c_golub_kahn_bidiagonalization.cpp

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/*
 *  SPDX-FileCopyrightText: Copyright 2021, Siavash Ameli <sameli@berkeley.edu>
 *  SPDX-License-Identifier: BSD-3-Clause
 *  SPDX-FileType: SOURCE
 *
 *  This program is free software: you can redistribute it and/or modify it
 *  under the terms of the license found in the LICENSE.txt file in the root
 *  directory of this source tree.
 */
 
 
// =======
// Headers
// =======
 
#include "./c_golub_kahn_bidiagonalization.h"
#include <cmath>  // sqrt
#include "./c_orthogonalization.h"  // cOrthogonalization
#include "../_c_basic_algebra/c_vector_operations.h"  // cVectorOperations
 
 
// ============================
// golub-kahn bidiagonalization
// ============================
 
 
template <typename DataType>
IndexType c_golub_kahn_bidiagonalization(
        cLinearOperator<DataType>* A,
        const DataType* v,
        const LongIndexType n,
        const IndexType m,
        const DataType lanczos_tol,
        const FlagType orthogonalize,
        DataType* alpha,
        DataType* beta)
{
    // buffer_size is number of last orthogonal vectors to keep in buffers U, V
    IndexType buffer_size;
    if (orthogonalize == 0)
    {
        // At least two vectors must be stored in buffer for Lanczos recursion
        buffer_size = 2;
    }
    else if ((orthogonalize < 0) ||
             (orthogonalize > static_cast<FlagType>(m) - 1))
    {
        // Using full reorthogonalization, keep all of the m vectors in buffer
        buffer_size = m;
    }
    else
    {
        // Orthogonalize with less than m vectors (0 < orthogonalize < m-1)
        // plus one vector for the latest (the j-th) vector
        buffer_size = orthogonalize + 1;
    }
 
    // Allocate 2D array (as 1D array, and coalesced row-wise) to store
    // the last buffer_size of orthogonalized vectors of length n. New vectors
    // are stored by cycling through the buffer to replace with old ones.
    DataType* U = new DataType[n * buffer_size];
    DataType* V = new DataType[n * buffer_size];
 
    // Normalize vector v and copy to v_old
    cVectorOperations<DataType>::normalize_vector_and_copy(v, n, &V[0]);
 
    // Declare iterators
    IndexType j;
    IndexType lanczos_size = 0;
    IndexType num_ortho;
 
    // Golub-Kahn iteration
    for (j=0; j < m; ++j)
    {
        // Counter for the non-zero size of alpha and beta
        ++lanczos_size;
 
        // u_new = A.dot(v_old)
        A->dot(&V[(j % buffer_size)*n], &U[(j % buffer_size)*n]);
 
        // Performing: u_new[i] = u_new[i] - beta[j] * u_old[i]
        if (j > 0)
        {
            cVectorOperations<DataType>::subtract_scaled_vector(
                    &U[((j-1) % buffer_size)*n], n, beta[j-1],
                    &U[(j % buffer_size)*n]);
        }
 
        // orthogonalize u_new against previous vectors
        if (orthogonalize != 0)
        {
            // Find how many column vectors are filled so far in the buffer V
            if (j < buffer_size)
            {
                num_ortho = j;
            }
            else
            {
                num_ortho = buffer_size - 1;
            }
 
            // Gram-Schmidt process
            if (j > 0)
            {
                cOrthogonalization<DataType>::gram_schmidt_process(
                        &U[0], n, buffer_size, (j-1)%buffer_size, num_ortho,
                        &U[(j % buffer_size)*n]);
            }
        }
 
        // Normalize u_new and set its norm to alpha[j]
        alpha[j] = cVectorOperations<DataType>::normalize_vector_in_place(
                &U[(j % buffer_size)*n], n);
 
        // Performing: v_new = A.T.dot(u_new) - alpha[j] * v_old
        A->transpose_dot(&U[(j % buffer_size)*n], &V[((j+1) % buffer_size)*n]);
 
        // Performing: v_new[i] = v_new[i] - alpha[j] * v_old[i]
        cVectorOperations<DataType>::subtract_scaled_vector(
                &V[(j % buffer_size)*n], n, alpha[j],
                &V[((j+1) % buffer_size)*n]);
 
        // orthogonalize v_new against previous vectors
        if (orthogonalize != 0)
        {
            cOrthogonalization<DataType>::gram_schmidt_process(
                    &V[0], n, buffer_size, j%buffer_size, num_ortho,
                    &V[((j+1) % buffer_size)*n]);
        }
 
        // Update beta as the norm of v_new
        beta[j] = cVectorOperations<DataType>::normalize_vector_in_place(
                &V[((j+1) % buffer_size)*n], n);
 
        // Exit criterion when the vector r is zero. If each component of a
        // zero vector has the tolerance epsilon, (which is called lanczos_tol
        // here), the tolerance of norm of r is epsilon times sqrt of n.
        if (beta[j] < lanczos_tol * sqrt(n))
        {
            break;
        }
    }
 
    // Free dynamic memory
    delete[] U;
    delete[] V;
 
    return lanczos_size;
}
 
 
// ===============================
// Explicit template instantiation
// ===============================
 
// golub kahn bidiagonalization
template IndexType c_golub_kahn_bidiagonalization<float>(
        cLinearOperator<float>* A,
        const float* v,
        const LongIndexType n,
        const IndexType m,
        const float lanczos_tol,
        const FlagType orthogonalize,
        float* alpha,
        float* beta);
 
template IndexType c_golub_kahn_bidiagonalization<double>(
        cLinearOperator<double>* A,
        const double* v,
        const LongIndexType n,
        const IndexType m,
        const double lanczos_tol,
        const FlagType orthogonalize,
        double* alpha,
        double* beta);
 
template IndexType c_golub_kahn_bidiagonalization<long double>(
        cLinearOperator<long double>* A,
        const long double* v,
        const LongIndexType n,
        const IndexType m,
        const long double lanczos_tol,
        const FlagType orthogonalize,
        long double* alpha,
        long double* beta);