add Gram-Schmidt process #11

.gitignore

@@ -4,3 +4,5 @@ include/Eigen/*
 .idea/
 cmake-build*/
 build/*
+.vscode
+*~

include/Spectra/LinAlg/Orthogonalization.h

@@ -0,0 +1,98 @@
+// Copyright (C) 2020 Netherlands eScience Center <[email protected]>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at https://mozilla.org/MPL/2.0/.
+
+#ifndef SPECTRA_ORTHOGONALIZATION_H
+#define SPECTRA_ORTHOGONALIZATION_H
+
+#include <Eigen/Core>
+#include <Eigen/Dense>
+
+namespace Spectra {
+
+template <typename Matrix>
+Eigen::Index sanity_check(Matrix& in_output, Eigen::Index leftColsToSkip = 0)
+{
+    assert(in_output.cols() > leftColsToSkip && "leftColsToSkip is larger than columns of matrix");
+    assert(leftColsToSkip >= 0 && "leftColsToSkip is negative");
+    if (leftColsToSkip == 0)
+    {
+        in_output.col(0).normalize();
+        leftColsToSkip = 1;
+    }
+    return leftColsToSkip;
+}
+
+template <typename Matrix>
+void QR_orthogonalisation(Matrix& in_output)
+{
+    using InternalMatrix= Eigen::Matrix<typename Matrix::Scalar,Eigen::Dynamic, Eigen::Dynamic>;
+    Eigen::Index nrows = in_output.rows();
+    Eigen::Index ncols = in_output.cols();
+    ncols = std::min(nrows, ncols);
+    InternalMatrix I = InternalMatrix::Identity(nrows, ncols);
+    Eigen::HouseholderQR<Matrix> qr(in_output);
+    in_output = qr.householderQ() * I;
+}
+
+template <typename Matrix>
+void MGS_orthogonalisation(Matrix& in_output, Eigen::Index leftColsToSkip = 0)
+{
+    leftColsToSkip = sanity_check(in_output, leftColsToSkip);
+
+    for (Eigen::Index k = leftColsToSkip; k < in_output.cols(); ++k)
+    {
+        for (Eigen::Index j = 0; j < k; j++)
+        {
+            in_output.col(k) -= in_output.col(j).dot(in_output.col(k)) / (in_output.col(j).dot(in_output.col(j))) * in_output.col(j);
+        }
+        in_output.col(k).normalize();
+    }
+}
+
+template <typename Matrix>
+void GS_orthogonalisation(Matrix& in_output, Eigen::Index leftColsToSkip = 0)
+{
+    leftColsToSkip = sanity_check(in_output, leftColsToSkip);
+
+    for (Eigen::Index j = leftColsToSkip; j < in_output.cols(); ++j)
+    {
+        in_output.col(j) -= in_output.leftCols(j) * (in_output.leftCols(j).transpose() * in_output.col(j));
+        in_output.col(j).normalize();
+    }
+}
+
+template <typename Matrix>
+void twice_is_enough_orthogonalisation(Matrix& in_output, Eigen::Index leftColsToSkip = 0)
+{
+    GS_orthogonalisation(in_output, leftColsToSkip);
+    GS_orthogonalisation(in_output, leftColsToSkip);
+}
+
+template <typename Matrix>
+void partial_orthogonalisation(Matrix& in_output, Eigen::Index leftColsToSkip = 0)
+{
+    leftColsToSkip = sanity_check(in_output, leftColsToSkip);
+
+    Eigen::Index rightColToOrtho = in_output.cols() - leftColsToSkip;
+    in_output.rightCols(rightColToOrtho) -= (in_output.leftCols(leftColsToSkip) * in_output.leftCols(leftColsToSkip).transpose()) * in_output.rightCols(rightColToOrtho);
+    in_output.rightCols(rightColToOrtho).colwise().normalize();
+}
+
+template <typename Matrix>
+void JensWehner_orthogonalisation(Matrix& in_output, Eigen::Index leftColsToSkip = 0)
+{
+    leftColsToSkip = sanity_check(in_output, leftColsToSkip);
+
+    Eigen::Index rightColToOrtho = in_output.cols() - leftColsToSkip;
+    partial_orthogonalisation(in_output, leftColsToSkip);
+    Eigen::Ref<Matrix> right_cols = in_output.rightCols(rightColToOrtho);
+    QR_orthogonalisation(right_cols);
+    in_output.rightCols(rightColToOrtho) = right_cols;
+}
+
+}  // namespace Spectra
+
+#endif  //SPECTRA_ORTHOGONALIZATION_H

test/CMakeLists.txt

@@ -9,6 +9,7 @@ list(APPEND test_target_sources
         GenEigs.cpp
         GenEigsRealShift.cpp
         GenEigsComplexShift.cpp
+        Orthogonalization.cpp
         SymGEigsCholesky.cpp
         SymGEigsRegInv.cpp
         SVD.cpp

test/Orthogonalization.cpp

@@ -0,0 +1,99 @@
+#include <Eigen/Core>
+#include <Spectra/LinAlg/Orthogonalization.h>
+#include <iostream>
+using namespace Spectra;
+
+#define CATCH_CONFIG_MAIN
+#include "catch.hpp"
+
+using Eigen::MatrixXd;
+using Eigen::VectorXd;
+using Eigen::Index;
+
+template <typename Matrix>
+void check_orthogonality(const Matrix& basis)
+{
+    const double tol = 1e-12;
+    Matrix xs = basis.transpose() * basis;
+    INFO("The orthonormalized basis must fulfill that basis.T * basis = I");
+    INFO("Matrix is\n " << basis);
+    INFO("Overlap is\n " << xs);
+    CHECK(xs.isIdentity(tol));
+}
+
+TEST_CASE("complete orthonormalization", "[orthogonalisation]")
+{
+    std::srand(123);
+    const Index n = 20;
+
+    MatrixXd mat = MatrixXd::Random(n, n);
+
+    SECTION("MGS")
+    {
+        MGS_orthogonalisation(mat);
+        check_orthogonality(mat);
+    }
+
+    SECTION("GS")
+    {
+        GS_orthogonalisation(mat);
+        check_orthogonality(mat);
+    }
+
+    SECTION("QR")
+    {
+        QR_orthogonalisation(mat);
+        check_orthogonality(mat);
+    }
+
+    SECTION("twice_is_enough")
+    {
+        twice_is_enough_orthogonalisation(mat);
+        check_orthogonality(mat);
+    }
+
+    SECTION("JensWehner")
+    {
+        JensWehner_orthogonalisation(mat);
+        check_orthogonality(mat);
+    }
+}
+
+TEST_CASE("Partial orthonormalization", "[orthogonalisation]")
+{
+    std::srand(123);
+    const Index n = 20;
+    const Index sub = 5;
+    Index start = n - sub;
+
+    // Create a n x 20 orthonormal basis
+    MatrixXd mat = MatrixXd::Random(n, start);
+    QR_orthogonalisation(mat);
+
+    mat.conservativeResize(Eigen::NoChange, n);
+    mat.rightCols(sub) = MatrixXd::Random(n, sub);
+
+    SECTION("MGS")
+    {
+        MGS_orthogonalisation(mat, start);
+        check_orthogonality(mat);
+    }
+
+    SECTION("GS")
+    {
+        GS_orthogonalisation(mat, start);
+        check_orthogonality(mat);
+    }
+
+    SECTION("twice_is_enough")
+    {
+        twice_is_enough_orthogonalisation(mat, start);
+        check_orthogonality(mat);
+    }
+
+    SECTION("JensWehner")
+    {
+        JensWehner_orthogonalisation(mat, start);
+        check_orthogonality(mat);
+    }
+}