public void SolveWideMatrixThrowsArgumentException()
{
var matrix = new SparseMatrix(2, 3);
var input = new DenseVector(2);
var solver = new TFQMR();
Assert.Throws<ArgumentException>(() => matrix.SolveIterative(input, solver));
}
public void SolveLongMatrixThrowsArgumentException()
{
var matrix = new SparseMatrix(3, 2);
var input = new DenseVector(3);
var solver = new MlkBiCgStab();
Assert.Throws<ArgumentException>(() => matrix.SolveIterative(input, solver));
}
public void SolveWideMatrixThrowsArgumentException()
{
var matrix = new SparseMatrix(2, 3);
var input = new DenseVector(2);
var solver = new MlkBiCgStab();
Assert.That(() => matrix.SolveIterative(input, solver), Throws.ArgumentException);
}
public void SolveLongMatrixThrowsArgumentException()
{
var matrix = new SparseMatrix(3, 2);
var input = new DenseVector(3);
var solver = new GpBiCg();
Assert.That(() => matrix.SolveIterative(input, solver), Throws.ArgumentException);
}
/// <summary>
/// Create unit matrix.
/// </summary>
/// <param name="size">Matrix size.</param>
/// <returns>New unit matrix.</returns>
internal SparseMatrix CreateUnitMatrix(int size)
{
var matrix = new SparseMatrix(size);
for (var i = 0; i < size; i++)
{
matrix[i, i] = 2;
}
return matrix;
}
/// <summary>
/// Check the result.
/// </summary>
/// <param name="preconditioner">Specific preconditioner.</param>
/// <param name="matrix">Source matrix.</param>
/// <param name="vector">Initial vector.</param>
/// <param name="result">Result vector.</param>
protected override void CheckResult(IPreconditioner<float> preconditioner, SparseMatrix matrix, Vector<float> vector, Vector<float> result)
{
Assert.AreEqual(typeof (UnitPreconditioner<float>), preconditioner.GetType(), "#01");
// Unit preconditioner is doing nothing. Vector and result should be equal
for (var i = 0; i < vector.Count; i++)
{
Assert.IsTrue(vector[i] == result[i], "#02-" + i);
}
}
/// <summary>
/// Check the result.
/// </summary>
/// <param name="preconditioner">Specific preconditioner.</param>
/// <param name="matrix">Source matrix.</param>
/// <param name="vector">Initial vector.</param>
/// <param name="result">Result vector.</param>
protected override void CheckResult(IPreconditioner<float> preconditioner, SparseMatrix matrix, Vector<float> vector, Vector<float> result)
{
Assert.AreEqual(typeof (DiagonalPreconditioner), preconditioner.GetType(), "#01");
// Compute M * result = product
// compare vector and product. Should be equal
var product = new DenseVector(result.Count);
matrix.Multiply(result, product);
for (var i = 0; i < product.Count; i++)
{
Assert.IsTrue(((double) vector[i]).AlmostEqualNumbersBetween(product[i], -Epsilon.Magnitude()), "#02-" + i);
}
}
public void CanAddSparseMatricesBothWays()
{
var m1 = new SparseMatrix(1, 3);
var m2 = SparseMatrix.OfArray(new float[,] { { 0, 1, 1 } });
var sum1 = m1 + m2;
var sum2 = m2 + m1;
Assert.IsTrue(sum1.Equals(m2));
Assert.IsTrue(sum1.Equals(sum2));
var sparseResult = new SparseMatrix(1, 3);
sparseResult.Add(m2, sparseResult);
Assert.IsTrue(sparseResult.Equals(sum1));
sparseResult = SparseMatrix.OfArray(new float[,] { { 0, 1, 1 } });
sparseResult.Add(m1, sparseResult);
Assert.IsTrue(sparseResult.Equals(sum1));
sparseResult = SparseMatrix.OfArray(new float[,] { { 0, 1, 1 } });
m1.Add(sparseResult, sparseResult);
Assert.IsTrue(sparseResult.Equals(sum1));
sparseResult = SparseMatrix.OfArray(new float[,] { { 0, 1, 1 } });
sparseResult.Add(sparseResult, sparseResult);
Assert.IsTrue(sparseResult.Equals(2*sum1));
var denseResult = new DenseMatrix(1, 3);
denseResult.Add(m2, denseResult);
Assert.IsTrue(denseResult.Equals(sum1));
denseResult = DenseMatrix.OfArray(new float[,] {{0, 1, 1}});
denseResult.Add(m1, denseResult);
Assert.IsTrue(denseResult.Equals(sum1));
var m3 = DenseMatrix.OfArray(new float[,] {{0, 1, 1}});
var sum3 = m1 + m3;
var sum4 = m3 + m1;
Assert.IsTrue(sum3.Equals(m3));
Assert.IsTrue(sum3.Equals(sum4));
}
/// <summary>
/// Creates a new <see cref="SparseMatrix"/> and inserts the given column at the given index.
/// </summary>
/// <param name="columnIndex">The index of where to insert the column.</param>
/// <param name="column">The column to insert.</param>
/// <returns>A new <see cref="SparseMatrix"/> with the inserted column.</returns>
/// <exception cref="ArgumentNullException">If <paramref name="column "/> is <see langword="null" />. </exception>
/// <exception cref="ArgumentOutOfRangeException">If <paramref name="columnIndex"/> is < zero or > the number of columns.</exception>
/// <exception cref="ArgumentException">If the size of <paramref name="column"/> != the number of rows.</exception>
public override Matrix<float> InsertColumn(int columnIndex, Vector<float> column)
{
if (column == null)
{
throw new ArgumentNullException("column");
}
if (columnIndex < 0 || columnIndex > ColumnCount)
{
throw new ArgumentOutOfRangeException("columnIndex");
}
if (column.Count != RowCount)
{
throw new ArgumentException(Resources.ArgumentMatrixSameRowDimension, "column");
}
var result = new SparseMatrix(RowCount, ColumnCount + 1);
for (var i = 0; i < columnIndex; i++)
{
result.SetColumn(i, Column(i));
}
result.SetColumn(columnIndex, column);
for (var i = columnIndex + 1; i < ColumnCount + 1; i++)
{
result.SetColumn(i, Column(i - 1));
}
return result;
}
/// <summary>
/// Initializes a square <see cref="SparseMatrix"/> with all zero's except for ones on the diagonal.
/// </summary>
/// <param name="order">the size of the square matrix.</param>
/// <returns>Identity <c>SparseMatrix</c></returns>
/// <exception cref="ArgumentException">
/// If <paramref name="order"/> is less than one.
/// </exception>
public static SparseMatrix Identity(int order)
{
var m = new SparseMatrix(order);
var mStorage = m.Raw;
mStorage.ValueCount = order;
mStorage.Values = new float[order];
mStorage.ColumnIndices = new int[order];
for (var i = 0; i < order; i++)
{
mStorage.Values[i] = 1f;
mStorage.ColumnIndices[i] = i;
mStorage.RowPointers[i] = i;
}
return m;
}
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);
// Due to datatype "float" it can happen that solution will not converge for specific random starting vectors
// That's why we will do 3 tries
for (var iteration = 0; iteration <= 3; iteration++)
{
// Create an iteration monitor which will keep track of iterative convergence
var monitor = new Iterator<float>(
new IterationCountStopCriterium<float>(MaximumIterations),
new ResidualStopCriterium<float>(ConvergenceBoundary),
new DivergenceStopCriterium<float>(),
new FailureStopCriterium<float>());
var solver = new MlkBiCgStab();
// Solve equation Ax = y
Vector<float> x;
try
{
x = matrix.SolveIterative(y, solver, monitor);
}
catch (Exception)
{
continue;
}
if (monitor.Status != IterationStatus.Converged)
{
continue;
}
// Now compare the results
Assert.IsNotNull(x, "#02");
Assert.AreEqual(y.Count, x.Count, "#03");
// Back multiply the vector
var z = matrix.Multiply(x);
// Now compare the vectors
for (var i = 0; i < y.Count; i++)
{
Assert.GreaterOrEqual(ConvergenceBoundary, Math.Abs(y[i] - z[i]), "#05-" + i);
}
return;
}
}
/// <summary> /// Check the result. /// </summary> /// <param name="preconditioner">Specific preconditioner.</param> /// <param name="matrix">Source matrix.</param> /// <param name="vector">Initial vector.</param> /// <param name="result">Result vector.</param> protected abstract void CheckResult(IPreconditioner<float> preconditioner, SparseMatrix matrix, Vector<float> vector, Vector<float> result);
/// <summary>
/// Create a new sparse matrix with the diagonal as a copy of the given vector.
/// This new matrix will be independent from the vector.
/// A new memory block will be allocated for storing the matrix.
/// </summary>
public static SparseMatrix OfDiagonalVector(int rows, int columns, Vector<float> diagonal)
{
var m = new SparseMatrix(rows, columns);
m.SetDiagonal(diagonal);
return m;
}
/// <summary>
/// Create a new sparse matrix with the diagonal as a copy of the given array.
/// This new matrix will be independent from the array.
/// A new memory block will be allocated for storing the matrix.
/// </summary>
public static SparseMatrix OfDiagonalArray(int rows, int columns, float[] diagonal)
{
var m = new SparseMatrix(rows, columns);
m.SetDiagonal(diagonal);
return m;
}
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<float>(
new IterationCountStopCriterium<float>(MaximumIterations),
new ResidualStopCriterium(ConvergenceBoundary),
new DivergenceStopCriterium(),
new FailureStopCriterium());
var solver = new TFQMR();
// Solve equation Ax = y
var x = matrix.SolveIterative(y, solver, monitor);
// 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 == IterationStatus.Converged, "#04");
// Now compare the vectors
for (var i = 0; i < y.Count; i++)
{
Assert.IsTrue(Math.Abs(y[i] - z[i]).IsSmaller(ConvergenceBoundary, 1), "#05-" + i);
}
}
/// <summary>
/// Creates a matrix that contains the values from the requested sub-matrix.
/// </summary>
/// <param name="rowIndex">The row to start copying from.</param>
/// <param name="rowLength">The number of rows to copy. Must be positive.</param>
/// <param name="columnIndex">The column to start copying from.</param>
/// <param name="columnLength">The number of columns to copy. Must be positive.</param>
/// <returns>The requested sub-matrix.</returns>
/// <exception cref="ArgumentOutOfRangeException">If: <list><item><paramref name="rowIndex"/> is
/// negative, or greater than or equal to the number of rows.</item>
/// <item><paramref name="columnIndex"/> is negative, or greater than or equal to the number
/// of columns.</item>
/// <item><c>(columnIndex + columnLength) >= Columns</c></item>
/// <item><c>(rowIndex + rowLength) >= Rows</c></item></list></exception>
/// <exception cref="ArgumentException">If <paramref name="rowLength"/> or <paramref name="columnLength"/>
/// is not positive.</exception>
public override Matrix<float> SubMatrix(int rowIndex, int rowLength, int columnIndex, int columnLength)
{
if (rowIndex >= RowCount || rowIndex < 0)
{
throw new ArgumentOutOfRangeException("rowIndex");
}
if (columnIndex >= ColumnCount || columnIndex < 0)
{
throw new ArgumentOutOfRangeException("columnIndex");
}
if (rowLength < 1)
{
throw new ArgumentException(Resources.ArgumentMustBePositive, "rowLength");
}
if (columnLength < 1)
{
throw new ArgumentException(Resources.ArgumentMustBePositive, "columnLength");
}
var colMax = columnIndex + columnLength;
var rowMax = rowIndex + rowLength;
if (rowMax > RowCount)
{
throw new ArgumentOutOfRangeException("rowLength");
}
if (colMax > ColumnCount)
{
throw new ArgumentOutOfRangeException("columnLength");
}
var result = new SparseMatrix(rowLength, columnLength);
if (rowIndex > columnIndex && columnIndex + columnLength > rowIndex)
{
for (var i = 0; rowIndex - columnIndex + i < Math.Min(columnLength, rowLength); i++)
{
result[i, rowIndex - columnIndex + i] = Data[rowIndex + i];
}
}
else if (rowIndex < columnIndex && rowIndex + rowLength > columnIndex)
{
for (var i = 0; rowIndex - columnIndex + i < Math.Min(columnLength, rowLength); i++)
{
result[columnIndex - rowIndex + i, i] = Data[columnIndex + i];
}
}
else
{
for (var i = 0; i < Math.Min(columnLength, rowLength); i++)
{
result[i, i] = Data[rowIndex + i];
}
}
return result;
}
/// <summary>
/// Stacks this matrix on top of the given matrix and places the result into the result <see cref="SparseMatrix"/>.
/// </summary>
/// <param name="lower">The matrix to stack this matrix upon.</param>
/// <returns>The combined <see cref="SparseMatrix"/>.</returns>
/// <exception cref="ArgumentNullException">If lower is <see langword="null" />.</exception>
/// <exception cref="ArgumentException">If <strong>upper.Columns != lower.Columns</strong>.</exception>
public override Matrix<float> Stack(Matrix<float> lower)
{
if (lower == null)
{
throw new ArgumentNullException("lower");
}
if (lower.ColumnCount != ColumnCount)
{
throw new ArgumentException(Resources.ArgumentMatrixSameColumnDimension, "lower");
}
var result = new SparseMatrix(RowCount + lower.RowCount, ColumnCount);
Stack(lower, result);
return result;
}
public void SolveLongMatrixThrowsArgumentException()
{
var matrix = new SparseMatrix(3, 2);
var input = new DenseVector(3);
var solver = new TFQMR();
Assert.Throws<ArgumentException>(() => solver.Solve(matrix, input));
}
/// <summary>
/// Create a matrix based on this vector in row form (one single row).
/// </summary>
/// <returns>This vector as a row matrix.</returns>
public override Matrix<float> ToRowMatrix()
{
var matrix = new SparseMatrix(1, Count);
for (var i = 0; i < _storage.ValueCount; i++)
{
matrix.At(0, _storage.Indices[i], _storage.Values[i]);
}
return matrix;
}
/// <summary>
/// Create a matrix based on this vector in column form (one single column).
/// </summary>
/// <returns>This vector as a column matrix.</returns>
public override Matrix<float> ToColumnMatrix()
{
var matrix = new SparseMatrix(Count, 1);
for (var i = 0; i < _storage.ValueCount; i++)
{
matrix.At(_storage.Indices[i], 0, _storage.Values[i]);
}
return matrix;
}
/// <summary>
/// Creates a new <see cref="SparseMatrix"/> and inserts the given row at the given index.
/// </summary>
/// <param name="rowIndex">The index of where to insert the row.</param>
/// <param name="row">The row to insert.</param>
/// <returns>A new <see cref="SparseMatrix"/> with the inserted column.</returns>
/// <exception cref="ArgumentNullException">If <paramref name="row"/> is <see langword="null" />. </exception>
/// <exception cref="ArgumentOutOfRangeException">If <paramref name="rowIndex"/> is < zero or > the number of rows.</exception>
/// <exception cref="ArgumentException">If the size of <paramref name="row"/> != the number of columns.</exception>
public override Matrix<float> InsertRow(int rowIndex, Vector<float> row)
{
if (row == null)
{
throw new ArgumentNullException("row");
}
if (rowIndex < 0 || rowIndex > RowCount)
{
throw new ArgumentOutOfRangeException("rowIndex");
}
if (row.Count != ColumnCount)
{
throw new ArgumentException(Resources.ArgumentMatrixSameRowDimension, "row");
}
var result = new SparseMatrix(RowCount + 1, ColumnCount);
for (var i = 0; i < rowIndex; i++)
{
result.At(i, i, At(i, i));
}
result.SetRow(rowIndex, row);
for (var i = rowIndex + 1; i < result.RowCount; i++)
{
result.At(i, i - 1, At(i - 1, i - 1));
}
return result;
}
public void CanCreateLargeSparseMatrix()
{
var matrix = new SparseMatrix(500, 1000);
var nonzero = 0;
var rnd = new System.Random(0);
for (var i = 0; i < matrix.RowCount; i++)
{
for (var j = 0; j < matrix.ColumnCount; j++)
{
var value = rnd.Next(10)*rnd.Next(10)*rnd.Next(10)*rnd.Next(10)*rnd.Next(10);
if (value != 0)
{
nonzero++;
}
matrix[i, j] = value;
}
}
Assert.AreEqual(matrix.NonZerosCount, nonzero);
}
/// <summary>
/// Create a new sparse matrix with the diagonal as a copy of the given array.
/// This new matrix will be independent from the array.
/// A new memory block will be allocated for storing the matrix.
/// </summary>
public static SparseMatrix OfDiagonalArray(float[] diagonal)
{
var m = new SparseMatrix(diagonal.Length, diagonal.Length);
m.SetDiagonal(diagonal);
return m;
}
/// <summary>
/// Can multiply a matrix with matrix.
/// </summary>
/// <param name="nameA">Matrix A name.</param>
/// <param name="nameB">Matrix B name.</param>
public override void CanMultiplyMatrixWithMatrixIntoResult(string nameA, string nameB)
{
var matrixA = TestMatrices[nameA];
var matrixB = TestMatrices[nameB];
var matrixC = new SparseMatrix(matrixA.RowCount, matrixB.ColumnCount);
matrixA.Multiply(matrixB, matrixC);
Assert.AreEqual(matrixC.RowCount, matrixA.RowCount);
Assert.AreEqual(matrixC.ColumnCount, matrixB.ColumnCount);
for (var i = 0; i < matrixC.RowCount; i++)
{
for (var j = 0; j < matrixC.ColumnCount; j++)
{
AssertHelpers.AlmostEqualRelative(matrixA.Row(i)*matrixB.Column(j), matrixC[i, j], 15);
}
}
}
/// <summary>
/// Create a new sparse matrix with the diagonal as a copy of the given vector.
/// This new matrix will be independent from the vector.
/// A new memory block will be allocated for storing the matrix.
/// </summary>
public static SparseMatrix OfDiagonalVector(Vector<float> diagonal)
{
var m = new SparseMatrix(diagonal.Count, diagonal.Count);
m.SetDiagonal(diagonal);
return m;
}
public void CanCreateLargeMatrix()
{
const int Order = 1000000;
var matrix = new SparseMatrix(Order);
Assert.AreEqual(Order, matrix.RowCount);
Assert.AreEqual(Order, matrix.ColumnCount);
Assert.DoesNotThrow(() => matrix[0, 0] = 1);
}
/// <summary>
/// Concatenates this matrix with the given matrix.
/// </summary>
/// <param name="right">The matrix to concatenate.</param>
/// <returns>The combined <see cref="SparseMatrix"/>.</returns>
public override Matrix<float> Append(Matrix<float> right)
{
if (right == null)
{
throw new ArgumentNullException("right");
}
if (right.RowCount != RowCount)
{
throw new ArgumentException(Resources.ArgumentMatrixSameRowDimension);
}
var result = new SparseMatrix(RowCount, ColumnCount + right.ColumnCount);
Append(right, result);
return result;
}
public void CanSubtractSparseMatricesBothWays()
{
var m1 = new SparseMatrix(1, 3);
var m2 = SparseMatrix.OfArray(new float[,] { { 0, 1, 1 } });
var diff1 = m1 - m2;
var diff2 = m2 - m1;
Assert.IsTrue(diff1.Equals(m2.Negate()));
Assert.IsTrue(diff1.Equals(diff2.Negate()));
var sparseResult = new SparseMatrix(1, 3);
sparseResult.Subtract(m2, sparseResult);
Assert.IsTrue(sparseResult.Equals(diff1));
sparseResult = SparseMatrix.OfArray(new float[,] { { 0, 1, 1 } });
sparseResult.Subtract(m1, sparseResult);
Assert.IsTrue(sparseResult.Equals(diff2));
sparseResult = SparseMatrix.OfArray(new float[,] { { 0, 1, 1 } });
m1.Subtract(sparseResult, sparseResult);
Assert.IsTrue(sparseResult.Equals(diff1));
sparseResult = SparseMatrix.OfArray(new float[,] { { 0, 1, 1 } });
sparseResult.Subtract(sparseResult, sparseResult);
Assert.IsTrue(sparseResult.Equals(0*diff1));
var denseResult = new DenseMatrix(1, 3);
denseResult.Subtract(m2, denseResult);
Assert.IsTrue(denseResult.Equals(diff1));
denseResult = DenseMatrix.OfArray(new float[,] {{0, 1, 1}});
denseResult.Subtract(m1, denseResult);
Assert.IsTrue(denseResult.Equals(diff2));
var m3 = DenseMatrix.OfArray(new float[,] {{0, 1, 1}});
var diff3 = m1 - m3;
var diff4 = m3 - m1;
Assert.IsTrue(diff3.Equals(m3.Negate()));
Assert.IsTrue(diff3.Equals(diff4.Negate()));
}
/// <summary>
/// Outer product of two vectors
/// </summary>
/// <param name="u">First vector</param>
/// <param name="v">Second vector</param>
/// <returns>Matrix M[i,j] = u[i]*v[j] </returns>
/// <exception cref="ArgumentNullException">If the u vector is <see langword="null" />.</exception>
/// <exception cref="ArgumentNullException">If the v vector is <see langword="null" />.</exception>
public static Matrix<float> OuterProduct(SparseVector u, SparseVector v)
{
if (u == null)
{
throw new ArgumentNullException("u");
}
if (v == null)
{
throw new ArgumentNullException("v");
}
var matrix = new SparseMatrix(u.Count, v.Count);
for (var i = 0; i < u._storage.ValueCount; i++)
{
for (var j = 0; j < v._storage.ValueCount; j++)
{
if (u._storage.Indices[i] == v._storage.Indices[j])
{
matrix.At(i, j, u._storage.Values[i] * v._storage.Values[j]);
}
}
}
return matrix;
}
/// <summary>
/// Diagonally stacks his matrix on top of the given matrix. The new matrix is a M-by-N matrix,
/// where M = this.Rows + lower.Rows and N = this.Columns + lower.Columns.
/// The values of off the off diagonal matrices/blocks are set to zero.
/// </summary>
/// <param name="lower">The lower, right matrix.</param>
/// <exception cref="ArgumentNullException">If lower is <see langword="null" />.</exception>
/// <returns>the combined matrix</returns>
public override Matrix<float> DiagonalStack(Matrix<float> lower)
{
if (lower == null)
{
throw new ArgumentNullException("lower");
}
var result = new SparseMatrix(RowCount + lower.RowCount, ColumnCount + lower.ColumnCount);
DiagonalStack(lower, result);
return result;
}