public void CanCheckRankOfSquareSingular(int order)
        {
            var matrixA = new DenseMatrix(order, order);
            matrixA[0, 0] = 1;
            matrixA[order - 1, order - 1] = 1;
            for (var i = 1; i < order - 1; i++)
            {
                matrixA[i, i - 1] = 1;
                matrixA[i, i + 1] = 1;
                matrixA[i - 1, i] = 1;
                matrixA[i + 1, i] = 1;
            }

            var factorSvd = matrixA.Svd();

            Assert.AreEqual(factorSvd.Determinant, Complex32.Zero);
            Assert.AreEqual(factorSvd.Rank, order - 1);
        }

        public void CanCheckRankOfSquareSingular([Values(10, 50, 100)] int order)
        {
            var A = new DenseMatrix(order, order);
            A[0, 0] = 1;
            A[order - 1, order - 1] = 1;
            for (var i = 1; i < order - 1; i++)
            {
                A[i, i - 1] = 1;
                A[i, i + 1] = 1;
                A[i - 1, i] = 1;
                A[i + 1, i] = 1;
            }

            var factorEvd = A.Evd();

            Assert.AreEqual(factorEvd.Determinant, Complex32.Zero);
            Assert.AreEqual(factorEvd.Rank, order - 1);
        }

        public void CanAddSparseMatricesBothWays()
        {
            var m1 = new SparseMatrix(1, 3);
            var m2 = SparseMatrix.OfArray(new Complex32[,] { { 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 Complex32[,] { { 0, 1, 1 } });
            sparseResult.Add(m1, sparseResult);
            Assert.IsTrue(sparseResult.Equals(sum1));

            sparseResult = SparseMatrix.OfArray(new Complex32[,] { { 0, 1, 1 } });
            m1.Add(sparseResult, sparseResult);
            Assert.IsTrue(sparseResult.Equals(sum1));

            sparseResult = SparseMatrix.OfArray(new Complex32[,] { { 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 Complex32[,] {{0, 1, 1}});
            denseResult.Add(m1, denseResult);
            Assert.IsTrue(denseResult.Equals(sum1));

            var m3 = DenseMatrix.OfArray(new Complex32[,] {{0, 1, 1}});
            var sum3 = m1 + m3;
            var sum4 = m3 + m1;
            Assert.IsTrue(sum3.Equals(m3));
            Assert.IsTrue(sum3.Equals(sum4));
        }

        /// <summary>
        /// Initializes a square <see cref="DenseMatrix"/> with all zero's except for ones on the diagonal.
        /// </summary>
        /// <param name="order">the size of the square matrix.</param>
        /// <returns>A dense identity matrix.</returns>
        /// <exception cref="ArgumentException">
        /// If <paramref name="order"/> is less than one.
        /// </exception>
        public static DenseMatrix Identity(int order)
        {
            var m = new DenseMatrix(order);
            for (var i = 0; i < order; i++)
            {
                m.Data[(i * order) + i] = 1.0f;
            }

            return m;
        }

 /// <summary>
 /// Create a new dense 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 DenseMatrix OfDiagonalVector(int rows, int columns, Vector<Complex32> diagonal)
 {
     var m = new DenseMatrix(rows, columns);
     m.SetDiagonal(diagonal);
     return m;
 }

 /// <summary>
 /// Create a new dense 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 DenseMatrix OfDiagonalArray(int rows, int columns, Complex32[] diagonal)
 {
     var m = new DenseMatrix(rows, columns);
     m.SetDiagonal(diagonal);
     return m;
 }

 public void LUFailsWithNonSquareMatrix()
 {
     var matrix = new DenseMatrix(3, 1);
     Assert.That(() => matrix.LU(), Throws.ArgumentException);
 }

        /// <summary>
        /// Adds another matrix to this matrix.
        /// </summary>
        /// <param name="other">The matrix to add to this matrix.</param>
        /// <returns>The result of the addition.</returns>
        /// <exception cref="ArgumentNullException">If the other matrix is <see langword="null"/>.</exception>
        /// <exception cref="ArgumentOutOfRangeException">If the two matrices don't have the same dimensions.</exception>
        public override Matrix<Complex32> Add(Matrix<Complex32> other)
        {
            if (other == null)
            {
                throw new ArgumentNullException("other");
            }

            if (other.RowCount != RowCount || other.ColumnCount != ColumnCount)
            {
                throw new ArgumentOutOfRangeException("other", Resources.ArgumentMatrixDimensions);
            }

            Matrix<Complex32> result;
            if (other is DiagonalMatrix)
            {
                result = new DiagonalMatrix(RowCount, ColumnCount);
            }
            else
            {
                result = new DenseMatrix(RowCount, ColumnCount);
            }

            Add(other, result);
            return result;
        }

        /// <summary>
        /// Subtracts another matrix from this matrix.
        /// </summary>
        /// <param name="other">The matrix to subtract.</param>
        /// <returns>The result of the subtraction.</returns>
        /// <exception cref="ArgumentNullException">If the other matrix is <see langword="null"/>.</exception>
        /// <exception cref="ArgumentOutOfRangeException">If the two matrices don't have the same dimensions.</exception>
        public override Matrix<Complex32> Subtract(Matrix<Complex32> other)
        {
            if (other == null)
            {
                throw new ArgumentNullException("other");
            }

            if (other.RowCount != RowCount || other.ColumnCount != ColumnCount)
            {
                throw DimensionsDontMatch<ArgumentOutOfRangeException>(this, other, "other");
            }

            Matrix<Complex32> result;
            if (other is DiagonalMatrix)
            {
                result = new DiagonalMatrix(RowCount, ColumnCount);
            }
            else
            {
                result = new DenseMatrix(RowCount, ColumnCount);
            }

            Subtract(other, result);
            return result;
        }

        public void CanComputeThinQRFactorTallMatrixWithWorkArray()
        {
            var matrix = _matrices["Tall3x2"];
            var r = new Complex32[matrix.ColumnCount*matrix.ColumnCount];
            var tau = new Complex32[3];
            var q = new Complex32[matrix.RowCount*matrix.ColumnCount];
            Array.Copy(matrix.Values, q, q.Length);

            var work = new Complex32[matrix.ColumnCount*Control.BlockSize];
            Control.LinearAlgebraProvider.ThinQRFactor(q, matrix.RowCount, matrix.ColumnCount, r, tau, work);

            var mq = new DenseMatrix(matrix.RowCount, matrix.ColumnCount, q);
            var mr = new DenseMatrix(matrix.ColumnCount, matrix.ColumnCount, r);
            var a = mq*mr;
            for (var row = 0; row < matrix.RowCount; row++)
            {
                for (var col = 0; col < matrix.ColumnCount; col++)
                {
                    AssertHelpers.AlmostEqualRelative(matrix[row, col], a[row, col], 5);
                }
            }
        }

        public void CanComputeQRFactorWideMatrix()
        {
            var matrix = _matrices["Wide2x3"];
            var r = new Complex32[matrix.RowCount*matrix.ColumnCount];
            Array.Copy(matrix.Values, r, r.Length);

            var tau = new Complex32[3];
            var q = new Complex32[matrix.RowCount*matrix.RowCount];
            Control.LinearAlgebraProvider.QRFactor(r, matrix.RowCount, matrix.ColumnCount, q, tau);

            var mr = new DenseMatrix(matrix.RowCount, matrix.ColumnCount, r).UpperTriangle();
            var mq = new DenseMatrix(matrix.RowCount, matrix.RowCount, q);
            var a = mq*mr;

            for (var row = 0; row < matrix.RowCount; row++)
            {
                for (var col = 0; col < matrix.ColumnCount; col++)
                {
                    AssertHelpers.AlmostEqualRelative(matrix[row, col], a[row, col], 5);
                }
            }
        }

        public void CanSolveUsingCholesky()
        {
            var matrix = new DenseMatrix(3, 3, new Complex32[] {1, 1, 1, 1, 2, 3, 1, 3, 6});
            var a = new Complex32[] {1, 1, 1, 1, 2, 3, 1, 3, 6};

            var b = new[] {new Complex32(1.0f, 0.0f), 2.0f, 3.0f, 4.0f, 5.0f, 6.0f};
            Control.LinearAlgebraProvider.CholeskySolve(a, 3, b, 2);

            AssertHelpers.AlmostEqualRelative(b[0], 0, 5);
            AssertHelpers.AlmostEqualRelative(b[1], 1, 5);
            AssertHelpers.AlmostEqualRelative(b[2], 0, 5);
            AssertHelpers.AlmostEqualRelative(b[3], 3, 5);
            AssertHelpers.AlmostEqualRelative(b[4], 1, 5);
            AssertHelpers.AlmostEqualRelative(b[5], 0, 5);

            NotModified(3, 3, a, matrix);
        }

        public void CanMultiplyTallAndWideMatricesWithUpdate()
        {
            var x = _matrices["Tall3x2"];
            var y = _matrices["Wide2x3"];
            var c = new DenseMatrix(x.RowCount, y.ColumnCount);

            Control.LinearAlgebraProvider.MatrixMultiplyWithUpdate(Transpose.DontTranspose, Transpose.DontTranspose, 2.2f, x.Values, x.RowCount, x.ColumnCount, y.Values, y.RowCount, y.ColumnCount, 1.0f, c.Values);

            for (var i = 0; i < c.RowCount; i++)
            {
                for (var j = 0; j < c.ColumnCount; j++)
                {
                    var test = 2.2f*x.Row(i)*y.Column(j);

                    // if they are both close to zero, skip
                    if (Math.Abs(test.Real) < 1e-7 && Math.Abs(c[i, j].Real) < 1e-7)
                    {
                        continue;
                    }

                    AssertHelpers.AlmostEqualRelative(2.2f*x.Row(i)*y.Column(j), c[i, j], 5);
                }
            }
        }

        public void CanMultiplyWideAndTallMatricesWithUpdate()
        {
            var x = _matrices["Wide2x3"];
            var y = _matrices["Tall3x2"];
            var c = new DenseMatrix(x.RowCount, y.ColumnCount);

            Control.LinearAlgebraProvider.MatrixMultiplyWithUpdate(Transpose.DontTranspose, Transpose.DontTranspose, 2.2f, x.Values, x.RowCount, x.ColumnCount, y.Values, y.RowCount, y.ColumnCount, 1.0f, c.Values);

            for (var i = 0; i < c.RowCount; i++)
            {
                for (var j = 0; j < c.ColumnCount; j++)
                {
                    AssertHelpers.AlmostEqualRelative(2.2f*x.Row(i)*y.Column(j), c[i, j], 5);
                }
            }
        }

        public void CanMultiplyTallAndWideMatrices()
        {
            var x = _matrices["Tall3x2"];
            var y = _matrices["Wide2x3"];
            var c = new DenseMatrix(x.RowCount, y.ColumnCount);

            Control.LinearAlgebraProvider.MatrixMultiply(x.Values, x.RowCount, x.ColumnCount, y.Values, y.RowCount, y.ColumnCount, c.Values);

            for (var i = 0; i < c.RowCount; i++)
            {
                for (var j = 0; j < c.ColumnCount; j++)
                {
                    AssertHelpers.AlmostEqualRelative(x.Row(i)*y.Column(j), c[i, j], 5);
                }
            }
        }

        public void CanSolveUsingSvdTallMatrixOnFactoredMatrix()
        {
            var matrix = _matrices["Tall3x2"];
            var a = new Complex32[matrix.RowCount*matrix.ColumnCount];
            Array.Copy(matrix.Values, a, a.Length);

            var s = new Complex32[matrix.ColumnCount];
            var u = new Complex32[matrix.RowCount*matrix.RowCount];
            var vt = new Complex32[matrix.ColumnCount*matrix.ColumnCount];

            Control.LinearAlgebraProvider.SingularValueDecomposition(true, a, matrix.RowCount, matrix.ColumnCount, s, u, vt);

            var b = new[] {new Complex32(1.0f, 0.0f), 2.0f, 3.0f, 4.0f, 5.0f, 6.0f};
            var x = new Complex32[matrix.ColumnCount*2];
            Control.LinearAlgebraProvider.SvdSolveFactored(matrix.RowCount, matrix.ColumnCount, s, u, vt, b, 2, x);

            var mb = new DenseMatrix(matrix.RowCount, 2, b);
            var test = (matrix.Transpose()*matrix).Inverse()*matrix.Transpose()*mb;

            AssertHelpers.AlmostEqual(test[0, 0], x[0], 5);
            AssertHelpers.AlmostEqual(test[1, 0], x[1], 5);
            AssertHelpers.AlmostEqual(test[0, 1], x[2], 5);
            AssertHelpers.AlmostEqual(test[1, 1], x[3], 5);
        }

        /// <summary>
        /// Returns the conjugate transpose of this matrix.
        /// </summary>        
        /// <returns>The conjugate transpose of this matrix.</returns>
        public override Matrix<Complex32> ConjugateTranspose()
        {
            var ret = new DenseMatrix(_columnCount, _rowCount);
            for (var j = 0; j < _columnCount; j++)
            {
                var index = j * _rowCount;
                for (var i = 0; i < _rowCount; i++)
                {
                    ret._data[(i * _columnCount) + j] = _data[index + i].Conjugate();
                }
            }

            return ret;
        }

        public void CanSolveUsingQRTallMatrixUsingWorkArray()
        {
            var matrix = _matrices["Tall3x2"];
            var a = new Complex32[matrix.RowCount*matrix.ColumnCount];
            Array.Copy(matrix.Values, a, a.Length);

            var b = new[] {new Complex32(1.0f, 0.0f), 2.0f, 3.0f, 4.0f, 5.0f, 6.0f};
            var x = new Complex32[matrix.ColumnCount*2];
            var work = new Complex32[matrix.RowCount*matrix.RowCount];
            Control.LinearAlgebraProvider.QRSolve(a, matrix.RowCount, matrix.ColumnCount, b, 2, x, work);

            NotModified(3, 2, a, matrix);

            var mb = new DenseMatrix(matrix.RowCount, 2, b);
            var test = (matrix.Transpose()*matrix).Inverse()*matrix.Transpose()*mb;

            AssertHelpers.AlmostEqualRelative(test[0, 0], x[0], 5);
            AssertHelpers.AlmostEqualRelative(test[1, 0], x[1], 5);
            AssertHelpers.AlmostEqualRelative(test[0, 1], x[2], 5);
            AssertHelpers.AlmostEqualRelative(test[1, 1], x[3], 5);
        }

        /// <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 DenseMatrix OuterProduct(DenseVector u, DenseVector v)
        {
            if (u == null)
            {
                throw new ArgumentNullException("u");
            }

            if (v == null)
            {
                throw new ArgumentNullException("v");
            }

            var matrix = new DenseMatrix(u.Count, v.Count);
            CommonParallel.For(
                0,
                u.Count,
                i =>
                {
                    for (var j = 0; j < v.Count; j++)
                    {
                        matrix.At(i, j, u._values[i] * v._values[j]);
                    }
                });
            return matrix;
        }

        public void CanSolveUsingQRSquareMatrixOnFactoredMatrix()
        {
            var matrix = _matrices["Square3x3"];
            var a = new Complex32[matrix.RowCount*matrix.RowCount];
            Array.Copy(matrix.Values, a, a.Length);

            var tau = new Complex32[matrix.ColumnCount];
            var q = new Complex32[matrix.ColumnCount*matrix.ColumnCount];
            Control.LinearAlgebraProvider.QRFactor(a, matrix.RowCount, matrix.ColumnCount, q, tau);

            var b = new[] {new Complex32(1.0f, 0.0f), 2.0f, 3.0f, 4.0f, 5.0f, 6.0f};
            var x = new Complex32[matrix.ColumnCount*2];
            Control.LinearAlgebraProvider.QRSolveFactored(q, a, matrix.RowCount, matrix.ColumnCount, tau, b, 2, x);

            var mx = new DenseMatrix(matrix.ColumnCount, 2, x);
            var mb = matrix*mx;

            AssertHelpers.AlmostEqualRelative(mb[0, 0], b[0], 5);
            AssertHelpers.AlmostEqualRelative(mb[1, 0], b[1], 5);
            AssertHelpers.AlmostEqualRelative(mb[2, 0], b[2], 5);
            AssertHelpers.AlmostEqualRelative(mb[0, 1], b[3], 5);
            AssertHelpers.AlmostEqualRelative(mb[1, 1], b[4], 4);
            AssertHelpers.AlmostEqualRelative(mb[2, 1], b[5], 4);
        }