C# TFQMR.Solve方法代码示例

        /// <summary>
        /// The main method that runs the TFQMR iterative solver.
        /// </summary>
        public void UseSolver()
        {
            // Create a sparse matrix. For now the size will be 10 x 10 elements
            Matrix matrix = CreateMatrix(10);

            // Create the right hand side vector. The size is the same as the matrix
            // and all values will be 2.0.
            Vector rightHandSideVector = new DenseVector(10, 2.0);

            // Create a preconditioner. The possibilities are:
            // 1) No preconditioner - Simply do not provide the solver with a preconditioner.
            // 2) A simple diagonal preconditioner - Create an instance of the Diagonal class.
            // 3) A ILU preconditioner - Create an instance of the IncompleteLu class.
            // 4) A ILU preconditioner with pivoting and drop tolerances - Create an instance of the Ilutp class.

            // Here we'll use the simple diagonal preconditioner.
            // We need a link to the matrix so the pre-conditioner can do it's work.
            IPreConditioner preconditioner = new Diagonal();

            // Create a new iterator. This checks for convergence of the results of the
            // iterative matrix solver.
            // In this case we'll create the default iterator
            IIterator iterator = Iterator.CreateDefault();

            // Create the solver
            TFQMR solver = new TFQMR(preconditioner, iterator);

            // Now that all is set we can solve the matrix equation.
            Vector solutionVector = solver.Solve(matrix, rightHandSideVector);

            // Another way to get the values is by using the overloaded solve method
            // In this case the solution vector needs to be of the correct size.
            solver.Solve(matrix, rightHandSideVector, solutionVector);

            // Finally you can check the reason the solver finished the iterative process
            // by calling the SolutionStatus property on the iterator
            ICalculationStatus status = iterator.Status;
            if (status is CalculationCancelled)
                Console.WriteLine("The user cancelled the calculation.");

            if (status is CalculationIndetermined)
                Console.WriteLine("Oh oh, something went wrong. The iterative process was never started.");

            if (status is CalculationConverged)
                Console.WriteLine("Yippee, the iterative process converged.");

            if (status is CalculationDiverged)
                Console.WriteLine("I'm sorry the iterative process diverged.");

            if (status is CalculationFailure)
                Console.WriteLine("Oh dear, the iterative process failed.");

            if (status is CalculationStoppedWithoutConvergence)
                Console.WriteLine("Oh dear, the iterative process did not converge.");
        }

        public void CanSolveForRandomMatrix(int order)
        {
            var matrixA = MatrixLoader.GenerateRandomDenseMatrix(order, order);
            var matrixB = MatrixLoader.GenerateRandomDenseMatrix(order, order);

            var monitor = new Iterator<double>(new IIterationStopCriterium<double>[]
                {
                    new IterationCountStopCriterium<double>(1000),
                    new ResidualStopCriterium(1e-10)
                });
            var solver = new TFQMR(monitor);
            var matrixX = solver.Solve(matrixA, matrixB);

            // The solution X row dimension is equal to the column dimension of A
            Assert.AreEqual(matrixA.ColumnCount, matrixX.RowCount);

            // The solution X has the same number of columns as B
            Assert.AreEqual(matrixB.ColumnCount, matrixX.ColumnCount);

            var matrixBReconstruct = matrixA*matrixX;

            // Check the reconstruction.
            for (var i = 0; i < matrixB.RowCount; i++)
            {
                for (var j = 0; j < matrixB.ColumnCount; j++)
                {
                    Assert.AreEqual(matrixB[i, j], matrixBReconstruct[i, j], 1.0e-7);
                }
            }
        }

        public void SolveLongMatrixThrowsArgumentException()
        {
            var matrix = new SparseMatrix(3, 2);
            Vector input = new DenseVector(3);

            var solver = new TFQMR();
            Assert.Throws<ArgumentException>(() => solver.Solve(matrix, input));
        }

        public void CanSolveForRandomMatrix(int order)
        {
            // Due to datatype "float" it can happen that solution will not converge for specific random matrix
            // That's why we will do 4 tries and downgrade stop criterium each time
            for (var iteration = 6; iteration > 3; iteration--)
            {
                var matrixA = MatrixLoader.GenerateRandomDenseMatrix(order, order);
                var matrixB = MatrixLoader.GenerateRandomDenseMatrix(order, order);

                var monitor = new Iterator(new IIterationStopCriterium<float>[]
                                           {
                                               new IterationCountStopCriterium(MaximumIterations),
                                               new ResidualStopCriterium((float)Math.Pow(1.0/10.0, iteration))
                                           });
                var solver = new TFQMR(monitor);
                var matrixX = solver.Solve(matrixA, matrixB);

                if (!(monitor.Status is CalculationConverged))
                {
                    // Solution was not found, try again downgrading convergence boundary
                    continue;
                }

                // The solution X row dimension is equal to the column dimension of A
                Assert.AreEqual(matrixA.ColumnCount, matrixX.RowCount);
                // The solution X has the same number of columns as B
                Assert.AreEqual(matrixB.ColumnCount, matrixX.ColumnCount);

                var matrixBReconstruct = matrixA * matrixX;

                // Check the reconstruction.
                for (var i = 0; i < matrixB.RowCount; i++)
                {
                    for (var j = 0; j < matrixB.ColumnCount; j++)
                    {
                        Assert.AreApproximatelyEqual(matrixB[i, j], matrixBReconstruct[i, j], (float)Math.Pow(1.0 / 10.0, iteration - 3));
                    }
                }

                return;
            }

            Assert.Fail("Solution was not found in 3 tries");
        }

        public void CanSolveForRandomMatrix(int order)
        {
            for (var iteration = 5; iteration > 3; iteration--)
            {
                var matrixA = MatrixLoader.GenerateRandomDenseMatrix(order, order);
                var matrixB = MatrixLoader.GenerateRandomDenseMatrix(order, order);

                var monitor = new Iterator(new IIterationStopCriterium[]
                                           {
                                               new IterationCountStopCriterium(1000),
                                               new ResidualStopCriterium((float)Math.Pow(1.0 / 10.0, iteration))
                                           });
                var solver = new TFQMR(monitor);
                var matrixX = solver.Solve(matrixA, matrixB);

                if (!(monitor.Status is CalculationConverged))
                {
                    // Solution was not found, try again downgrading convergence boundary
                    continue;
                }

                // The solution X row dimension is equal to the column dimension of A
                Assert.AreEqual(matrixA.ColumnCount, matrixX.RowCount);

                // The solution X has the same number of columns as B
                Assert.AreEqual(matrixB.ColumnCount, matrixX.ColumnCount);

                var matrixBReconstruct = matrixA * matrixX;

                // Check the reconstruction.
                for (var i = 0; i < matrixB.RowCount; i++)
                {
                    for (var j = 0; j < matrixB.ColumnCount; j++)
                    {
                        Assert.AreEqual(matrixB[i, j].Real, matrixBReconstruct[i, j].Real, (float)Math.Pow(1.0 / 10.0, iteration - 3));
                        Assert.AreEqual(matrixB[i, j].Imaginary, matrixBReconstruct[i, j].Imaginary, (float)Math.Pow(1.0 / 10.0, iteration - 3));
                    }
                }

                return;
            }

            Assert.Fail("Solution was not found in 3 tries");
        }

        public void CanSolveForRandomVector(int order)
        {
            var matrixA = MatrixLoader.GenerateRandomDenseMatrix(order, order);
            var vectorb = MatrixLoader.GenerateRandomDenseVector(order);

            var monitor = new Iterator(new IIterationStopCriterium<double>[]
                                       {
                                           new IterationCountStopCriterium(1000),
                                           new ResidualStopCriterium(1e-10),
                                       });
            var solver = new TFQMR(monitor);

            var resultx = solver.Solve(matrixA, vectorb);
            Assert.AreEqual(matrixA.ColumnCount, resultx.Count);

            var bReconstruct = matrixA * resultx;

            // Check the reconstruction.
            for (var i = 0; i < order; i++)
            {
                Assert.AreApproximatelyEqual(vectorb[i], bReconstruct[i], 1e-7);
            }
        }

        /// <summary>
        /// Run example
        /// </summary>
        public void Run()
        {
            // Format matrix output to console
            var formatProvider = (CultureInfo)CultureInfo.InvariantCulture.Clone();
            formatProvider.TextInfo.ListSeparator = " ";

            // Solve next system of linear equations (Ax=b):
            // 5*x + 2*y - 4*z = -7
            // 3*x - 7*y + 6*z = 38
            // 4*x + 1*y + 5*z = 43

            // Create matrix "A" with coefficients
            var matrixA = DenseMatrix.OfArray(new[,] { { 5.00, 2.00, -4.00 }, { 3.00, -7.00, 6.00 }, { 4.00, 1.00, 5.00 } });
            Console.WriteLine(@"Matrix 'A' with coefficients");
            Console.WriteLine(matrixA.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // Create vector "b" with the constant terms.
            var vectorB = new DenseVector(new[] { -7.0, 38.0, 43.0 });
            Console.WriteLine(@"Vector 'b' with the constant terms");
            Console.WriteLine(vectorB.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // Create stop criteriums to monitor an iterative calculation. There are next available stop criteriums:
            // - DivergenceStopCriterium: monitors an iterative calculation for signs of divergence;
            // - FailureStopCriterium: monitors residuals for NaN's;
            // - IterationCountStopCriterium: monitors the numbers of iteration steps;
            // - ResidualStopCriterium: monitors residuals if calculation is considered converged;

            // Stop calculation if 1000 iterations reached during calculation
            var iterationCountStopCriterium = new IterationCountStopCriterium<double>(1000);

            // Stop calculation if residuals are below 1E-10 --> the calculation is considered converged
            var residualStopCriterium = new ResidualStopCriterium(1e-10);

            // Create monitor with defined stop criteriums
            var monitor = new Iterator<double>(new IIterationStopCriterium<double>[] { iterationCountStopCriterium, residualStopCriterium });

            // Create Transpose Free Quasi-Minimal Residual solver
            var solver = new TFQMR(monitor);

            // 1. Solve the matrix equation
            var resultX = solver.Solve(matrixA, vectorB);
            Console.WriteLine(@"1. Solve the matrix equation");
            Console.WriteLine();

            // 2. Check solver status of the iterations.
            // Solver has property IterationResult which contains the status of the iteration once the calculation is finished.
            // Possible values are:
            // - CalculationCancelled: calculation was cancelled by the user;
            // - CalculationConverged: calculation has converged to the desired convergence levels;
            // - CalculationDiverged: calculation diverged;
            // - CalculationFailure: calculation has failed for some reason;
            // - CalculationIndetermined: calculation is indetermined, not started or stopped;
            // - CalculationRunning: calculation is running and no results are yet known;
            // - CalculationStoppedWithoutConvergence: calculation has been stopped due to reaching the stopping limits, but that convergence was not achieved;
            Console.WriteLine(@"2. Solver status of the iterations");
            Console.WriteLine(solver.IterationResult);
            Console.WriteLine();

            // 3. Solution result vector of the matrix equation
            Console.WriteLine(@"3. Solution result vector of the matrix equation");
            Console.WriteLine(resultX.ToString("#0.00\t", formatProvider));
            Console.WriteLine();

            // 4. Verify result. Multiply coefficient matrix "A" by result vector "x"
            var reconstructVecorB = matrixA * resultX;
            Console.WriteLine(@"4. Multiply coefficient matrix 'A' by result vector 'x'");
            Console.WriteLine(reconstructVecorB.ToString("#0.00\t", formatProvider));
            Console.WriteLine();
        }

        public void SolveUnitMatrixAndBackMultiply()
        {
            // Create the identity matrix
            var matrix = SparseMatrix.Identity(100);

            // Create the y vector
            var y = DenseVector.Create(matrix.RowCount, i => 1);

            // Create an iteration monitor which will keep track of iterative convergence
            var monitor = new Iterator<Complex32>(new IIterationStopCriterium<Complex32>[]
                {
                    new IterationCountStopCriterium<Complex32>(MaximumIterations),
                    new ResidualStopCriterium(ConvergenceBoundary),
                    new DivergenceStopCriterium(),
                    new FailureStopCriterium()
                });

            var solver = new TFQMR(monitor);

            // Solve equation Ax = y
            var x = solver.Solve(matrix, y);

            // Now compare the results
            Assert.IsNotNull(x, "#02");
            Assert.AreEqual(y.Count, x.Count, "#03");

            // Back multiply the vector
            var z = matrix.Multiply(x);

            // Check that the solution converged
            Assert.IsTrue(monitor.HasConverged, "#04");

            // Now compare the vectors
            for (var i = 0; i < y.Count; i++)
            {
                Assert.IsTrue((y[i] - z[i]).Magnitude.IsSmaller(ConvergenceBoundary, 1), "#05-" + i);
            }
        }

        public void SolvePoissonMatrixAndBackMultiply()
        {
            // Create the matrix
            var matrix = new SparseMatrix(25);

            // Assemble the matrix. We assume we're solving the Poisson equation
            // on a rectangular 5 x 5 grid
            const int GridSize = 5;

            // The pattern is:
            // 0 .... 0 -1 0 0 0 0 0 0 0 0 -1 4 -1 0 0 0 0 0 0 0 0 -1 0 0 ... 0
            for (var i = 0; i < matrix.RowCount; i++)
            {
                // Insert the first set of -1's
                if (i > (GridSize - 1))
                {
                    matrix[i, i - GridSize] = -1;
                }

                // Insert the second set of -1's
                if (i > 0)
                {
                    matrix[i, i - 1] = -1;
                }

                // Insert the centerline values
                matrix[i, i] = 4;

                // Insert the first trailing set of -1's
                if (i < matrix.RowCount - 1)
                {
                    matrix[i, i + 1] = -1;
                }

                // Insert the second trailing set of -1's
                if (i < matrix.RowCount - GridSize)
                {
                    matrix[i, i + GridSize] = -1;
                }
            }

            // Create the y vector
            var y = DenseVector.Create(matrix.RowCount, i => 1);

            // Create an iteration monitor which will keep track of iterative convergence
            var monitor = new Iterator<Complex32>(new IIterationStopCriterium<Complex32>[]
                {
                    new IterationCountStopCriterium<Complex32>(MaximumIterations),
                    new ResidualStopCriterium(ConvergenceBoundary),
                    new DivergenceStopCriterium(),
                    new FailureStopCriterium()
                });
            var solver = new TFQMR(monitor);

            // Solve equation Ax = y
            var x = solver.Solve(matrix, y);

            // Now compare the results
            Assert.IsNotNull(x, "#02");
            Assert.AreEqual(y.Count, x.Count, "#03");

            // Back multiply the vector
            var z = matrix.Multiply(x);

            // Check that the solution converged
            Assert.IsTrue(monitor.HasConverged, "#04");

            // Now compare the vectors
            for (var i = 0; i < y.Count; i++)
            {
                Assert.IsTrue((y[i] - z[i]).Magnitude.IsSmaller(1e-4f, 1), "#05-" + i);
            }
        }

        public void CanSolveForRandomVector(int order)
        {
            for (var iteration = 5; iteration > 3; iteration--)
            {
                var matrixA = MatrixLoader.GenerateRandomDenseMatrix(order, order);
                var vectorb = MatrixLoader.GenerateRandomDenseVector(order);

                var monitor = new Iterator<Complex32>(new IIterationStopCriterium<Complex32>[]
                    {
                        new IterationCountStopCriterium<Complex32>(1000),
                        new ResidualStopCriterium((float) Math.Pow(1.0/10.0, iteration))
                    });
                var solver = new TFQMR(monitor);

                var resultx = solver.Solve(matrixA, vectorb);

                if (!monitor.HasConverged)
                {
                    // Solution was not found, try again downgrading convergence boundary
                    continue;
                }

                Assert.AreEqual(matrixA.ColumnCount, resultx.Count);
                var matrixBReconstruct = matrixA*resultx;

                // Check the reconstruction.
                for (var i = 0; i < order; i++)
                {
                    Assert.AreEqual(vectorb[i].Real, matrixBReconstruct[i].Real, (float) Math.Pow(1.0/10.0, iteration - 3));
                    Assert.AreEqual(vectorb[i].Imaginary, matrixBReconstruct[i].Imaginary, (float) Math.Pow(1.0/10.0, iteration - 3));
                }

                return;
            }

            Assert.Fail("Solution was not found in 3 tries");
        }

        public void CanSolveForRandomVector([Values(4, 8, 10)] int order)
        {
            var matrixA = MatrixLoader.GenerateRandomDenseMatrix(order, order);
            var vectorb = MatrixLoader.GenerateRandomDenseVector(order);

            var monitor = new Iterator(new IIterationStopCriterium[]
                                       {
                                           new IterationCountStopCriterium(1000),
                                           new ResidualStopCriterium(1e-10),
                                       });
            var solver = new TFQMR(monitor);

            var resultx = solver.Solve(matrixA, vectorb);
            Assert.AreEqual(matrixA.ColumnCount, resultx.Count);

            var matrixBReconstruct = matrixA * resultx;

            // Check the reconstruction.
            for (var i = 0; i < order; i++)
            {
                Assert.AreEqual(vectorb[i].Real, matrixBReconstruct[i].Real, 1e-5);
                Assert.AreEqual(vectorb[i].Imaginary, matrixBReconstruct[i].Imaginary, 1e-5);
            }
        }

        public void SolveWideMatrix()
        {
            var matrix = new SparseMatrix(2, 3);
            Vector<Complex32> input = new DenseVector(2);

            var solver = new TFQMR();
            solver.Solve(matrix, input);
        }

        public void CanSolveForRandomVector(int order)
        {
            // Due to datatype "float" it can happen that solution will not converge for specific random matrix
            // That's why we will do 3 tries and downgrade stop criterium each time
            for (var iteration = 6; iteration > 3; iteration--)
            {
                var matrixA = MatrixLoader.GenerateRandomDenseMatrix(order, order);
                var vectorb = MatrixLoader.GenerateRandomDenseVector(order);

                var monitor = new Iterator<float>(new IIterationStopCriterium<float>[]
                    {
                        new IterationCountStopCriterium<float>(MaximumIterations),
                        new ResidualStopCriterium((float) Math.Pow(1.0/10.0, iteration))
                    });
                var solver = new TFQMR(monitor);
                var resultx = solver.Solve(matrixA, vectorb);

                if (!monitor.HasConverged)
                {
                    // Solution was not found, try again downgrading convergence boundary
                    continue;
                }

                Assert.AreEqual(matrixA.ColumnCount, resultx.Count);
                var matrixBReconstruct = matrixA*resultx;

                // Check the reconstruction.
                for (var i = 0; i < order; i++)
                {
                    Assert.AreEqual(vectorb[i], matrixBReconstruct[i], (float) Math.Pow(1.0/10.0, iteration - 3));
                }

                return;
            }
        }

        public void SolveScaledUnitMatrixAndBackMultiply()
        {
            // Create the identity matrix
            Matrix<float> matrix = SparseMatrix.Identity(100);

            // Scale it with a funny number
            matrix.Multiply((float)Math.PI);

            // Create the y vector
            Vector<float> y = new DenseVector(matrix.RowCount, 1);

            // Create an iteration monitor which will keep track of iterative convergence
            var monitor = new Iterator(new IIterationStopCriterium<float>[]
                                       {
                                           new IterationCountStopCriterium(MaximumIterations),
                                           new ResidualStopCriterium(ConvergenceBoundary),
                                           new DivergenceStopCriterium(),
                                           new FailureStopCriterium()
                                       });
            var solver = new TFQMR(monitor);

            // Solve equation Ax = y
            var x = solver.Solve(matrix, y);

            // Now compare the results
            Assert.IsNotNull(x, "#02");
            Assert.AreEqual(y.Count, x.Count, "#03");

            // Back multiply the vector
            var z = matrix.Multiply(x);

            // Check that the solution converged
            Assert.IsTrue(monitor.Status is CalculationConverged, "#04");

            // Now compare the vectors
            for (var i = 0; i < y.Count; i++)
            {
                Assert.IsTrue((y[i] - z[i]).IsSmaller(ConvergenceBoundary, 1), "#05-" + i);
            }
        }

        public void SolveLongMatrix()
        {
            var matrix = new SparseMatrix(3, 2);
            Vector<float> input = new DenseVector(3);

            var solver = new TFQMR();
            solver.Solve(matrix, input);
        }