My master's thesis deals with dense and sparse solvers for generalized eigenvalue problems (GEPs) with Hermitian positive semidefinite matrices. Key results are
- structure-preserving backward error bounds computable in linear time,
- the runtime of GSVD-based dense GEP solvers is within factor 5 of the fastest GEP solver with Netlib LAPACK in my tests,
- computing the GSVD directly is up to 20 times slower than the computation by means of QR factorizations and the CS decomposition with Netlib LAPACK in my tests,
- given a pair of matrices with 2x2 block structure, I show how to minimize eigenvalue perturbation by off-diagonal blocks with the aid of graph algorithms, and
- I propose a new multilevel eigensolver for sparse GEPs that is able to compute up to 1000 eigenpairs on a cluster node with two dual-core CPUs and 16 GB virtual memory limit for problems with up to 150,000 degrees of freedom in less than eleven hours.
The revised edition of the thesis with fixed typos is here (PDF), the source code is available here, and the abstract is below. In February, I already gave a talk on the preliminary thesis results; more details can be found in the corresponding blog post.
This thesis treats the numerical solution of generalized eigenvalue problems (GEPs) , where , are Hermitian positive semidefinite (HPSD). We discuss problem and solution properties, accuracy assessment of solutions, aspect of computations in finite precision, the connection to the finite element method (FEM), dense solvers, and projection methods for these GEPs. All results are directly applicable to real-world problems.
We present properties and origins of GEPs with HPSD matrices and briefly mention the FEM as a source of such problems.
With respect to accuracy assessment of solutions, we address quickly computable and structure-preserving backward error bounds and their corresponding condition numbers for GEPs with HPSD matrices. There is an abundance of literature on backward error measures possessing one of these features; the backward error in this thesis provides both.
In Chapter 3, we elaborate on dense solvers for GEPs with HPSD matrices. The standard solver reduces the GEP to a standard eigenvalue problem; it is fast but requires positive definite mass matrices and is only conditionally backward stable. The QZ algorithm for general GEPs is backward stable but it is also much slower and does not preserve any problem properties. We present two new backward stable and structure preserving solvers, one using deflation of infinite eigenvalues, the other one using the generalized singular value decomposition (GSVD). We analyze backward stability and computational complexity. In comparison to the QZ algorithm, both solvers are competitive with the standard solver in our tests. Finally, we propose a new solver combining the speed of deflation with the ability of GSVD-based solvers to handle singular matrix pencils.
Finally, we consider black-box solvers based on projection methods to compute the eigenpairs with the smallest eigenvalues of large, sparse GEPs with Hermitian positive definite matrices (HPD). After reviewing common methods for spectral approximation, we briefly mention ways to improve numerical stability. We discuss the automated multilevel substructuring method (AMLS) before analyzing the impact of off-diagonal blocks in block matrices on eigenvalues. We use the results of this thesis and insights in recent papers to propose a new divide-and-conquer eigensolver and to suggest a change that makes AMLS more robust. We test the divide-and-conquer eigensolver on sparse structural engineering matrices with 10,000 to 150,000 degrees of freedom.
2010 Mathematics Subject Classification. 65F15, 65F50, 65Y04, 65Y20.
Edit: Revised master's thesis from April 2016 (PDF)
Until recently, my blog felt slow although it contained only static content. In this post, I describe how I decreased loading times and the size of this website by
- using WordPress as a static website generator,
- not loading unused scripts and fonts, and
- employing compression and client-side caches.
According to WebPagetest, a Firefox client in Frankurt with a DSL connection and an empty cache needed to download 666 kB (28 requests) and had to wait approximately 7.7 seconds before being able to view my frontpage from April 4. With the static website, the same client has to wait about 3.1 seconds and transfer 165 kB (18 requests). As a side-effect, the website offers considerably less attack surface now and user privacy was improved.
Continue reading A Fast, Static WordPress Blog
Finally, I managed to upload code to gitlab.com:
The repository contains:
Many commits were modified on April 12; I retrofitted all files of my WordPress theme with license headers in order to avoid licensing issues.
For my master's thesis I implemented multiple solvers for structured generalized eigenvalue problems using LAPACK. In this post, I will briefly discuss a method to simplify the memory management and ways to catch programming errors as early as possible when implementing a solver for linear algebra problems that uses LAPACK. The advice in this post only supplements good programming practices like using version control systems and automated tests.
Continue reading Advice: Implementing a Solver using LAPACK
Consider a language where we only allow automatic, implicit type conversions (coercion) among numeric types if every value of the source type can be represented as a value of the target type, i.e., there is no truncation and no round-off. Let us call this kind of coercion value-preserving coercion. With such a strict coercion rule, an unsigned integer with four bytes can be coerced into a signed integer with eight bytes but the compiler (interpreter) will not coerce signed integers to unsigned integers. In this post, we highlight a reason why coercions from single-precision to double-precision floating-point types may be undesirable although they are value preserving.
Continue reading Type Coercions and Floating-Point Types
NumPy and SciPy rely on BLAS and LAPACK for basic linear algebra functionality like matrix-vector multiplication, linear system solves, or routines for eigenvalue computation. The Intel Math Kernel Library (MKL) is a mathematics library providing amongst other things fast and multithreaded implementations of BLAS and LAPACK. In this blog post, I describe how to compile NumPy and SciPy with the Intel compilers using Intel MKL on Linux. Note NumPy and SciPy can be linked to MKL without the Intel compilers by providing the proper linker options to, e.g., GCC, and I will briefly explain this as well.
Continue reading Building NumPy and SciPy with Intel Compilers and Intel MKL
Yesterday I gave a talk on the preliminary results of my master's thesis on Projection Methods for Generalized Eigenvalue Problems. Below is the announcement for the talk; it mentions only complex (Hermitian) matrices but all results are immediately applicable to real matrices. The slides are here (PDF).
Projection Methods for Generalized Eigenvalue Problems
In this talk, we will discuss backward errors, dense solvers, and a new projection method for generalized eigenvalue problems (GEPs) with Hermitian positive semidefinite (HPSD) matrices.
We will present properties and origins of such GEPs. We will also address quickly computable and structure preserving backward error bounds for these kinds of GEPs. There is an abundance of literature on backward error measures possessing one of these features but only recently, the author came across a backward error providing both.
We will elaborate on dense solvers for GEPs with HPSD matrices. The standard solver for GEPs with Hermitian matrices is fast but requires positive definite mass matrices and is only conditionally backward stable; the QZ algorithm for general GEPs is backward stable but it is also magnitudes slower and does not preserve any problem properties. In the talk, we will present two new backward stable and structure preserving solvers, one using deflation, the other one using the generalized singular value decomposition (GSVD). In comparison to the QZ algorithm, both solvers are competitive with the standard solver in our tests.
Finally, we will touch on a new solver for large, sparse GEPs.
In Matlab, a directory called
foo can be created by calling
mkdir is successful, then the function returns logical 1; otherwise it returns logical 0. Now consider the case where Matlab is run on Linux and there is already a file called
foo in the current directory. Linux does not permit the existence of a file and a directory of the same name in the same folder so the call to
mkdir should fail. Instead,
mkdir will leave the file intact, return logical 1, and print the misleading warning "Directory already exists". That is, there is no directory called
foo after this supposedly successful function call.
The peculiarity of this behavior has been reported to Mathworks in September 2015. The example above was tested with Matlab R2015b and R2014a on Linux.
Consider the following setup:
using namespace std;
size_t n = 100;
size_t* array = new size_t[n];
The goal is to set every element of the array to the fixed value
In C++, an idiomatic way to solve this problem are the functions
std::fill (link to documentation) and
std::fill_n (link to documentation) from the STL header
algorithm. In addition to the fixed value
std::fill requires a pair of output iterators specifying a half-open range as its arguments whereas the additional argument to
std::fill_n is an output iterator pointing to the first element and the length of the container. That is, we could assign to the elements of the array above as follows:
using namespace std;
size_t n = 100;
size_t* array = new size_t[n];
fill(array, array+n, x);
fill_n(array, n, x);
What made me write this blog post is the following line of code:
fill_n(array, array+n, x);
Here, I accidentally used
std::fill_n in place of
std::fill but the code still compiles because of the automatic, implicit type conversion (type coercion) from
std::size_t. With warnings enabled (
-Wextra -Wall -std=c++11 -pedantic) neither g++ 4.8.5 nor clang++ 3.5 warn about this line and yet this piece of code causes a segmentation fault on my computer whenever it is executed.
In this post, I will explain why you should start to phase out DECT and how you can do this with VoIP (Voice over IP, also known as Internet telephony) over Wi-Fi. This guide allows you to attach several wireless devices to your landline and it permits calls among all telephones in your household. Moreover, the guide provides encrypted wireless communication, it uses off-the-shelf consumer products, and works worldwide. On the downside, the voice quality may be worse unless your router supports WMM (Wireless Multimedia Extensions) and you may need a WLAN repeater because DECT tends to have higher range than WLAN. (Surprisingly, the electronics markets in my vicinity offer dual-band WLAN repeaters with 600 MBit/s transfer rate fitting into a wall socket costing no more than the cheapest DECT repeater.) If so desired, you can continue to use your existing DECT phones but communication with these devices is not guaranteed to be encrypted.
You can follow this guide if you have
- an account with an Internet telephony provider,
- your Internet router is a Wi-Fi hotspot,
- the Internet router manages your landline,
- the Internet router allows wireless VoIP telephony using SIP, and
- a mobile device with Wi-Fi as well Android, iOS, or Windows.
Continue reading More Secure Wireless Telephony with VoIP over Wi-Fi