; nrm = nrm1; } } } // Complex version. Higham's paper does not deal with complex case, so we // use a simple extension. First, guess the magnitudes as in real version, // then try to rotate lambda to improve further. template <typename ColVectorT, typename R> static void higham_subp (const ColVectorT& y, const ColVectorT& col, octave_idx_type nsamp, R p, std::complex<R>& lambda, std::complex<R>& mu) { typedef std::complex<R> CR; R nrm = 0; lambda = 1.0; CR lamcu = lambda / std::abs (lambda); // Probe magnitudes for (octave_idx_type i = 0; i < nsamp; i++) { octave_quit (); R fi = i * static_cast<R> (M_PI) / nsamp; R lambda1 = cos (fi); R mu1 = sin (fi); R lmnr = std::pow (std::pow (std::abs (lambda1), p) + std::pow (std::abs (mu1), p), 1/p); lambda1 /= lmnr; mu1 /= lmnr; R nrm1 = vector_norm (lambda1 * lamcu * y + mu1 * col, p); if (nrm1 > nrm) { lambda = lambda1 * lamcu; mu = mu1; nrm = nrm1; } } R lama = std::abs (lambda); // Probe orientation for (octave_idx_type i = 0; i < nsamp; i++) { octave_quit (); R fi = i * static_cast<R> (M_PI) / nsamp; lamcu = CR (cos (fi), sin (fi)); R nrm1 = vector_norm (lama * lamcu * y + mu * col, p); if (nrm1 > nrm) { lambda = lama * lamcu; nrm = nrm1; } } } // the p-dual element (should work for both real and complex) template <typename T, typename R> inline T elem_dual_p (T x, R p) { return math::signum (x) * std::pow (std::abs (x), p-1); } // the VectorT is used for vectors, but actually it has to be // a Matrix type to allow all the operations. For instance SparseMatrix // does not support multiplication with column/row vectors. // the dual vector template <typename VectorT, typename R> VectorT dual_p (const VectorT& x, R p, R q) { VectorT res (x.dims ()); for (octave_idx_type i = 0; i < x.numel (); i++) res.xelem (i) = elem_dual_p (x(i), p); return res / vector_norm (res, q); } // Higham's hybrid method template <typename MatrixT, typename VectorT, typename R> R higham (const MatrixT& m, R p, R tol, int maxiter, VectorT& x) { x.resize (m.columns (), 1); // the OSE part VectorT y(m.rows (), 1, 0), z(m.rows (), 1); typedef typename VectorT::element_type RR; RR lambda = 0; RR mu = 1; for (octave_idx_type k = 0; k < m.columns (); k++) { octave_quit (); VectorT col (m.column (k)); if (k > 0) higham_subp (y, col, 4*k, p, lambda, mu); for (octave_idx_type i = 0; i < k; i++) x(i) *= lambda; x(k) = mu; y = lambda * y + mu * col; } // the PM part x = x / vector_norm (x, p); R q = p/(p-1); R gamma = 0, gamma1; int iter = 0; while (iter < maxiter) { octave_quit (); y = m*x; gamma1 = gamma; gamma = vector_norm (y, p); z = dual_p (y, p, q); z = z.hermitian (); z = z * m; if (iter > 0 && (vector_norm (z, q) <= gamma || (gamma - gamma1) <= tol*gamma)) break; z = z.hermitian (); x = dual_p (z, q, p); iter++; } return gamma; } // derive column vector and SVD types static const char *p_less1_gripe = "xnorm: p must be >= 1"; // Static constant to control the maximum number of iterations. 100 seems to // be a good value. Eventually, we can provide a means to change this // constant from Octave. static int max_norm_iter = 100; // version with SVD for dense matrices template <typename MatrixT, typename VectorT, typename R> R svd_matrix_norm (const MatrixT& m, R p, VectorT) { // NOTE: The octave:: namespace tags are needed for the following // function calls until the deprecated inline functions are removed // from oct-norm.h. R res = 0; if (p == 2) { math::svd<MatrixT> fact (m, math::svd<MatrixT>::Type::sigma_only); res = fact.singular_values () (0, 0); } else if (p == 1) res = octave::xcolnorms (m, static_cast<R> (1)).max (); else if (lo_ieee_isinf (p) && p > 1) res = octave::xrownorms (m, static_cast<R> (1)).max (); else if (p > 1) { VectorT x; const R sqrteps = std::sqrt (std::numeric_limits<R>::epsilon ()); res = higham (m, p, sqrteps, max_norm_iter, x); } else (*current_liboctave_error_handler) ("%s", p_less1_gripe); return res; } // SVD-free version for sparse matrices template <typename MatrixT, typename VectorT, typename R> R matrix_norm (const MatrixT& m, R p, VectorT) { // NOTE: The octave:: namespace tags are needed for the following // function calls until the deprecated inline functions are removed // from oct-norm.h. R res = 0; if (p == 1) res = octave::xcolnorms (m, static_cast<R> (1)).max (); else if (lo_ieee_isinf (p) && p > 1) res = octave::xrownorms (m, static_cast<R> (1)).max (); else if (p > 1) { VectorT x; const R sqrteps = std::sqrt (std::numeric_limits<R>::epsilon ()); res = higham (m, p, sqrteps, max_norm_iter, x); } else (*current_liboctave_error_handler) ("%s", p_less1_gripe); return res; } // and finally, here's what we've promised in the header file #define DEFINE_XNORM_FCNS(PREFIX, RTYPE) \ RTYPE xnorm (const PREFIX##ColumnVector& x, RTYPE p) \ { \ return vector_norm (x, p); \ } \ RTYPE xnorm (const PREFIX##RowVector& x, RTYPE p) \ { \ return vector_norm (x, p); \ } \ RTYPE xnorm (const PREFIX##Matrix& x, RTYPE p) \ { \ return svd_matrix_norm (x, p, PREFIX##Matrix ()); \ } \ RTYPE xfrobnorm (const PREFIX##Matrix& x) \ { \ return vector_norm (x, static_cast<RTYPE> (2)); \ } DEFINE_XNORM_FCNS(, double) DEFINE_XNORM_FCNS(Complex, double) DEFINE_XNORM_FCNS(Float, float) DEFINE_XNORM_FCNS(FloatComplex, float) // this is needed to avoid copying the sparse matrix for xfrobnorm template <typename T, typename R> inline void array_norm_2 (const T *v, octave_idx_type n, R& res) { norm_accumulator_2<R> acc; for (octave_idx_type i = 0; i < n; i++) acc.accum (v[i]); res = acc; } #define DEFINE_XNORM_SPARSE_FCNS(PREFIX, RTYPE) \ RTYPE xnorm (const Sparse##PREFIX##Matrix& x, RTYPE p) \ { \ return matrix_norm (x, p, PREFIX##Matrix ()); \ } \ RTYPE xfrobnorm (const Sparse##PREFIX##Matrix& x) \ { \ RTYPE res; \ array_norm_2 (x.data (), x.nnz (), res); \ return res; \ } DEFINE_XNORM_SPARSE_FCNS(, double) DEFINE_XNORM_SPARSE_FCNS(Complex, double) #define DEFINE_COLROW_NORM_FCNS(PREFIX, RPREFIX, RTYPE) \ RPREFIX##RowVector \ xcolnorms (const PREFIX##Matrix& m, RTYPE p) \ { \ return column_norms (m, p); \ } \ RPREFIX##ColumnVector \ xrownorms (const PREFIX##Matrix& m, RTYPE p) \ { \ return row_norms (m, p); \ } \ DEFINE_COLROW_NORM_FCNS(, , double) DEFINE_COLROW_NORM_FCNS(Complex, , double) DEFINE_COLROW_NORM_FCNS(Float, Float, float) DEFINE_COLROW_NORM_FCNS(FloatComplex, Float, float) DEFINE_COLROW_NORM_FCNS(Sparse, , double) DEFINE_COLROW_NORM_FCNS(SparseComplex, , double) }