European Mathematical Society - 65-01
https://euro-math-soc.eu/msc-full/65-01
en Linear Algebra and Optimization with Applications to Machine Learning. Volume I: Linear Algebra for Computer Vision, Robotics, and Machine Learning
https://euro-math-soc.eu/review/linear-algebra-and-optimization-applications-machine-learning-volume-i-linear-algebra
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>The "with applications" in the title of the book should be read as "applicable" because it provides the fundamental mathematics, but it does not explicitly treat the applications that are mentioned. In this volume I, these fundamentals are basically linear algebra, and in volume II it is promised to cover optimization. I can imagine though that there is some interaction like for example the important subject of linear programming. Except for a few illustrative examples, the applications themselves are supposed to be covered in other courses or textbooks. There is a bit of statistics, which I think is also important for the applications mentioned, but probability and statistics as well as calculus is assumed known, but anyway, whatever is needed from these is recalled briefly.</p>
<p>Since this volume introduces the fundamentals with applications in mind, it is in some sense similar to a first course in linear algebra that I have been teaching to engineering students for many years. This volume has also some numerical procedures and even matlab codes. Those I taught in a separate numerical course. The procedures discussed include Gaussian elimination, Cholesky factorization, QR decomposition, eigenvalue and SVD algorithms, Krylov and Lanczos methods, but it skips the basic numerics of rounding errors, error propagation, numerical stability, and the more analytic problems such as numerical quadrature, differential equations, zero finding, etc. The latter also have a definite link with linear algebra, but clearly including all applications of linear algebra is an interminable task.</p>
<p>I wrote my own lecture notes, not satisfied with the existing books that were not providing the desired abstraction and that spent too many glossy pages on the introductory level with many examples and applications. In many ways my notes were very similar to the material covered here, which is why I like this book so much. But since I was trying to cover as much as possible in the most efficient way restricted by the amount of credits that were assigned to the course, my notes were much more concise. This is quite different in this book since, at its introductory level, it is almost encyclopedic and it is like the text I would have liked to write if I were not restricted by a time limit to cover all the material. For example, I spent some time explaining that finite dimensional real vector spaces and linear maps can be treated in an isomorphic way by discussing $\mathbb{R}^n$ and matrix algebra, and then I could just do matrix algebra. Not so in this book. The abstract vector spaces remain present throughout the book. Infinite dimensional vector spaces are a problem because there you need infinite sums and convergence, which require topology to define convergence, and the maps are operators. The authors here maintain some elements of function spaces but certain analysis aspects are not really covered in detail. But otherwise almost all the proofs are fully written out.</p>
<p>Thanks to LaTeX it is nowadays no problem to produce a professionally looking text. The illustrations however require different tools and producing good quality graphics is a challenge. Clearly the authors of this book had the same problem with graphics that I also had. The text is excellent, but the graphics are definitely of lesser quality. Unfortunately it is not only a problem of how they are generated and reproduced, also they do not always make very clear what they are supposed to illustrate.</p>
<p>What would one expect in a basic linear algebra course for engineering-type students? I think that should include vector spaces and linear maps and how they relate to matrices, the rank of a matrix with range and null space, determinants (in my opinion as little as possible), linear systems with Gaussian elimination, normed and Euclidean spaces, orthogonalization and QR, eigenvalue and singular values with generalized inverses and least squares, and geometric interpretation of all these concepts. All this is extensively discussed in this book. The numerical and matlab algorithms and certainly the iterative methods, I would rather expect in a more specialized numerical course, but it is not completely unexpected that they are found here in this book. More unexpected are the following: A discussion about the Haar wavelet with some signal and image processing; the chapter on linear systems is introduced with a discussion about the computation of interpolating Bézier curves; and the computation of the matrix exponential, important for the solution of differential equations, is discussed to some extent. There is also an extensive discussion of groups (SU(2), SO(3), and quaternions. This is important for robotics. Finally, there is much material related to graphs: graph Laplacians, clusters, and graph drawing. The final chapter about polynomial factorization and the Jordan form is less elementary and not always found with the same detail in a basic course. So there is a lot of material, and I can imagine that one wants to make a selection. No advise is given by the authors and it would be difficult anyway since successive chapters rely on previous ones. The authors have earmarked only some sections that they considered to be more advanced and these can be skipped initially.</p>
<p>To conclude, I can definitely recommend this extensive book on linear algebra that is both self contained and thorough and mostly remaining at a basic level. I have always considered linear algebra a basic tool in many applications. The applications mentioned in the title are computer vision, robotics and machine learning but they are not really discussed. However the Haar wavelet is a hint to image and signal processing, robotics are related to the study of the quaternions and the rotation groups, the linear algebra and the graphs are useful for a lot of applications, thus also for machine learning. To come to the actual applications though will need extra material, continuing the basics given here, but also some extra material for example statistics, and optimization (the latter is promised in volume II). Every chapter has a list of exercises (no answers are provided). There is a bibliography which lists mostly books and an index (12 pages). With a book of this size, the index can never be extensive enough. I tried to look up some topics that were not listed. On the other hand, I noted separate entries for "Jordan block" and "Jordan blocks", which makes no sense of course. I know from experience that collecting an index in an artisanal way is, just like generating good graphics, a time consuming task that needs patience and many iterations.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This is a extensive and thorough introduction to linear algebra that includes some extras like wavelets, Bézier curves, groups of rotations, quaternions, and applications in graph theory, that are of particular interest for applications in computer vision, robotics, and machine learning.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/jean-gallier" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jean Gallier</a></li><li class="vocabulary-links field-item odd"><a href="/author/jocelyn-quaintance" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jocelyn Quaintance</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/world-scientific" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">world scientific</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2020</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9789811206399 (hbk), 9789811207716 (pbk), 9789811206412 (ebk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">GBP 175.00 (hbk), GBP 85.00 (pbk), GBP 70.00 (ebk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">824</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/algebra" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Algebra</a></li><li class="vocabulary-links field-item odd"><a href="/imu/numerical-analysis-and-scientific-computing" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Numerical Analysis and Scientific Computing</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://www.worldscientific.com/worldscibooks/10.1142/11446" title="Link to web page">https://www.worldscientific.com/worldscibooks/10.1142/11446</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/15-linear-and-multilinear-algebra-matrix-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">15 Linear and multilinear algebra, matrix theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/15axx" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">15Axx</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/65-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/65d19" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65D19</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/65t05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65T05</a></li></ul></span>Fri, 10 Apr 2020 06:07:24 +0000Adhemar Bultheel50668 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/linear-algebra-and-optimization-applications-machine-learning-volume-i-linear-algebra#commentsNumerical methods : design, analysis, and computer implementation of algorithms
https://euro-math-soc.eu/review/numerical-methods-design-analysis-and-computer-implementation-algorithms
<div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Juan Antonio Infante</div></div></div><div class="field field-name-field-review-appendix field-type-file field-label-hidden"><div class="field-items"><div class="field-item even"><span class="file"><img class="file-icon" alt="PDF icon" title="application/pdf" src="/modules/file/icons/application-pdf.png" /> <a href="https://euro-math-soc.eu/sites/default/files/book-review/GreenbaumChartier_0.pdf" type="application/pdf; length=43207">GreenbaumChartier.pdf</a></span></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/anne-greenbaum" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Anne Greenbaum</a></li><li class="vocabulary-links field-item odd"><a href="/author/timothy-p-chartier" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Timothy P. Chartier</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2012</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-691-15122-9</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">454</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://press.princeton.edu/titles/9763.html" title="Link to web page">http://press.princeton.edu/titles/9763.html</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/65-numerical-analysis" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65 Numerical analysis</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/65-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65-01</a></li></ul></span>Tue, 22 Nov 2016 12:07:56 +0000Juan Antonio Infante47288 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/numerical-methods-design-analysis-and-computer-implementation-algorithms#commentsNumerical Methods in Engineering with Python 3
https://euro-math-soc.eu/review/numerical-methods-engineering-python-3
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Python is a general purpose programming language supporting object-oriented and structured programming. It is free, simple to use and implement, and well structured, and equally useful for non-numerical as for numerical applications. For these reasons it is sometimes chosen as a language for a first programming course. It is clear that in such case, a first course on numerical methods prefers using Python as a tool for implementing and testing the algorithms over an alternative such as MATLAB, or <em>scilab</em>, the open source MATLAB clone.</p>
<p>
The present book is written in line with the previous observations. It is assumed that the student/reader is familiar with some programming language that is preferably, but not necessarily, Python. For those who do not know Python yet, a short introduction is given, and it as assumed that their programming skills suffice to learn the details along the way. Thus the book should not be considered as a course on Python. Only part of the language is used. For example, the only object that is used is an array. On the other hand, modules with many predefined functionality beyond the Python core such as <em>numpy</em> and <em>matplotlib.pyplot</em> are essential. Source code for the algorithms in the book are available for download at the CUP website.</p>
<p>
After the introductory chapter on Python, the book follows the classical structure and items of a first numerical analysis course: Solution of linear systems, interpolation and curve fitting, roots of equations, numerical differentiation and integration, (ordinary) differential equations (initial value and boundary value problems), (symmetric) eigenvalue problems and (elementary) optimization. It is a true hands-on course where formal proofs are almost always omitted or just replaced by some motivating examples. There is an ample number of exercises, mostly numerical (i.e., not asking for a proof or a generalization or the derivation of a formula) and they are often taken from engineering applications.</p>
<p>
Some peculiarities that struck me are the following.</p>
<ul>
<li>
There is no list of references. With proofs and more formal aspects missing, some students may be interested in further reading or a more advanced approach. They are left on their own, although these references hould be readily available in any library or even on the Internet.</li>
<li>
There is no treatment of the mechanism of rounding errors in floating point computations and neither is error analysis or the numerical stability of a method formally included (except for stability and stiffness of ODE solvers). Remarks on rounding errors, and hence on stability, are downgraded to remarks sometimes with computing in double precision as a possible cure.</li>
<li>
Polynomial interpolation is included in extension, even rational and spline interpolation are discussed, but not the choice of the interpolation points like adaptive interpolation or the use of Chebyshev points for a finite interval. A simple idea that may influence the error and the convergence considerably.</li>
<li>
Speed of convergence for iterative methods is mentioned but not formally discussed.</li>
<li>
Multivariate optimization is very concise as compared to the extensive discussion of less important issues. Discussion includes only a method of M. Powell and the simplex method (not to be confused with the simplex method of linear programming which is not included).</li>
</ul>
<p>
So we may conclude that this is a practical introduction, pushing the theory as far in the background as possible. There is a lot of material, probably far too much for a single course. Some sections that could be deleted for a shorter course are marked with an asterisk. For those who are familiar with the second edition, the changes (apart from the necessary adaptation of the code to Python 3.0) include: an introduction to <em>matplotlib.pyplot</em> in chapter 1; an interpolating polynomial plot routine is added to the interpolation chapter; in the chapter on differential equations, the Taylor series method is dropped in favor of Euler's method; an improved implementation is given for the Jacobi method for eigenvalues as well as for the Runge-Kutta method for differential equations; and some new problems are added.</p>
<ul>
</ul>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">A. Bultheel</div></div></div><div class="field field-name-field-review-legacy-affiliation field-type-text field-label-inline clearfix"><div class="field-label">Affiliation: </div><div class="field-items"><div class="field-item even">KU Leuven</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a revised edition of previous version (2010) with adaptations for the new release of Python 3.0 (2008). It treats the classical subjects of an introductory course on numerical methods for engineering students with chunks of Python code implementing the algorithms. Python is just a working tool subordinate to the numerical techniques. So this should not be mistaken as a programming or a Python course.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/jaan-kiusalaas" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">jaan kiusalaas</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/cambridge-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">cambridge university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2013</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-1070-3385-6 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 110.00, GBP 65.00 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">432</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9781107033856" title="Link to web page">http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9781107033856</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/65-numerical-analysis" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65 Numerical analysis</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/65-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65-01</a></li></ul></span>Tue, 29 Oct 2013 16:21:54 +0000Adhemar Bultheel45526 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/numerical-methods-engineering-python-3#commentsMathematics of Approximation
https://euro-math-soc.eu/review/mathematics-approximation
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The idea of the new series <em>Mathematical Textbooks for Science and Engineering</em> (MSTE) is to publish self contained textbooks on applied or applicable mathematics at undergraduate and graduate level of mathematics science and engineering. It should be easy reading, even for self study. This book is volume 1 and it is devoted to the standard basics of approximation in the broad sense.</p>
<p>
The subjects are not a surprise and line up as one would expect: Polynomial interpolation, best approximation (in $L_2$ and $L_\infty$-sense by polynomials), elementary orthogonal polynomials (Chebyshev, Legendre), numerical quadrature, Fourier series, and spline approximation. A more detailed table of contents can be found on the publisher's website. Note that it involves only approximation by polynomials, trigonometric polynomials or piecewise polynomials (no rationals or other function systems) and only real approximation (complex analysis is avoided completely).</p>
<p>
All the material is standard and fully proved in the book, and no historical notes are given about the origin of these results. Hence no references to research literature are needed and thus there is no list of references. As with most textbooks, it is an endeavour resulting from many years of experience while teaching the course to many students and it is not different in this case. Such a ripening process is needed to make all the things explicit that a teacher considers obvious but may actually be a source of wild and unexpected speculations by students. So it is a remarkable achievement to introduce the right amount of rigour to avoid any misunderstanding from the reader's side and yet not to overload the text and keep it very readable also for self studying. In that respect de Villiers has made some specific choices that allowed to deliver a thoroughly thought over product that serves these characteristics very well.</p>
<p>
Some of these choices have their consequences. Although the series aims at an audience from `science and engineering', this book is in my opinion more oriented towards science than towards engineering. It is practical in the sense that a lot of attention goes to error estimates and that it handles approximation techniques that are important and actually used in practical applications, but it is not so practical that proper attention is paid to the pure numerical and algorithmic aspects. Many of the subjects discussed will also appear in a textbook on numerical analysis, but there much more attention will be given to rounding errors and efficient implementation. This particular choice also shows by a total absence of graphs. A simple graph of an orthogonal polynomial, the location of the Chebyshev points, a spline, or whatever could relax the reader a bit from the rigorous and sometimes complicated formulas to explain a relatively simple idea. Another exponent of this choice of approach is a lack of numerical examples. There are some examples included, but mostly quite simple and of an academic nature. So there are also no graphs or tables that illustrate the error or the convergence behaviour of some of these methods. Of course a particular phenomenon might require a numerical explanation that has been deliberately avoided in this book. Another example is the incidental mentioning of FFT in the chapter on Fourier series: <em>`[...] indeed (9.3.73)-(9.3.76) form the basis of the widely used Fast Fourier Transform (FFT), a detailed presentation of which is beyond the scope of this book.'</em>. In many computational issues, the linear algebra aspect takes an important role when it comes down to implementation (e.g. the computation of Gaussian quadrature formulas by solving an eigenvalue problem). This is another aspect that has not been included in this book.</p>
<p>
But even, if the contents has a strong theoretical component, the scrupulous attention paid to error estimates, estimates of Lebesgue constants, and the different ways in which a solution can be represented and computed have a definite `applicable' component as well. So, I am somewhat surprised that, besides Lagrange and Newton forms of interpolation polynomials, there is no mentioning of the barycentric representation which allows some nice theory and has definitely great practical importance. Another nice piece of theory could have been devoted to wavelets, which are absent as well (except for a brief note in an exercise about splines). Let me make it clear that I do not consider my enumeration of what is not in this book as a critique. After all, any course that is actually taught has to transfer knowledge in a finite number of teaching hours, and there is enough material left to teach a major course on approximation. My only intention is to inform potential buyers of what is and what is not to be expected.</p>
<p>
A completely different approach is for example taken in the matlab based package <a href="http://www2.maths.ox.ac.uk/chebfun/">Chebfun</a> developed by Nick Trefethen and his team. This is less broad, less theoretical, and obviously much more hands-on. Learn by experience, not by formulas. A book on Chebfun is announced as a SIAM publication for early 2013. That book will be in many aspects different from the present book under review: probably not the opposite but certainly a complement.</p>
<p>
</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">A. Bultheel</div></div></div><div class="field field-name-field-review-legacy-affiliation field-type-text field-label-inline clearfix"><div class="field-label">Affiliation: </div><div class="field-items"><div class="field-item even">KU Leuven</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is the first volume of a new series <em>Mathematical Textbooks for Science and Engineering</em>. It builds up standard approximation theory from scratch (requiring only some advanced calculus and linear algebra) up to a reasonably advanced level. No complex analysis or advanced functional analysis is needed. As a textbook it includes all the proofs and has many exercises following each chapter.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/johan-de-villiers" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">johan de villiers</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/atlantis-press-springer-verlag" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Atlantis Press / Springer Verlag</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2012</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-94-91216-49-7 (P), 978-94-91216-50-3 (E)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">59.95 € (net)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">427</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/analysis-and-its-applications" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Analysis and its Applications</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.springer.com/mathematics/book/978-94-91216-49-7" title="Link to web page">http://www.springer.com/mathematics/book/978-94-91216-49-7</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/65-numerical-analysis" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65 Numerical analysis</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/65-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65-01</a></li></ul></span>Tue, 28 Aug 2012 07:17:30 +0000Adhemar Bultheel45458 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/mathematics-approximation#comments