Math/Algorithms: update algorithm docs and make code snippets compiled.

And add a bunch more math equations. Of course there was deprecated
functionality being used and some stuff not even compiling.

doc/snippets/CMakeLists.txt

@@ -42,7 +42,8 @@ add_library(snippets-Magnum STATIC
     Magnum.cpp
     MagnumAnimation.cpp
     MagnumAnimation-custom.cpp
-    MagnumMath.cpp)
+    MagnumMath.cpp
+    MagnumMathAlgorithms.cpp)
 target_link_libraries(snippets-Magnum PRIVATE Magnum)
 set_target_properties(snippets-Magnum PROPERTIES FOLDER "Magnum/doc/snippets")
 

doc/snippets/MagnumMathAlgorithms.cpp

@@ -0,0 +1,85 @@
+/*
+    This file is part of Magnum.
+
+    Copyright © 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018
+              Vladimír Vondruš <mosra@centrum.cz>
+
+    Permission is hereby granted, free of charge, to any person obtaining a
+    copy of this software and associated documentation files (the "Software"),
+    to deal in the Software without restriction, including without limitation
+    the rights to use, copy, modify, merge, publish, distribute, sublicense,
+    and/or sell copies of the Software, and to permit persons to whom the
+    Software is furnished to do so, subject to the following conditions:
+
+    The above copyright notice and this permission notice shall be included
+    in all copies or substantial portions of the Software.
+
+    THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+    IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+    FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
+    THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+    LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
+    FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
+    DEALINGS IN THE SOFTWARE.
+*/
+
+#include <numeric>
+#include <vector>
+#include <Corrade/Containers/ArrayView.h>
+
+#include "Magnum/Magnum.h"
+#include "Magnum/Math/Algorithms/KahanSum.h"
+#include "Magnum/Math/Algorithms/Svd.h"
+#include "Magnum/Math/Packing.h"
+
+using namespace Magnum;
+using namespace Magnum::Math::Literals;
+
+int main() {
+{
+/* [kahanSum] */
+std::vector<Float> data(100000000, 1.0f);
+Float a = std::accumulate(data.begin(), data.end(), 0.0f);      // 1.667e7f
+Float b = Math::Algorithms::kahanSum(data.begin(), data.end()); // 1.000e8f
+/* [kahanSum] */
+static_cast<void>(a);
+static_cast<void>(b);
+}
+
+{
+/* [kahanSum-iterative] */
+Containers::ArrayView<UnsignedByte> pixels;
+Float sum = 0.0f, c = 0.0f;
+for(UnsignedByte pixel: pixels) {
+    Float value = Math::unpack<Float>(pixel);
+    sum = Math::Algorithms::kahanSum(&value, &value + 1, sum, &c);
+}
+/* [kahanSum-iterative] */
+}
+
+{
+enum: std::size_t { cols = 3, rows = 4 };
+/* [svd] */
+Math::RectangularMatrix<cols, rows, Double> m;
+
+Math::RectangularMatrix<cols, rows, Double> uPart;
+Math::Vector<cols, Double> wDiagonal;
+Math::Matrix<cols, Double> v;
+
+std::tie(uPart, wDiagonal, v) = Math::Algorithms::svd(m);
+
+// Extend U
+Math::Matrix<rows, Double> u{Math::ZeroInit};
+for(std::size_t i = 0; i != rows; ++i)
+    u[i] = uPart[i];
+
+// Diagonal W
+Math::RectangularMatrix<cols, rows, Double> w =
+    Math::RectangularMatrix<cols, rows, Double>::fromDiagonal(wDiagonal);
+
+// u*w*v.transposed() == m
+/* [svd] */
+static_cast<void>(w);
+}
+
+}

src/Magnum/Math/Algorithms/GaussJordan.h

@@ -40,10 +40,10 @@ namespace Magnum { namespace Math { namespace Algorithms {
 @return True if @p a is regular, false if @p a is singular (and thus the
     system cannot be solved).
 
-As Gauss-Jordan elimination works on rows and matrices in OpenGL are
-column-major, it is more efficient to operate on transposed matrices and treat
-columns as rows. See also @ref gaussJordanInPlace() which works with
-non-transposed matrices.
+As [Gauss-Jordan elimination](https://en.wikipedia.org/wiki/Gaussian_elimination)
+works on rows and matrices in OpenGL are column-major, it is more efficient to
+operate on transposed matrices and treat columns as rows. See also
+@ref gaussJordanInPlace() which works with non-transposed matrices.
 
 The function eliminates matrix @p a and solves @p t in place. For efficiency
 reasons, only pure Gaussian elimination is done on @p a and the final
@@ -114,8 +114,9 @@ template<std::size_t size, std::size_t cols, class T> bool gaussJordanInPlace(Re
 /**
 @brief Gauss-Jordan matrix inversion
 
-Since @f$ (\boldsymbol{A}^{-1})^T = (\boldsymbol{A}^T)^{-1} @f$, passes
-@p matrix and an identity matrix to @ref gaussJordanInPlaceTransposed();
+Uses the [Gauss-Jordan elimination](https://en.wikipedia.org/wiki/Gaussian_elimination#Finding_the_inverse_of_a_matrix) to perform a matrix
+inversion. Since @f$ (\boldsymbol{A}^{-1})^T = (\boldsymbol{A}^T)^{-1} @f$,
+passes @p matrix and an identity matrix to @ref gaussJordanInPlaceTransposed();
 returning the inverted matrix. Expects that the matrix is invertible.
 @see @ref Matrix::inverted()
 */

src/Magnum/Math/Algorithms/GramSchmidt.h

@@ -36,6 +36,25 @@ namespace Magnum { namespace Math { namespace Algorithms {
 /**
 @brief In-place Gram-Schmidt matrix orthogonalization
 @param[in,out] matrix   Matrix to perform orthogonalization on
+
+Performs the [Gram-Schmidt process](https://en.wikipedia.org/wiki/Gram–Schmidt_process).
+With a *projection operator* defined as
+@f[
+    \operatorname{proj}_{\boldsymbol{u}}\,(\boldsymbol{v}) = \frac{\boldsymbol{u} \cdot \boldsymbol{v}}{\boldsymbol{u} \cdot \boldsymbol{u}} \boldsymbol{u}
+@f]
+
+@m_class{m-noindent}
+
+the process works as follows, with @f$ \boldsymbol{v}_k @f$ being columns of
+@p matrix and @f$ \boldsymbol{u}_k @f$ columns of the output: @f[
+    \boldsymbol{u}_k = \boldsymbol{v}_k - \sum_{j = 1}^{k - 1} \operatorname{proj}_{\boldsymbol{u}_j}\,(\boldsymbol{v}_k)
+@f]
+
+Note that the above is not performed directly due to numerical instability,
+the stable [modified Gram-Schmidt](https://en.wikipedia.org/wiki/Gram–Schmidt_process#Numerical_stability)
+algorithm is used instead.
+@see @ref gramSchmidtOrthogonalize(), @ref gramSchmidtOrthonormalizeInPlace(),
+    @ref Vector::projected()
 */
 template<std::size_t cols, std::size_t rows, class T> void gramSchmidtOrthogonalizeInPlace(RectangularMatrix<cols, rows, T>& matrix) {
     static_assert(cols <= rows, "Unsupported matrix aspect ratio");
@@ -59,6 +78,25 @@ template<std::size_t cols, std::size_t rows, class T> RectangularMatrix<cols, ro
 /**
 @brief In-place Gram-Schmidt matrix orthonormalization
 @param[in,out] matrix   Matrix to perform orthonormalization on
+
+Performs the [Gram-Schmidt process](https://en.wikipedia.org/wiki/Gram–Schmidt_process).
+With a *projection operator* defined as
+@f[
+    \operatorname{proj}_{\boldsymbol{u}}\,(\boldsymbol{v}) = \frac{\boldsymbol{u} \cdot \boldsymbol{v}}{\boldsymbol{u} \cdot \boldsymbol{u}} \boldsymbol{u}
+@f]
+
+@m_class{m-noindent}
+
+the Gram-Schmidt process works as follows, with @f$ \boldsymbol{v}_k @f$ being
+columns of @p matrix and @f$ \boldsymbol{e}_k @f$ columns of the output: @f[
+    \boldsymbol{u}_k = \boldsymbol{v}_k - \sum_{j = 1}^{k - 1} \operatorname{proj}_{\boldsymbol{u}_j}\,(\boldsymbol{v}_k), ~~~~~~ \boldsymbol{e}_k = \frac{\boldsymbol{u}_k}{|\boldsymbol{u}_k|}
+@f]
+
+In particular, this adds the normalization step on top of
+@ref gramSchmidtOrthogonalizeInPlace(). Note that the above is not performed
+directly due to numerical instability, the stable [modified Gram-Schmidt](https://en.wikipedia.org/wiki/Gram–Schmidt_process#Numerical_stability)
+algorithm is used instead.
+@see @ref gramSchmidtOrthonormalize(), @ref Vector::projectedOntoNormalized()
 */
 template<std::size_t cols, std::size_t rows, class T> void gramSchmidtOrthonormalizeInPlace(RectangularMatrix<cols, rows, T>& matrix) {
     static_assert(cols <= rows, "Unsupported matrix aspect ratio");

src/Magnum/Math/Algorithms/KahanSum.h

@@ -39,35 +39,24 @@ namespace Magnum { namespace Math { namespace Algorithms {
 @param[in] end              Range end
 @param[in] sum              Initial value for the sum
 @param[in,out] compensation Floating-point roundoff error compensation value.
-    If non-`nullptr`, value behind the pointer is used as initial compensation
-    value and the resulting value is
+    If non-@cpp nullptr @ce, value behind the pointer is used as initial
+    compensation value and the resulting value is
 
 Calculates a sum of a large range of floating-point numbers with roundoff error
-compensation. Compared to for example `std::accumulate()` the algorithm
-significantly reduces numerical error in the total. See Wikipedia for an
-in-depth explanation: https://en.wikipedia.org/wiki/Kahan_summation_algorithm
+compensation. Compared to for example @ref std::accumulate() the algorithm
+significantly reduces numerical error in the total. See the
+[Kahan summation algorithm](https://en.wikipedia.org/wiki/Kahan_summation_algorithm)
+article on Wikipedia for an in-depth explanation. Example with summation of a
+hundred million ones:
 
-Example with summation of a hundred million ones:
-
-@code{.cpp}
-std::vector<Float> data(100000000, 1.0f);
-Float a = std::accumulate(data.begin(), data.end());            // 1.667e7f
-Float b = Math::Algorithms::kahanSum(data.begin(), data.end()); // 1.000e8f
-@endcode
+@snippet MagnumMathAlgorithms.cpp kahanSum
 
 If required, it is also possible to use this algorithm on non-contiguous ranges
 or single values (for example when calculating sum of pixel values in an image
 with some row padding or when the inputs are generated / converted from other
 values):
 
-@code{.cpp}
-Containers::ArrayView<UnsignedByte> pixels;
-Float sum = 0.0f, c = 0.0f;
-for(UnsignedByte pixel: pixels) {
-    Float value = Math::normalize<Float>(pixel);
-    sum = Math::Algorithms::kahanSum(&value, &value + 1, sum, &c);
-}
-@endcode
+@snippet MagnumMathAlgorithms.cpp kahanSum-iterative
 */
 template<class Iterator, class T = typename std::decay<decltype(*std::declval<Iterator>())>::type> T kahanSum(Iterator begin, Iterator end, T sum = T(0), T* compensation = nullptr) {
     T c = compensation ? *compensation : T(0);

src/Magnum/Math/Algorithms/Qr.h

@@ -36,7 +36,7 @@ namespace Magnum { namespace Math { namespace Algorithms {
 /**
 @brief QR decomposition
 
-Calculated using classic Gram-Schmidt process.
+Calculated using the classic [Gram-Schmidt process](https://en.wikipedia.org/wiki/QR_decomposition#Using_the_Gram–Schmidt_process).
 */
 template<std::size_t size, class T> std::pair<Matrix<size, T>, Matrix<size, T>> qr(const Matrix<size, T>& matrix) {
     Matrix<size, T> q, r;

src/Magnum/Math/Algorithms/Svd.h

@@ -58,10 +58,11 @@ template<> constexpr Double smallestDelta<Double>() { return 1.0e-64; }
 /**
 @brief Singular Value Decomposition
 
-Performs Thin SVD on given matrix where @p rows >= @p cols:
-@f[
+Performs [Thin SVD](https://en.wikipedia.org/wiki/Singular-value_decomposition#Thin_SVD)
+on given matrix where @p rows >= `cols`: @f[
     M = U \Sigma V^*
 @f]
+
 Returns first @p cols column vectors of @f$ U @f$, diagonal of @f$ \Sigma @f$
 and non-transposed @f$ V @f$. If the solution doesn't converge, returns
 zero matrices.
@@ -69,26 +70,7 @@ zero matrices.
 Full @f$ U @f$, @f$ \Sigma @f$ matrices and original @f$ M @f$ matrix can be
 reconstructed from the values as following:
 
-@code{.cpp}
-Math::RectangularMatrix<cols, rows, Double> m;
-
-Math::RectangularMatrix<cols, rows, Double> uPart;
-Math::Vector<cols, Double> wDiagonal;
-Math::Matrix<cols, Double> v;
-
-std::tie(uPart, wDiagonal, v) = Math::Algorithms::svd(m);
-
-// Extend U
-Math::Matrix<rows, Double> u(Matrix<rows, Double>::Zero);
-for(std::size_t i = 0; i != rows; ++i)
-    u[i] = uPart[i];
-
-// Diagonal W
-Math::RectangularMatrix<cols, rows, Double> w =
-    Math::RectangularMatrix<cols, rows, Double>::fromDiagonal(wDiagonal);
-
-// u*w*v.transposed() == m
-@endcode
+@snippet MagnumMathAlgorithms.cpp svd
 
 Implementation based on *Golub, G. H.; Reinsch, C. (1970). "Singular value
 decomposition and least squares solutions"*.