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lambda lanczos

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Lambda Lanczos

C++ adaptive and header-only Lanczos algorithm library

Overview

Lambda Lanczos calculates the smallest or largest eigenvalue and the corresponding eigenvector of a symmetric (Hermitian) matrix.

The characteristic feature is the matrix-vector multiplication routine used in the Lanczos algorithm is adaptable:

#include <lambda_lanczos/lambda_lanczos.hpp>using lambda_lanczos::LambdaLanczos;/* Some include and using declarations are omitted */void sample() {  const int n = 3;  double matrix[n][n] = { {2.0, 1.0, 1.0},
                          {1.0, 2.0, 1.0},
                          {1.0, 1.0, 2.0} };  // Its eigenvalues are {4, 1, 1}

  /* Prepare matrix-vector multiplication routine used in Lanczos algorithm */
  auto mv_mul = [&](const vector<double>& in, vector<double>& out) {for(int i = 0;i < n;i++) {      for(int j = 0;j < n;j++) {
        out[i] += matrix[i][j]*in[j];
      }
    }
  };  /* Execute Lanczos algorithm */
  LambdaLanczos<double> engine(mv_mul, n, true, 1); // Find 1 maximum eigenvalue
  vector<double> eigenvalues;
  vector<vector<double>> eigenvectors;
  engine.run(eigenvalues, eigenvectors);  //// If C++17 is available, the following notation does the same thing:
  // auto [eigenvalues, eigenvectors] = engine.run()


  /* Print result */
  cout << "Eigenvalue: " << setprecision(16) << eigenvalues[0] << endl;
  cout << "Eigenvector:";  for(int i = 0; i < n; i++) {
    cout << eigenvectors[0][i] << " ";
  }
  cout << endl;
}

This feature allows you to

easily combine Lambda Lanczos with existing matrix libraries (e.g. Eigen; see a sample code).
use a matrix whose elements are partially given, e.g. a sparse matrix whose non-zero elements are stored as a list of {row-index, column-index, value} tuples.

Interfaces

Lambda Lanczos provides the following two interfaces:

1. Eigenvalue problem

LambdaLanczos class computes maximum (minimum) eigenvalues and corresponding eigenvectors. Degenerate eigenvalues are taken into account. This class aims problems that require one or a few eigenpairs (e.g. low-energy excitation of a quantum system).

2. Exponentiation

Exponentiator class computes the following type of matrix-vector multiplication:

$$boldsymbol{v}'=e^{delta A} boldsymbol{v},$$

where$A$is a symmetric (Hermitian) matrix and$delta$is a scalar parameter. This class is based on the same theory of the Lanczos algorithm (Krylov subspace method).

As an application, this class may be used for time evolution of a quantum many-body system:

$$ ket{psi(t+Delta t)} = e^{-iHDelta t} ket{psi(t)},$$

and more sophisticated algorithms, such as TDVP and other tensor networks.