Merge branch 'master' into development

.gitlab-ci.yml

@@ -10,7 +10,7 @@ stages:
 
 job_build_homepage:
   stage: build_homepage
-  script: "make -C doc doc; scp -r doc/html xerusweb:libxerus.org-443"
+  script: "cp .config.mk.ci.gcc config.mk; make -C doc doc; scp -r doc/html xerusweb:libxerus.org-443"
   when: always
   only: 
     - master

ChangeLog.md

@@ -2,6 +2,11 @@
 
 Potentially breaking changes are marked with an exclamation point '!' at the begin of their description.
 
+* 2017-05-31 v3.0.1
+ * Added TTNetwork::use_dense_representations() to convert all components to dense representation.
+ * Exporting Tensor::dense_copy() and Tensor::sparse_copy() to python.
+ * More examples on the homepage and checks for code-coverage.
+
 * 2017-05-29 v3.0.0
  * Python wrapper now stable.
  * ! REQUIRE macro now logs as error instead of fatal error.

Makefile

@@ -224,7 +224,7 @@ fullTest: $(TUTORIALS) $(TEST_NAME)
 
 
 .FORCE:
-doc: .FORCE
+doc: .FORCE doc/parseDoxytags doc/findDoxytag
 	make -C doc doc
 
 
@@ -232,14 +232,13 @@ clean:
 	rm -fr build
 	-rm -f $(TEST_NAME)
 	-rm -f include/xerus.h.gch
-	-rm -r doc/html
+	make -C doc clean
 
 
 
 benchmark: $(MINIMAL_DEPS) $(LOCAL_HEADERS) benchmark.cxx $(LIB_NAME_STATIC)
 	$(CXX) $(FLAGS) benchmark.cxx $(LIB_NAME_STATIC) $(SUITESPARSE) $(LAPACK_LIBRARIES) $(BLAS_LIBRARIES) $(CALLSTACK_LIBS) -lboost_filesystem -lboost_system -o Benchmark
 
-
 # Build rule for normal misc objects
 build/.miscObjects/%.o: %.cpp $(MINIMAL_DEPS)
 	mkdir -p $(dir $@) 

README.md

@@ -3,7 +3,7 @@
 The `xerus` library is a general purpose library for numerical calculations with higher order tensors, Tensor-Train Decompositions / Matrix Product States and other Tensor Networks.
 The focus of development was the simple usability and adaptibility to any setting that requires higher order tensors or decompositions thereof. 
 
-For tutorials and a documentation see <a href="http://libxerus.org">the doxygen documentation</a>.
+For tutorials and a documentation see <a href="http://libxerus.org">the documentation</a>.
 
 The source code is licenced under the AGPL v3.0. For more details see the LICENSE file.
 
@@ -20,6 +20,7 @@ The source code is licenced under the AGPL v3.0. For more details see the LICENS
 
 Copy the default configuration and modify it for your needs
 > cp config.mk.default config.mk
+
 > nano config.mk
 
 Test whether everything works correctly with
@@ -27,9 +28,10 @@ Test whether everything works correctly with
 
 build (and optionally install) the library with
 > make all -j4
+
 > sudo make install
 
-and you should be ready to use the library. For more details see <a href="https://www.libxerus.org/md_building_xerus.html">the "Building Xerus" page in the documentation</a>.
+and you should be ready to use the library. For more details see <a href="https://www.libxerus.org/building_xerus/">the "Building Xerus" page in the documentation</a>.
 
 
 # Issues #

VERSION

-3.0.0
+3.0.1

doc/Makefile

@@ -8,15 +8,34 @@ help:
 	\t\tserve \t\t -- Build the html documentation for the xerus library and offer it via 'jekyll serve'.\n \
 	\t\tclean \t\t -- Remove all documentation files.\n"
 
-doc:
+.FORCE:
+doc: .FORCE parseDoxytags findDoxytag
 	-mkdir html
 	doxygen doxygen/Doxyfile
+	./parseDoxytags
 	jekyll build --source jekyll/ --destination html/
 
 clean:
-	-rm -r html
+	-rm -rf html
+	-rm -f parseDoxytags findDoxytag
+	-rm -f xerus.tags xerus.tagfile
 
-serve:
+serve: .FORCE parseDoxytags findDoxytag
 	-mkdir html
 	doxygen doxygen/Doxyfile
+	./parseDoxytags
 	jekyll serve --source jekyll/ --destination html/
+
+include ../config.mk
+include ../makeIncludes/general.mk
+include ../makeIncludes/warnings.mk
+include ../makeIncludes/optimization.mk
+
+FLAGS = $(strip $(WARNINGS) $(OPTIMIZE) $(OTHER))
+
+parseDoxytags: ../src/docHelper/parseDoxytags.cpp
+	$(CXX) $(FLAGS) ../src/docHelper/parseDoxytags.cpp -o parseDoxytags
+
+
+findDoxytag: ../src/docHelper/findDoxytag.cpp
+	$(CXX) $(FLAGS) ../src/docHelper/findDoxytag.cpp -o findDoxytag

doc/doxygen/Doxyfile

@@ -740,7 +740,7 @@ WARN_IF_DOC_ERROR      = YES
 # parameter documentation, but not about the absence of documentation.
 # The default value is: NO.
 
-WARN_NO_PARAMDOC       = YES
+WARN_NO_PARAMDOC       = NO
 
 # The WARN_FORMAT tag determines the format of the warning messages that doxygen
 # can produce. The string should contain the $file, $line, and $text tags, which
@@ -2050,7 +2050,7 @@ TAGFILES               =
 # tag file that is based on the input files it reads. See section "Linking to
 # external documentation" for more information about the usage of tag files.
 
-GENERATE_TAGFILE       = 
+GENERATE_TAGFILE       = xerus.tagfile
 
 # If the ALLEXTERNALS tag is set to YES, all external class will be listed in
 # the class index. If set to NO, only the inherited external classes will be

doc/jekyll/_includes/examples/cascade.cpp

+#include <xerus.h>
+#include <iostream>
+#include <string>
+#include <fstream>
+
+using namespace xerus;
+
+const size_t MAX_NUM_PER_SITE = 32;
+
+Tensor create_M() {
+	Tensor M = -1*Tensor::identity({MAX_NUM_PER_SITE, MAX_NUM_PER_SITE});
+	for (size_t i = 0; i < MAX_NUM_PER_SITE-1; ++i) {
+		M[{i+1, i}] = 1.0;
+	}
+	return M;
+}
+
+Tensor create_L() {
+	Tensor L({MAX_NUM_PER_SITE, MAX_NUM_PER_SITE});
+	L.modify_diagonal_entries([](value_t& _x, const size_t _i) { 
+		_x = double(_i)/double(_i+5); 
+	});
+	return L;
+}
+
+Tensor create_S() {
+	Tensor S({MAX_NUM_PER_SITE, MAX_NUM_PER_SITE});
+	
+	// Set diagonal
+	for (size_t i = 0; i < MAX_NUM_PER_SITE; ++i) {
+		S[{i, i}] = -double(i);
+	}
+	
+	// Set offdiagonal
+	for (size_t i = 0; i < MAX_NUM_PER_SITE-1; ++i) {
+		S[{i, i+1}] = double(i+1);
+	}
+	return 0.07*S;
+}
+
+
+
+TTOperator create_operator(const size_t _degree) { 
+	const Index i, j, k, l;
+	
+	// Create matrices
+	const Tensor M = create_M();
+	const Tensor S = create_S();
+	const Tensor L = create_L();
+	const Tensor Sstar = 0.7*M+S;
+	const Tensor I = Tensor::identity({MAX_NUM_PER_SITE, MAX_NUM_PER_SITE});
+	
+	// Create empty TTOperator
+	TTOperator A(2*_degree);
+	
+	Tensor comp;
+	
+	// Create first component
+	comp(i, j, k, l) = 
+		Sstar(j, k)*Tensor::dirac({1, 3}, 0)(i, l) 
+		+   L(j, k)*Tensor::dirac({1, 3}, 1)(i, l) 
+		+   I(j, k)*Tensor::dirac({1, 3}, 2)(i, l);
+    
+	A.set_component(0, comp); 
+	
+	// Create middle components
+	comp(i, j, k, l) =
+		  I(j, k) * Tensor::dirac({3, 3}, {0, 0})(i, l)
+		+ M(j, k) * Tensor::dirac({3, 3}, {1, 0})(i, l)
+		+ S(j, k) * Tensor::dirac({3, 3}, {2, 0})(i, l)
+		+ L(j, k) * Tensor::dirac({3, 3}, {2, 1})(i, l)
+		+ I(j, k) * Tensor::dirac({3, 3}, {2, 2})(i, l);
+	
+	for(size_t c = 1; c+1 < _degree; ++c) {
+		A.set_component(c, comp);
+	}
+	
+	// Create last component
+	comp(i, j, k, l) = 
+		  I(j, k)*Tensor::dirac({3, 1}, 0)(i, l) 
+		+ M(j, k)*Tensor::dirac({3, 1}, 1)(i, l) 
+		+ S(j, k)*Tensor::dirac({3, 1}, 2)(i, l);
+    
+	A.set_component(_degree-1, comp);
+	
+	return A;
+}
+
+/// @brief calculates the one-norm of a TTTensor, assuming that all entries in it are positive
+double one_norm(const TTTensor &_x) {
+	Index j;
+	return double(_x(j&0) * TTTensor::ones(_x.dimensions)(j&0));
+}
+
+std::vector<TTTensor> implicit_euler(const TTOperator& _A, TTTensor _x, 
+			const double _stepSize, const size_t _n) 
+{ 
+	const TTOperator op = TTOperator::identity(_A.dimensions)-_stepSize*_A;
+	
+	Index j,k;
+	auto ourALS = ALS_SPD;
+	ourALS.convergenceEpsilon = 0;
+	ourALS.numHalfSweeps = 2;
+	
+	std::vector<TTTensor> results;
+	TTTensor nextX = _x;
+	results.push_back(_x);
+    
+	for(size_t i = 0; i < _n; ++i) {
+		ourALS(op, nextX, _x);
+        
+		// Normalize
+		double norm = one_norm(nextX);
+		nextX /= norm;
+		
+		XERUS_LOG(iter, "Done itr " << i 
+				<< " residual: " << frob_norm(op(j/2,k/2)*nextX(k&0) - _x(j&0)) 
+				<< " norm: " << norm);
+		
+		_x = nextX;
+		results.push_back(_x);
+	}
+	
+	return results;
+}
+
+double get_mean_concentration(const TTTensor& _res, const size_t _i) { 
+	const Index k,l;
+	TensorNetwork result(_res);
+		const Tensor weights({MAX_NUM_PER_SITE}, [](const size_t _k){ 
+		return double(_k); 
+	});
+	const Tensor ones = Tensor::ones({1, MAX_NUM_PER_SITE, 1});
+	
+	for (size_t j = 0; j < _res.degree(); ++j) {
+		if (j == _i) {
+			result(l&0) = result(k, l&1) * weights(k);
+		} else {
+			result(l&0) = result(k, l&1) * ones(k);
+		}
+	}
+	// at this point the degree of 'result' is 0, so there is only one entry
+	return result[{}]; 
+}
+
+void print_mean_concentrations_to_file(const std::vector<TTTensor> &_result) {
+	std::fstream out("mean.dat", std::fstream::out);
+	for (const auto& res : _result) {
+		for (size_t k = 0; k < res.degree(); ++k) {
+			out << get_mean_concentration(res, k) << ' ';
+		}
+		out << std::endl;
+	}
+}
+
+int main() {
+	const size_t numProteins = 10;
+	const size_t numSteps = 300;
+	const double stepSize = 1.0;
+	const size_t rankX = 3;
+	
+	auto start = TTTensor::dirac(
+			std::vector<size_t>(numProteins, MAX_NUM_PER_SITE), 
+			0
+	);
+	start.use_dense_representations();
+	start += 1e-14 * TTTensor::random(
+			start.dimensions, 
+			std::vector<size_t>(start.degree()-1, rankX-1)
+	);
+	const auto A = create_operator(numProteins);
+	
+	const auto results = implicit_euler(A, start, stepSize, numSteps);
+	
+	print_mean_concentrations_to_file(results);
+}
+

doc/jekyll/_includes/examples/cascade.gnuplot

+#!/usr/bin/gnuplot
+
+set term pngcairo size 600, 400
+unset key
+set grid
+set output "cascade.png"
+set xlabel "timestep"
+set ylabel "mean concentration"
+
+plot 'mean.dat' u 1 w l, \
+     'mean.dat' u 2 w l, \
+     'mean.dat' u 3 w l, \
+     'mean.dat' u 4 w l, \
+     'mean.dat' u 5 w l, \
+     'mean.dat' u 6 w l, \
+     'mean.dat' u 7 w l, \
+     'mean.dat' u 8 w l, \
+     'mean.dat' u 9 w l, \
+     'mean.dat' u 10 w l

doc/jekyll/_includes/examples/cascade.py

+import xerus as xe
+
+MAX_NUM_PER_SITE = 32
+
+def create_M():
+	M = -1 * xe.Tensor.identity([MAX_NUM_PER_SITE, MAX_NUM_PER_SITE])
+	for i in xrange(MAX_NUM_PER_SITE-1) :
+		M[[i+1, i]] = 1.0
+	return M
+
+def create_L():
+	L = xe.Tensor([MAX_NUM_PER_SITE, MAX_NUM_PER_SITE])
+	for i in xrange(MAX_NUM_PER_SITE) :
+		L[[i,i]] = i / (i+5.0)
+	return L
+
+def create_S():
+	S = xe.Tensor([MAX_NUM_PER_SITE, MAX_NUM_PER_SITE])
+	
+	# set diagonal
+	for i in xrange(MAX_NUM_PER_SITE) :
+		S[[i,i]] = -i
+	
+	# set offdiagonal
+	for i in xrange(MAX_NUM_PER_SITE-1) :
+		S[[i,i+1]] = i+1
+	
+	return 0.07*S
+
+
+def create_operator(degree):
+	i,j,k,l = xe.indices(4)
+	
+	# create matrices
+	M = create_M()
+	S = create_S()
+	L = create_L()
+	Sstar = 0.7*M + S;
+	I = xe.Tensor.identity([MAX_NUM_PER_SITE, MAX_NUM_PER_SITE])
+	
+	# create empty TTOperator
+	A = xe.TTOperator(2*degree)
+	
+	# create first component
+	comp = xe.Tensor()
+	comp(i, j, k, l) << \
+		Sstar(j, k) * xe.Tensor.dirac([1, 3], 0)(i, l) \
+		+   L(j, k) * xe.Tensor.dirac([1, 3], 1)(i, l) \
+		+   I(j, k) * xe.Tensor.dirac([1, 3], 2)(i, l)
+	
+	A.set_component(0, comp)
+	
+	# create middle components
+	comp(i, j, k, l) << \
+		  I(j, k) * xe.Tensor.dirac([3, 3], [0, 0])(i, l) \
+		+ M(j, k) * xe.Tensor.dirac([3, 3], [1, 0])(i, l) \
+		+ S(j, k) * xe.Tensor.dirac([3, 3], [2, 0])(i, l) \
+		+ L(j, k) * xe.Tensor.dirac([3, 3], [2, 1])(i, l) \
+		+ I(j, k) * xe.Tensor.dirac([3, 3], [2, 2])(i, l)
+	
+	for c in xrange(1, degree-1) :
+		A.set_component(c, comp)
+	
+	# create last component
+	comp(i, j, k, l) << \
+		  I(j, k) * xe.Tensor.dirac([3, 1], 0)(i, l) \
+		+ M(j, k) * xe.Tensor.dirac([3, 1], 1)(i, l) \
+		+ S(j, k) * xe.Tensor.dirac([3, 1], 2)(i, l)
+	
+	A.set_component(degree-1, comp)
+	
+	return A
+
+
+def one_norm(x):
+	i = xe.Index()
+	return float(x(i&0) * xe.TTTensor.ones(x.dimensions)(i&0))
+
+def implicit_euler(A, x, stepSize, n):
+	op = xe.TTOperator.identity(A.dimensions) - stepSize*A
+	
+	j,k = xe.indices(2)
+	ourALS = xe.ALS_SPD
+	ourALS.convergenceEpsilon = 1e-4
+	ourALS.numHalfSweeps = 100
+	
+	results = [x]
+	nextX = xe.TTTensor(x)
+	
+	for i in xrange(n) :
+		ourALS(op, nextX, x)
+		
+		# normalize
+		norm = one_norm(nextX)
+		nextX /= norm
+		
+		print("done itr", i, \
+			"residual:", xe.frob_norm(op(j/2,k/2)*nextX(k&0) - x(j&0)), \
+			"one-norm:", norm)
+		
+		x = xe.TTTensor(nextX) # ensure it is a copy
+		results.append(x)
+	
+	return results
+
+
+def get_mean_concentration(x, i):
+	k,l = xe.indices(2)
+	result = xe.TensorNetwork(x)
+	weights = xe.Tensor.from_function([MAX_NUM_PER_SITE], lambda idx: idx[0])
+	ones = xe.Tensor.ones([MAX_NUM_PER_SITE])
+	
+	for j in xrange(x.degree()) :
+		if j == i :
+			result(l&0) << result(k, l&1) * weights(k)
+		else :
+			result(l&0) << result(k, l&1) * ones(k)
+	
+	# at this point the degree of 'result' is 0, so there is only one entry
+	return result[[]]
+
+def print_mean_concentrations_to_file(results):
+	f = open("mean.dat", 'w')
+	for res in results :
+		for k in xrange(res.degree()) :
+			f.write(str(get_mean_concentration(res, k))+' ')
+		f.write('\n')
+	f.close()
+
+
+numProteins = 10
+numSteps = 200
+stepSize = 1.0
+rankX = 3
+
+A = create_operator(numProteins)
+
+start = xe.TTTensor.dirac([MAX_NUM_PER_SITE]*numProteins, 0)
+for i in xrange(numProteins) :
+	start.set_component(i, start.get_component(i).dense_copy())
+
+start += 1e-14 * xe.TTTensor.random(start.dimensions, [rankX-1]*(start.degree()-1))
+
+results = implicit_euler(A, start, stepSize, numSteps)
+
+print_mean_concentrations_to_file(results)

doc/jekyll/_includes/examples/qtt.cpp

@@ -25,10 +25,8 @@ int main() {
 	std::cout << "ttA ranks: " << ttA.ranks() << std::endl;
 	
 	// the right hand side of the equation both as Tensor and in (Q)TT format
-	auto b = xerus::Tensor::ones({512});
-	
-	b.reinterpret_dimensions(std::vector<size_t>(9, 2));
-	xerus::TTTensor ttb(b);
+	auto b = xerus::Tensor::ones(std::vector<size_t>(9, 2));
+	auto ttb = xerus::TTTensor::ones(b.dimensions);
 	
 	// construct a random initial guess of rank 3 for the ALS algorithm
 	xerus::TTTensor ttx = xerus::TTTensor::random(std::vector<size_t>(9, 2), std::vector<size_t>(8, 3));

doc/jekyll/_includes/examples/qtt.py

@@ -23,9 +23,8 @@ ttA = xe.TTOperator(A)
 print("ttA ranks:", ttA.ranks())
 
 # the right hand side of the equation both as Tensor and in (Q)TT format
-b = xe.Tensor.ones([512])
-b.reinterpret_dimensions([2,]*9)
-ttb = xe.TTTensor(b)
+b = xe.Tensor.ones([2,]*9)
+ttb = xe.TTTensor.ones(b.dimensions)
 
 # construct a random initial guess of rank 3 for the ALS algorithm
 ttx = xe.TTTensor.random([2,]*9, [3,]*8)

doc/jekyll/_includes/header.html

@@ -3,6 +3,7 @@
 
 <head>
     <title>Xerus Documentation{% if page.title %} - {{ page.title }} {% endif %}</title>
+    <link rel="icon" href="{{ site.baseurl }}/images/xerus.png">
     <meta charset="UTF-8">
     <meta name="viewport" content="width=device-width, initial-scale=1.0">
     <script src="{{ site.baseurl }}/js/jquery.min.js"></script>

doc/jekyll/_plugins/tabs.rb

@@ -20,9 +20,12 @@ class TabsConverter < Converter
 			.gsub('<p>__dangerEnd</p>', "</div>")
 			.gsub('<p>__warnStart</p>', "<div class=\"alert alert-warning\">")
 			.gsub('<p>__warnEnd</p>', "</div>")
+			.gsub('<p>__infoStart</p>', "<div class=\"alert alert-info\">")
+			.gsub('<p>__infoEnd</p>', "</div>")
 			.gsub('__breakFix1</a></p>', "")
 			.gsub('<p>__breakFix2', "</a>")
-			.gsub('__version', %x( git describe --tags --always --abbrev=0 ) )
+			.gsub('__version', %x( cat ../VERSION ) )
+			.gsub(/__doxyref\(([^\)]+)\)/){ |m| %x( ./findDoxytag #{$1} ) }
 	end
 end
 end

doc/jekyll/_posts/1000-11-10-minimal_als.md

 ---
 layout: post
 title: "The ALS Algorithm"
-date: 1000-12-10
+date: 1000-11-10
 topic: "Examples"
 section: "Examples"
 ---

doc/jekyll/_posts/1000-12-10-quickstart.md

@@ -13,7 +13,7 @@ demonstrate the general layout and is enough for very basic applications. It is
 at the more detailed guides for all classes one wishes to use though - or even have a look at the doxygen class documentation for details on all functions.
 
 It is assumed that you have already obtained and compiled the library itself as well as know how to link against it.
-If this is not the case, please refer to the [building xerus](building_xerus) page.
+If this is not the case, please refer to the [building xerus](/building_xerus) page.
 
 In the following we will solve a FEM equation arising from the heat equation using a QTT decomposition and the ALS algorithm.
 
@@ -90,22 +90,18 @@ print("ttA ranks:", ttA.ranks())
 ~~~
 __tabsEnd
 
-For the right-hand-side we perform similar operations to obtain a QTT decomposed vector $ b_i = 1 \forall i $.
-As the generating function needs no index information, we create a `[]()->double` lambda function:
+For the right-hand-side we will take a simple tensor that is equal to 1 at every position. As this is a commonly used tensor,
+we can simply use the named constructor provide dby `xerus`.
 
 __tabsStart
 ~~~ cpp
-auto b = xerus::Tensor::ones({512});
-
-b.reinterpret_dimensions(std::vector<size_t>(9, 2));
-xerus::TTTensor ttb(b);
+auto b = xerus::Tensor::ones(std::vector<size_t>(9, 2));
+auto ttb = xerus::TTTensor::ones(b.dimensions);
 ~~~
 __tabsMid
 ~~~ python
-b = xerus.Tensor.ones([512])
-
-b.reinterpret_dimensions([2,]*9)
-ttb = xerus.TTTensor(b)
+b = xerus.Tensor.ones([2,]*9)
+ttb = xerus.TTTensor.ones(b.dimensions)
 ~~~
 __tabsEnd
 

doc/jekyll/_posts/2000-11-26-decompositions.md

@@ -12,7 +12,7 @@ In matrix calculus the decomposition of a matrix into the matrix product of seve
 into an orthogonal and a triangular matrix (QR) or orthogonal matrices and diagonal of singular values (SVD)) are among the
 most powerful tools to devise numerical algorithms. In the case of tensors of higher degree it is necessary to indicate along
 which modes the decomposition is supposed to happen, so `xerus` uses the notation of indexed equations explained in the previous
-chapter ([Indices and Equations](indices)).
+chapter ([Indices and Equations](/indices)).
 
 ## QR Decompositions
 To provide an intuitive approach to decompositions, `xerus` uses the assignment of multiple tensors with a single operator to 

doc/jekyll/_posts/2000-11-28-indices.md

@@ -57,12 +57,12 @@ left hand side.
 
 The left hand side (`c` in the above example) is not required to have the right degree or dimensions. The type of `c` on the
 other hand does change the meaning of the equation and so has to be set correctly. E.g. if `c` is a `xerus::TensorNetwork`, no
-contraction is performed as `A(i,j)*b(j)` is in itself a valid tensor network. See also the tutorials on [TT-Tensors](tttensors)
-and [Tensor Networks](tensornetworks) for details on the respective assignments.
+contraction is performed as `A(i,j)*b(j)` is in itself a valid tensor network. See also the tutorials on [TT-Tensors](/tttensors)
+and [Tensor Networks](/tensornetworks) for details on the respective assignments.
 
 __warnStart
 
-Unless runtime checks have explicitely been disabled during compilation of `xerus` (see [Optimizations](optimization)), invalid
+Unless runtime checks have explicitely been disabled during compilation of `xerus` (see [Optimizations](/optimization)), invalid
 indexed expressions will produce runtime errors (in the form of `xerus::misc::generic_error`s being thrown as exceptions).
 
 __tabsStart
@@ -163,7 +163,7 @@ C(i,k) << A(i&1, j) * B(j, k&1)
 ~~~
 __tabsEnd
 
-The division `i/n` is useful for example to write equations with high dimensional operators such as [TT-Operators](tttensors)
+The division `i/n` is useful for example to write equations with high dimensional operators such as [TT-Operators](/tttensors)
 for which the indices are ordered per default such, that the application of the operator can be written in analogy to 
 matrix-vector products as:
 
@@ -188,7 +188,7 @@ that the degree of `v` is equal to half the degree of `A` and that its dimension
 A common use for indexed expressions is to construct tensors in a blockwise fashion. In the following example we were able to
 calculate the tensor `comp` whenever the first index was fixed, either by numerical construction (`A` and `B`) or by showing
 mathematically, that it is then equal to a well known tensor (here the identity matrix). The full tensor can thus be constructed
-with the help of the named constructors of `xerus::Tensor` (see the [Tensor tutorial](tensor)) as follows.