Merge branch 'v3' of git.hemio.de:xerus/xerus into v3

.gitignore

@@ -16,9 +16,14 @@
 
 !*.sh
 
+!*.rb
+!*.py
+
 !*.md
 !*.xml
 !*.dox
+!*.css
+!*.js
 
 # ...even if they are in subdirectories
 !*/

ChangeLog.md

@@ -2,13 +2,16 @@
 
 Potentially breaking changes are marked with an exclamation point '!' at the begin of their description.
 
-* 2016-??-?? v3.0.0
+* 2017-??-?? v3.0.0
  * Python wrapper now stable.
  * ! REQUIRE macro now logs as error instead of fatal error.
  * ! All macros and preprocessor defines now use the XERUS_ prefix. The config.mk file changed accordingly.
  * ! TT::find_largest_entry and TT::dyadic_product left the TT scope.
  * ! Tensor::modify_diag_elements renamed to Tensor::modify_diagonal_entries for naming consistency.
- * Some minor bugfixes.
+ * Much faster solve of matrix equations Ax=b by exploiting symmetry and definiteness where possible. This directly speeds up the ALS as well.
+ * Added a highly optimized minimal version of the ALS algorithm as xALS.
+ * Added Tensor.one_norm() and one_norm(Tensor) to calculate the one norm of a Tensor.
+ * Some minor bugfixes and performance improvements.
 
 * 2016-06-23 v2.4.0
  * Introduced nomeclature 'mode'. Marked all functions that will be renamed / removed in v3.0.0 as deprecated.
@@ -26,7 +29,7 @@ Potentially breaking changes are marked with an exclamation point '!' at the beg
  * Bug fix in the handling of fixed indices in TensorNetworks.
  * Several static member function now warn if their return type is not used.
  * Initial support for compilation with the intel ICC.
- 
+
 * 2016-03-11 v2.2.1
  * Added support for 32bit systems.
 

Makefile

@@ -107,7 +107,7 @@ define \n
 endef
 
 FLAGS = $(strip $(WARNINGS) $(OPTIMIZE) $(LOGGING) $(DEBUG) $(ADDITIONAL_INCLUDE) $(OTHER))
-PYTHON_FLAGS = $(strip $(WARNINGS) $(LOGGING) $(DEBUG) $(ADDITIONAL_INCLUDE) $(OTHER))
+PYTHON_FLAGS = $(strip $(WARNINGS) $(LOGGING) $(DEBUG) $(ADDITIONAL_INCLUDE) $(OTHER) -fno-var-tracking-assignments)
 MINIMAL_DEPS = Makefile config.mk makeIncludes/general.mk makeIncludes/warnings.mk makeIncludes/optimization.mk
 
 

config.mk.default

@@ -73,6 +73,7 @@ COMPILE_THREADS = 8						# Number of threads to use during link time optimizatio
 # DEBUG += -fsanitize=undefined			# GCC only
 # DEBUG += -fsanitize=memory			# Clang only
 # DEBUG += -fsanitize=address			# find out of bounds access
+# DEBUG += -pg							# adds profiling code for the 'gprof' analyzer
 
 
 # Xerus has a buildin logging system to provide runtime information. Here you can adjust the logging level used by the library.

doc/advanced.md

-# Notes for Advanced Users
-
-## Multi-Threading
-
-Please note, that `xerus` is only thread-safe up to 1024 threads at the time of this writing. This number is due to the internal handling of indices. To ensure that indices can 
-be compared and identified uniquely, they store a unique id of which the first 10 bits are reserved to denote the number of the current thread. With more than 1024 threads (when these
-10 bits overflow) it can thus lead to collisions and indices that were shared between threads and were meant to be seperate suddenly evaluate to be equal to all algorithms.
-
-

doc/building_xerus.md

@@ -44,7 +44,7 @@ The output of this executable should then list a number of passed tests and end
 -------------------------------------------------------------------------------
 ~~~
 Note in particular, that all tests were passed. Should this not be the case please file a bug report with as many details as you 
-can in our [issuetracker](https://git.hemio.de/xerus/xerus/issues).
+can in our [issuetracker](https://git.hemio.de/xerus/xerus/issues) or write us an [email](mailto:contact@libxerus.org).
 
 
 ## Building the Library

doc/indices.md

+# Indices and Equations
+
+## Blockwise Construction of Tensors
+

doc/mainpage.md

@@ -24,7 +24,9 @@ The current development version is also available in the same git repository (br
 There are a number of tutorials to get you started using the `xerus` library.
 * [Building xerus](@ref md_building_xerus) - instruction on how to build the library iteself and your first own program using it.
 * [Quick-Start guide](_quick-_start-example.html) - a short introduction into the basic `xerus` functionality.
+* [Tensor Class](@ref md_tensor) - an introduction into the most important functions relating to the `xerus::Tensor` class.
 * [TT Tensors](_t_t-_tensors_01_07_m_p_s_08-example.html) - using the MPS or Tensor-Train decomposition.
+* [Optimization](@ref md_optimization) - some hints how to achieve the fastest possible numerical code using `xerus`.
 * [Debugging](@ref md_tut_debugging) - using `xerus`'s capabilities to debug your own application.
 
 ## Issues

doc/new/.gitignore

+_site
+

doc/new/_config.yml

+# Jekyll configuration
+
+name: Xerus Documentation
+description: Documentation for the xerus library.
+
+# baseurl will often be '', but for a project page on gh-pages, it needs to
+# be the project name.
+# *** IMPORTANT: If your local "jekyll serve" throws errors change this to '' or
+#     run it like so: jekyll serve --baseurl=''
+baseurl: ''
+
+permalink: /:title
+# paginate: 3
+
+highlighter: rouge
+# highlighter: coderay
+markdown: kramdown
+# gems: ['TabsConverter']
+
+# exclude: ['README.md', 'LICENSE']

doc/new/_includes/examples/als.cpp

+#include <xerus.h>
+
+using namespace xerus;
+
+class InternalSolver {
+	const size_t d;
+	
+	std::vector<Tensor> leftAStack;
+	std::vector<Tensor> rightAStack;
+	
+	std::vector<Tensor> leftBStack;
+	std::vector<Tensor> rightBStack;
+	
+	TTTensor& x;
+	const TTOperator& A;
+	const TTTensor& b;
+	const double solutionsNorm;
+public:
+	size_t maxIterations;
+	
+	InternalSolver(const TTOperator& _A, TTTensor& _x, const TTTensor& _b) 
+		: d(_x.degree()), x(_x), A(_A), b(_b), solutionsNorm(frob_norm(_b)), maxIterations(1000)
+	{ 
+		leftAStack.emplace_back(Tensor::ones({1,1,1}));
+		rightAStack.emplace_back(Tensor::ones({1,1,1}));
+		leftBStack.emplace_back(Tensor::ones({1,1}));
+		rightBStack.emplace_back(Tensor::ones({1,1}));
+	}
+	
+	
+	void push_left_stack(const size_t _position) {
+		Index i1, i2, i3, j1 , j2, j3, k1, k2;
+		const Tensor &xi = x.get_component(_position);
+		const Tensor &Ai = A.get_component(_position);
+		const Tensor &bi = b.get_component(_position);
+		
+		Tensor tmpA, tmpB;
+		tmpA(i1, i2, i3) = leftAStack.back()(j1, j2, j3)
+				*xi(j1, k1, i1)*Ai(j2, k1, k2, i2)*xi(j3, k2, i3);
+		leftAStack.emplace_back(std::move(tmpA));
+		tmpB(i1, i2) = leftBStack.back()(j1, j2)
+				*xi(j1, k1, i1)*bi(j2, k1, i2);
+		leftBStack.emplace_back(std::move(tmpB));
+	}
+	
+	
+	void push_right_stack(const size_t _position) {
+		Index i1, i2, i3, j1 , j2, j3, k1, k2;
+		const Tensor &xi = x.get_component(_position);
+		const Tensor &Ai = A.get_component(_position);
+		const Tensor &bi = b.get_component(_position);
+		
+		Tensor tmpA, tmpB;
+		tmpA(i1, i2, i3) = xi(i1, k1, j1)*Ai(i2, k1, k2, j2)*xi(i3, k2, j3)
+				*rightAStack.back()(j1, j2, j3);
+		rightAStack.emplace_back(std::move(tmpA));
+		tmpB(i1, i2) = xi(i1, k1, j1)*bi(i2, k1, j2)
+				*rightBStack.back()(j1, j2);
+		rightBStack.emplace_back(std::move(tmpB));
+	}
+	
+	double calc_residual_norm() {
+		Index i,j;
+		return frob_norm(A(i/2, j/2)*x(j&0) - b(i&0)) / solutionsNorm;
+	}
+	
+	
+	void solve() {
+		// Build right stack
+		x.move_core(0, true);
+		for (size_t pos = d-1; pos > 0; --pos) {
+			push_right_stack(pos);
+		}
+		
+		Index i1, i2, i3, j1 , j2, j3, k1, k2;
+		std::vector<double> residuals(10, 1000.0);
+		
+		for (size_t itr = 0; itr < maxIterations; ++itr) {
+			// Calculate residual and check end condition
+			residuals.push_back(calc_residual_norm());
+			if (residuals.back()/residuals[residuals.size()-10] > 0.99) {
+				XERUS_LOG(simpleALS, "Done! Residual decrease from " << std::scientific << residuals[10] << " to " << std::scientific << residuals.back() << " in " << residuals.size()-10 << " iterations.");
+				return; // We are done!
+			}
+			XERUS_LOG(simpleALS, "Iteration: " << itr << " Residual: " << residuals.back());
+			
+			
+			// Sweep Left -> Right
+			for (size_t corePosition = 0; corePosition < d; ++corePosition) {
+				Tensor op, rhs;
+				
+				const Tensor &Ai = A.get_component(corePosition);
+				const Tensor &bi = b.get_component(corePosition);
+				
+				op(i1, i2, i3, j1, j2, j3) = leftAStack.back()(i1, k1, j1)*Ai(k1, i2, j2, k2)*rightAStack.back()(i3, k2, j3);
+				rhs(i1, i2, i3) =            leftBStack.back()(i1, k1) *   bi(k1, i2, k2) *   rightBStack.back()(i3, k2);
+				
+				xerus::solve(x.component(corePosition), op, rhs);
+				
+				if (corePosition+1 < d) {
+					x.move_core(corePosition+1, true);
+					push_left_stack(corePosition);
+					rightAStack.pop_back();
+					rightBStack.pop_back();
+				}
+			}
+			
+			
+			// Sweep Right -> Left : only move core and update stacks
+			x.move_core(0, true);
+			for (size_t corePosition = d-1; corePosition > 0; --corePosition) {
+				push_right_stack(corePosition);
+				leftAStack.pop_back();
+				leftBStack.pop_back();
+			}
+			
+		}
+	}
+	
+};
+
+void simpleALS(const TTOperator& _A, TTTensor& _x, const TTTensor& _b)  {
+	InternalSolver solver(_A, _x, _b);
+	return solver.solve();
+}
+
+
+int main() {
+	Index i,j,k;
+	
+	auto A = TTOperator::random(std::vector<size_t>(16, 4), std::vector<size_t>(7,2));
+	A(i/2,j/2) = A(i/2, k/2) * A(j/2, k/2);
+	
+	auto solution = TTTensor::random(std::vector<size_t>(8, 4), std::vector<size_t>(7,3));
+	TTTensor b;
+	b(i&0) = A(i/2, j/2) * solution(j&0);
+	
+	auto x = TTTensor::random(std::vector<size_t>(8, 4), std::vector<size_t>(7,3));
+	simpleALS(A, x, b);
+	
+	XERUS_LOG(info, "Residual: " << frob_norm(A(i/2, j/2) * x(j&0) - b(i&0))/frob_norm(b));
+	XERUS_LOG(info, "Error: " << frob_norm(solution-x)/frob_norm(x));
+}

doc/new/_includes/examples/als.py

+import xerus as xe
+
+class InternalSolver :
+	def __init__(self, A, x, b):
+		self.A = A
+		self.b = b
+		self.x = x
+		self.d = x.degree()
+		self.solutionsNorm = b.frob_norm()
+		
+		self.leftAStack = [ xe.Tensor.ones([1,1,1]) ]
+		self.leftBStack = [ xe.Tensor.ones([1,1]) ]
+		self.rightAStack = [ xe.Tensor.ones([1,1,1]) ]
+		self.rightBStack = [ xe.Tensor.ones([1,1]) ]
+		
+		self.maxIterations = 1000
+	
+	
+	def push_left_stack(self, pos) :
+		i1,i2,i3, j1,j2,j3, k1,k2 = xe.indices(8)
+		Ai = self.A.get_component(pos)
+		xi = self.x.get_component(pos)
+		bi = self.b.get_component(pos)
+		
+		tmpA = xe.Tensor()
+		tmpB = xe.Tensor()
+		tmpA(i1, i2, i3) << self.leftAStack[-1](j1, j2, j3)\
+				*xi(j1, k1, i1)*Ai(j2, k1, k2, i2)*xi(j3, k2, i3)
+		self.leftAStack.append(tmpA)
+		tmpB(i1, i2) << self.leftBStack[-1](j1, j2)\
+				*xi(j1, k1, i1)*bi(j2, k1, i2)
+		self.leftBStack.append(tmpB)
+	
+	def push_right_stack(self, pos) :
+		i1,i2,i3, j1,j2,j3, k1,k2 = xe.indices(8)
+		Ai = self.A.get_component(pos)
+		xi = self.x.get_component(pos)
+		bi = self.b.get_component(pos)
+		
+		tmpA = xe.Tensor()
+		tmpB = xe.Tensor()
+		tmpA(j1, j2, j3) << xi(j1, k1, i1)*Ai(j2, k1, k2, i2)*xi(j3, k2, i3) \
+				* self.rightAStack[-1](i1, i2, i3)
+		self.rightAStack.append(tmpA)
+		tmpB(j1, j2) << xi(j1, k1, i1)*bi(j2, k1, i2) \
+				* self.rightBStack[-1](i1, i2)
+		self.rightBStack.append(tmpB)
+	
+	
+	def calc_residual_norm(self) :
+		i,j = xe.indices(2)
+		return xe.frob_norm(self.A(i/2, j/2)*self.x(j&0) - self.b(i&0)) / self.solutionsNorm
+	
+	
+	def solve(self) :
+		# build right stack
+		self.x.move_core(0, True)
+		for pos in reversed(xrange(1, self.d)) :
+			self.push_right_stack(pos)
+		
+		i1,i2,i3, j1,j2,j3, k1,k2 = xe.indices(8)
+		residuals = [1000]*10
+		
+		for itr in xrange(self.maxIterations) :
+			residuals.append(self.calc_residual_norm())
+			if residuals[-1]/residuals[-10] > 0.99 :
+				print("Done! Residual decreased from:", residuals[10], "to", residuals[-1], "in", len(residuals)-10, "sweeps")
+				return
+			
+			print("Iteration:",itr, "Residual:", residuals[-1])
+			
+			# sweep left -> right
+			for pos in xrange(self.d):
+				op = xe.Tensor()
+				rhs = xe.Tensor()
+				
+				Ai = self.A.get_component(pos)
+				bi = self.b.get_component(pos)
+				
+				op(i1, i2, i3, j1, j2, j3) << self.leftAStack[-1](i1, k1, j1)*Ai(k1, i2, j2, k2)*self.rightAStack[-1](i3, k2, j3)
+				rhs(i1, i2, i3) <<            self.leftBStack[-1](i1, k1) *   bi(k1, i2, k2) *   self.rightBStack[-1](i3, k2)
+				
+				tmp = xe.Tensor()
+				tmp(i1&0) << rhs(j1&0) / op(j1/2, i1/2)
+				self.x.set_component(pos, tmp)
+				
+				if pos+1 < self.d :
+					self.x.move_core(pos+1, True)
+					self.push_left_stack(pos)
+					self.rightAStack.pop()
+					self.rightBStack.pop()
+			
+			
+			# right -> left, only move core and update stack
+			self.x.move_core(0, True)
+			for pos in reversed(xrange(1,self.d)) :
+				self.push_right_stack(pos)
+				self.leftAStack.pop()
+				self.leftBStack.pop()
+
+
+def simpleALS(A, x, b) :
+	solver = InternalSolver(A, x, b)
+	solver.solve()
+
+if __name__ == "__main__":
+	i,j,k = xe.indices(3)
+	
+	A = xe.TTOperator.random([4]*16, [2]*7)
+	A(i/2,j/2) << A(i/2, k/2) * A(j/2, k/2)
+	
+	solution = xe.TTTensor.random([4]*8, [3]*7)
+	b = xe.TTTensor()
+	b(i&0) << A(i/2, j/2) * solution(j&0)
+	
+	x = xe.TTTensor.random([4]*8, [3]*7)
+	simpleALS(A, x, b)
+	
+	print("Residual:", xe.frob_norm(A(i/2, j/2) * x(j&0) - b(i&0))/xe.frob_norm(b))
+	print("Error:", xe.frob_norm(solution-x)/xe.frob_norm(x))
+	

tutorials/quickStart.cpp → doc/new/_includes/examples/qtt.cpp

-/**
-* @file 
-* @short the source code to the "Quick-Start" guide
-*/
-
 #include <xerus.h>
 
 int main() {
@@ -30,9 +25,7 @@ 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
-	xerus::Tensor b({512}, []() {
-		return 1.0;
-	});
+	auto b = xerus::Tensor::ones({512});
 	
 	b.reinterpret_dimensions(std::vector<size_t>(9, 2));
 	xerus::TTTensor ttb(b);
@@ -56,8 +49,5 @@ int main() {
 	x(j^9) = b(i^9) / A(i^9, j^9);
 	
 	// and calculate the Frobenius norm of the difference
-	// here i&0 denotes a multiindex large enough to fully index the respective tensors
-	// the subtraction of different formats will default to Tensor subtraction such that
-	// the TTTensor ttx will be evaluated to a Tensor prior to subtraction.
-	std::cout << "error: " << frob_norm(x(i&0) - ttx(i&0)) << std::endl;
+	std::cout << "error: " << frob_norm(x - xerus::Tensor(ttx)) << std::endl;
 }

doc/new/_includes/examples/qtt.py

+import xerus as xe
+
+# construct the stiffness matrix A using a fill function
+def A_fill(idx):
+	if idx[0] == idx[1] :
+		return 2.0
+	elif idx[1] == idx[0]+1 or idx[1]+1 == idx[0] :
+		return -1.0
+	else:
+		return 0.0
+
+A = xe.Tensor.from_function([512,512], A_fill)
+
+# and dividing it by h^2 = multiplying it with N^2
+A *= 512*512
+
+# reinterpret the 512x512 tensor as a 2^18 tensor
+# and create (Q)TT decomposition of it
+A.reinterpret_dimensions([2,]*18)
+ttA = xe.TTOperator(A)
+
+# and verify its rank
+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)
+
+# construct a random initial guess of rank 3 for the ALS algorithm
+ttx = xe.TTTensor.random([2,]*9, [3,]*8)
+
+# and solve the system with the default ALS algorithm for symmetric positive operators
+xe.ALS_SPD(ttA, ttx, ttb)
+
+# to perform arithmetic operations we need to define some indices
+i,j,k = xe.indices(3)
+
+# calculate the residual of the just solved system to evaluate its accuracy
+# here i^9 denotes a multiindex named i of dimension 9 (ie. spanning 9 indices of the respective tensors)
+residual = xe.frob_norm( ttA(i^9,j^9)*ttx(j^9) - ttb(i^9) )
+print("residual:", residual)
+
+# as an comparison solve the system exactly using the Tensor / operator
+x = xe.Tensor()
+x(j^9) << b(i^9) / A(i^9, j^9)
+
+# and calculate the Frobenius norm of the difference
+print("error:", xe.frob_norm(x - xe.Tensor(ttx)))
+

doc/new/_plugins/tabs.rb

+module Jekyll
+  class TabsConverter < Converter
+    safe true
+    priority :low
+
+    def matches(ext)
+      ext =~ /^\.md$/i
+    end
+
+    def output_ext(ext)
+      ".html"
+    end
+
+    def convert(content)
+      content.gsub('<p>__tabsInit</p>', "<input id=\"tab1\" type=\"radio\" name=\"tabs\" checked><input id=\"tab2\" type=\"radio\" name=\"tabs\">")
+             .gsub('<p>__tabsStart</p>', "<div id=\"tabs\"><label for=\"tab1\">C++</label><label for=\"tab2\">Python</label><div id=\"content\"><section id=\"content1\">")
+             .gsub('<p>__tabsMid</p>', "</section><section id=\"content2\">")
+             .gsub('<p>__tabsEnd</p>', "</section></div></div>")
+			 .gsub('<p>__dangerStart</p>', "<div class=\"alert alert-danger\">")
+			 .gsub('<p>__dangerEnd</p>', "</div>")
+			 .gsub('<p>__warnStart</p>', "<div class=\"alert alert-warning\">")
+			 .gsub('<p>__warnEnd</p>', "</div>")
+    end
+  end
+end
+

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

+---
+layout: post
+title: "QTT Decomposition"
+date: 1000-12-10
+topic: "Examples"
+section: "Examples"
+---
+__tabsInit
+# QTT Decomposition
+
+This guide can be used as a quick start into `xerus`. It will introduce some basic functionality of the library,
+demonstrate the general layout and is enough for very basic applications. It is recommended to also have a look
+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.
+
+In the following we will solve a FEM equation arising from the heat equation using a QTT decomposition and the ALS algorithm.
+
+To start we create the stiffness matrix as a dense (ie. not sparse or decomposed) tensor.
+To define the entries we pass a function to the constructor of the `Tensor` object that will be
+called for every entry with a vector of size_t integers defining the indices of the current entry.
+As it is simpler to think of the stiffness matrix in its original form rather than the QTT form we will 
+set the dimensions to 512x512.
+
+__tabsStart
+~~~ cpp
+xerus::Tensor A({512,512}, [](const std::vector<size_t> &idx){
+	if (idx[0] == idx[1]) {
+		return 2.0;
+	} else if (idx[1] == idx[0]+1 || idx[1]+1 == idx[0]) {
+		return -1.0;
+	} else {
+		return 0.0;
+	}
+});
+~~~
+__tabsMid
+~~~ python
+def A_fill(idx):
+	if idx[0] == idx[1] :
+		return 2
+	elif idx[1] == idx[0]+1 or idx[1]+1 == idx[0] :
+		return -1
+	else :
+		return 0
+
+A = xerus.Tensor([512,512], A_fill)
+~~~
+__tabsEnd
+
+To account for the $ h^2 $ factor that we have ignored so far we simply multipy the operator by $ N^2 $.
+
+__tabsStart
+~~~ cpp
+A *= 512*512;
+~~~
+__tabsMid
+~~~ python
+A *= 512*512
+~~~
+__tabsEnd
+
+By reinterpreting the dimension and thus effectively treating the tensor as a $ 2^{18} $ instead of a $ 512^2 $ tensor,
+the decomposition into a `TTTensor` will give us the stiffness matrix in a QTT format.
+
+__tabsStart
+~~~ cpp
+A.reinterpret_dimensions(std::vector<size_t>(18, 2));
+xerus::TTOperator ttA(A);
+~~~
+__tabsMid
+~~~ python
+A.reinterpret_dimensions([2,]*18)
+ttA = xerus.TTOperator(A)
+~~~
+__tabsEnd
+
+As the Laplace operator is representable exactly in a low-rank QTT format, the rank of this `ttA` should be low after this construction.
+We can verify this by printing the ranks:
+
+__tabsStart
+~~~ cpp
+using xerus::misc::operator<<; // to be able to pipe vectors
+std::cout << "ttA ranks: " << ttA.ranks() << std::endl;
+~~~
+__tabsMid
+~~~ python
+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:
+
+__tabsStart
+~~~ cpp
+auto b = xerus::Tensor::ones({512});
+
+b.reinterpret_dimensions(std::vector<size_t>(9, 2));
+xerus::TTTensor ttb(b);
+~~~
+__tabsMid
+~~~ python
+b = xerus.Tensor.ones([512])
+
+b.reinterpret_dimensions([2,]*9)
+ttb = xerus.TTTensor(b)
+~~~
+__tabsEnd
+
+To have an initial vector for the ALS algorithm we create a random TTTensor of the desired rank 
+(3 in this case - note, that this is the exact rank of the solution).
+
+__tabsStart
+~~~ cpp
+xerus::TTTensor ttx = xerus::TTTensor::random(std::vector<size_t>(9, 2), std::vector<size_t>(8, 3));
+~~~
+__tabsMid
+~~~ python
+ttx = xerus.TTTensor.random([2,]*9, [3,]*8)
+~~~
+__tabsEnd
+
+With these three tensors (the operator `ttA`, the right-hand-side `ttb` and the initial guess `ttx`)
+we can now perform the ALS algorithm to solve for `ttx` (note here, that the _SPD suffix chooses the variant of the ALS
+that assumes that the given operator is symmetric positive definite)
+
+__tabsStart
+~~~ cpp
+xerus::ALS_SPD(ttA, ttx, ttb);
+~~~
+__tabsMid
+~~~ python
+xerus.ALS_SPD(ttA, ttx, ttb)
+~~~
+__tabsEnd
+
+To verify the calculation performed by the ALS we will need to perform some arithmetic operations. 
+As these require the definition of (relative) index orderings in the tensors, we define some indices
+
+__tabsStart
+~~~ cpp
+xerus::Index i,j,k;
+~~~
+__tabsMid
+~~~ python
+i,j,k = xerus.indices(3)
+~~~
+__tabsEnd
+
+and use these in calculations like `A(i,j)*x(j) - b(i)`. Note though, that our tensors are of a higher
+degree due to the QTT decomposition and we thus need to specify the corresponding dimension of the
+multiindices i,j, and k with eg. `i^9` to denote a multiindex of dimension 9.
+
+__tabsStart
+~~~ cpp
+double residual = frob_norm( ttA(i^9,j^9)*ttx(j^9) - ttb(i^9) );
+std::cout << "residual: " << residual << std::endl;
+~~~
+__tabsMid
+~~~ python
+residual = xerus.frob_norm( ttA(i^9,j^9)*ttx(j^9) - ttb(i^9) )
+print("residual:", residual)
+~~~
+__tabsEnd
+
+To get the actual error of the ALS solution (rather than just the residual) we can calculate the exact solution
+of the system using the FullTensors A, x and b and the / operator
+