Numerička analiza

Vavilonska glinena pločica (c. 1800–1600 p.n.e.) sa anotacijama. Aproksimacija kvadratnog korena iz 2 sa četiri šezdesetne cifre, koje su oko šest decimalnih cifara. 1 + 24/60 + 51/60² + 10/60³ = 1.41421296...

Numerička analiza je grana numeričke matematike koja se bavi pronalaženjem i unapređivanjem algoritama za numeričko izračunavanje vrijednosti vezanih uz matematičku analizu, poput numeričkog integriranja, numeričkog deriviranja i numeričkog rješavanja diferencijalnih jednadžbi. Sastavni dio numeričke analize je i ocjenjivanje grešaka metoda (algoritama) i to na dvije razine -- analiza grešaka same metode, te analiza grešaka koje nastaju izvrednjavanjem, a vezane su uz arhitekturu računala .

Potrebe za numeričkim rješavanjem matematičkih problema su višestruke. Kod nekih problema, dokazano je da analitičko rješenje (rješenje zapisano pomoću elementarnih funkcija) ne postoji -- primjerice rješenje integrala $\int e^{x^{2}}\,dx$ nemoguće je zapisati pomoću elementarnih funkcija. Pa ipak, određeni integral $\int _{a}^{b}e^{x^{2}}\,dx$ predstavlja konkretnu, jedinstveno određenu površinu. Do te vrijednosti, koja ima široku upotrebu npr. u statistici, moguće je doći samo numeričkim metodama. Osim toga, numeričke metode često se koriste za određivanje rješenja matematičkih problema koji bi zbog svoje veličine, kroz standardni postupak rješavanja, predugo trajali -- primjerice, kada je potrebno riješiti sustav od 10 000 jednadžbi s 10 000 nepoznanica. I konačno, numeričke metode su nezaobilazne u aproksimativnom računu, kada se aproksimacijama (i ocjenama pripadnih grešaka) zamjenjuje stvarna vrijednost funkcije do koje je nemoguće ili preteško doći. To su metode poput metode konačnih elemenata ili pak kubičnih splineova kojima se aproksimira ponašanje nepoznate funkcije o kojoj znamo tek konačan broj vrijednosti, najčešće dobijenih mjerenjima.

Opšti uvod

Sveukupni cilj polja numeričke analize je razvoj i analiza tehnika koje daju aproksimativna ali precizna rešenja teških problema. Njihova raznovrstnost je sumirana sledećim pregledom:

• Napredni numerički metodi su esencijalni u omogućavanju numeričkih metereoloških predviđanja.
• Proračunavanje putanje svemirske letilice se vrši putem numeričkog rešavanja sistema običnih diferencijalnih jednačina.
• Automobilske kompanije mogu da poprave bezbednost vozila koristeće računarske simulacije automobilskih udesa. Takve simulacije se esencijalno sastoje od numeričkog rešavanja parcijalnih diferencijalnih jednačina.
• Hedž fondovi (privatni investicioni fondovi) koriste oruđa iz svih polja numeričke analize u proračunima vrednosti akcija i derivata.
• Aviokompanije koriste sofisticirane optimizacione algoritme u određivanju ceni karti, raspodele aviona i posade, i potrebe za gorivom. Istorijski, takvi algoritmi su razvijeni u okviru preklapajućeg polja operacionih istraživanja.
• Družtva za osiguranje koriste numeričke programe za aktuarne analize.

Ostatak ove sekcije razmatra nekoliko važnih tema numeričke analize.

Istorija

The field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than 2000 years ago. Many great mathematicians of the past were preoccupied by numerical analysis, as is obvious from the names of important algorithms like Newton's method, Lagrange interpolation polynomial, Gaussian elimination, or Euler's method.

To facilitate computations by hand, large books were produced with formulas and tables of data such as interpolation points and function coefficients. Using these tables, often calculated out to 16 decimal places or more for some functions, one could look up values to plug into the formulas given and achieve very good numerical estimates of some functions. The canonical work in the field is the NIST publication edited by Abramowitz and Stegun, a 1000-plus page book of a very large number of commonly used formulas and functions and their values at many points. The function values are no longer very useful when a computer is available, but the large listing of formulas can still be very handy.

The mechanical calculator was also developed as a tool for hand computation. These calculators evolved into electronic computers in the 1940s, and it was then found that these computers were also useful for administrative purposes. But the invention of the computer also influenced the field of numerical analysis, since now longer and more complicated calculations could be done.

Direktni i iterativni metodi

Direct vs iterative methods Consider the problem of solving 3x3 + 4 = 28 for the unknown quantity x. Direct method 3x3 + 4 = 28. Subtract 4 3x3 = 24. Divide by 3 x3 = 8. Take cube roots x = 2. For the iterative method, apply the bisection method to f(x) = 3x3 − 24. The initial values are a = 0, b = 3, f(a) = −24, f(b) = 57. Iterative method a b mid f(mid) 0 3 1.5 −13.875 1.5 3 2.25 10.17... 1.5 2.25 1.875 −4.22... 1.875 2.25 2.0625 2.32... We conclude from this table that the solution is between 1.875 and 2.0625. The algorithm might return any number in that range with an error less than 0.2. Diskretizacija i numerička integracija In a two hour race, we have measured the speed of the car at three instants and recorded them in the following table. Time 0:20 1:00 1:40 km/h 140 150 180 A discretization would be to say that the speed of the car was constant from 0:00 to 0:40, then from 0:40 to 1:20 and finally from 1:20 to 2:00. For instance, the total distance traveled in the first 40 minutes is approximately (2/3h × 140 km/h) = 93.3 km. This would allow us to estimate the total distance traveled as 93.3 km + 100 km + 120 km = 313.3 km, which is an example of numerical integration (see below) using a Riemann sum, because displacement is the integral of velocity. Ill-conditioned problem: Take the function f(x) = 1/(x − 1). Note that f(1.1) = 10 and f(1.001) = 1000: a change in x of less than 0.1 turns into a change in f(x) of nearly 1000. Evaluating f(x) near x = 1 is an ill-conditioned problem. Well-conditioned problem: By contrast, evaluating the same function f(x) = 1/(x − 1) near x = 10 is a well-conditioned problem. For instance, f(10) = 1/9 ≈ 0.111 and f(11) = 0.1: a modest change in x leads to a modest change in f(x).

Direct methods compute the solution to a problem in a finite number of steps. These methods would give the precise answer if they were performed in infinite precision arithmetic. Examples include Gaussian elimination, the QR factorization method for solving systems of linear equations, and the simplex method of linear programming. In practice, finite precision is used and the result is an approximation of the true solution (assuming stability).

In contrast to direct methods, iterative methods are not expected to terminate in a finite number of steps. Starting from an initial guess, iterative methods form successive approximations that converge to the exact solution only in the limit. A convergence test, often involving the residual, is specified in order to decide when a sufficiently accurate solution has (hopefully) been found. Even using infinite precision arithmetic these methods would not reach the solution within a finite number of steps (in general). Examples include Newton's method, the bisection method, and Jacobi iteration. In computational matrix algebra, iterative methods are generally needed for large problems.

Iterative methods are more common than direct methods in numerical analysis. Some methods are direct in principle but are usually used as though they were not, e.g. GMRES and the conjugate gradient method. For these methods the number of steps needed to obtain the exact solution is so large that an approximation is accepted in the same manner as for an iterative method.

Diskretizacija

Furthermore, continuous problems must sometimes be replaced by a discrete problem whose solution is known to approximate that of the continuous problem; this process is called discretization. For example, the solution of a differential equation is a function. This function must be represented by a finite amount of data, for instance by its value at a finite number of points at its domain, even though this domain is a continuum.

Generacija i propagacija grešaka

The study of errors forms an important part of numerical analysis. There are several ways in which error can be introduced in the solution of the problem.

Zaokruživanje

Round-off errors arise because it is impossible to represent all real numbers exactly on a machine with finite memory (which is what all practical digital computers are).

Skraćivanje i greške diskretizacije

Truncation errors are committed when an iterative method is terminated or a mathematical procedure is approximated, and the approximate solution differs from the exact solution. Similarly, discretization induces a discretization error because the solution of the discrete problem does not coincide with the solution of the continuous problem. For instance, in the iteration in the sidebar to compute the solution of $''3x^{3}+4=28''$, after 10 or so iterations, we conclude that the root is roughly 1.99 (for example). We therefore have a truncation error of 0.01.

Once an error is generated, it will generally propagate through the calculation. For instance, we have already noted that the operation + on a calculator (or a computer) is inexact. It follows that a calculation of the type a+b+c+d+e is even more inexact.

What does it mean when we say that the truncation error is created when we approximate a mathematical procedure? We know that to integrate a function exactly requires one to find the sum of infinite trapezoids. But numerically one can find the sum of only finite trapezoids, and hence the approximation of the mathematical procedure. Similarly, to differentiate a function, the differential element approaches to zero but numerically we can only choose a finite value of the differential element.

Numerička stabilnost i dobro postulirani problemi

Numerical stability is an important notion in numerical analysis. An algorithm is called numerically stable if an error, whatever its cause, does not grow to be much larger during the calculation. This happens if the problem is well-conditioned, meaning that the solution changes by only a small amount if the problem data are changed by a small amount. To the contrary, if a problem is ill-conditioned, then any small error in the data will grow to be a large error.

Both the original problem and the algorithm used to solve that problem can be well-conditioned and/or ill-conditioned, and any combination is possible.

So an algorithm that solves a well-conditioned problem may be either numerically stable or numerically unstable. An art of numerical analysis is to find a stable algorithm for solving a well-posed mathematical problem. For instance, computing the square root of 2 (which is roughly 1.41421) is a well-posed problem. Many algorithms solve this problem by starting with an initial approximation x₁ to ${\sqrt {2}}$, for instance x₁=1.4, and then computing improved guesses x₂, x₃, etc.. One such method is the famous Babylonian method, which is given by x_k+1 = x_k/2 + 1/x_k. Another iteration, which we will call Method X, is given by x_{k + 1} = (x_k²−2)² + x_k. We have calculated a few iterations of each scheme in table form below, with initial guesses x₁ = 1.4 and x₁ = 1.42.

Babylonian	Babylonian	Method X	Method X
x1 = 1.4	x1 = 1.42	x1 = 1.4	x1 = 1.42
x2 = 1.4142857...	x2 = 1.41422535...	x2 = 1.4016	x2 = 1.42026896
x3 = 1.414213564...	x3 = 1.41421356242...	x3 = 1.4028614...	x3 = 1.42056...
		...	...
		x1000000 = 1.41421...	x28 = 7280.2284...

Observe that the Babylonian method converges fast regardless of the initial guess, whereas Method X converges extremely slowly with initial guess 1.4 and diverges for initial guess 1.42. Hence, the Babylonian method is numerically stable, while Method X is numerically unstable.

Numerical stability is affected by the number of the significant digits the machine keeps on, if we use a machine that keeps on the first four floating-point digits, a good example on loss of significance is given by these two equivalent functions

$f(x)=x\left({\sqrt {x+1}}-{\sqrt {x}}\right){\text{ and }}g(x)={\frac {x}{{\sqrt {x+1}}+{\sqrt {x}}}}.$

If we compare the results of

$f(500)=500\left({\sqrt {501}}-{\sqrt {500}}\right)=500\left(22.3830-22.3607\right)=500(0.0223)=11.1500$

and

${\begin{alignedat}{3}g(500)&={\frac {500}{{\sqrt {501}}+{\sqrt {500}}}}\\&={\frac {500}{22.3830+22.3607}}\\&={\frac {500}{44.7437}}=11.1748\end{alignedat}}$

by looking to the two results above, we realize that loss of significance which is also called Subtractive Cancelation has a huge effect on the results, even though both functions are equivalent; to show that they are equivalent simply we need to start by f(x) and end with g(x), and so

${\begin{alignedat}{4}f(x)&=x\left({\sqrt {x+1}}-{\sqrt {x}}\right)\\&=x\left({\sqrt {x+1}}-{\sqrt {x}}\right){\frac {{\sqrt {x+1}}+{\sqrt {x}}}{{\sqrt {x+1}}+{\sqrt {x}}}}\\&=x{\frac {({\sqrt {x+1}})^{2}-({\sqrt {x}})^{2}}{{\sqrt {x+1}}+{\sqrt {x}}}}\\&=x{\frac {x+1-x}{{\sqrt {x+1}}+{\sqrt {x}}}}\\&=x{\frac {1}{{\sqrt {x+1}}+{\sqrt {x}}}}\\&={\frac {x}{{\sqrt {x+1}}+{\sqrt {x}}}}\end{alignedat}}$

The true value for the result is 11.174755..., which is exactly g(500) = 11.1748 after rounding the result to 4 decimal digits.

Now imagine that lots of terms like these functions are used in the program; the error will increase as one proceeds in the program, unless one uses the suitable formula of the two functions each time one evaluates either f(x), or g(x); the choice is dependent on the parity of x.

• The example is taken from Mathew; Numerical methods using matlab, 3rd ed.

Oblasti izučavanja

The field of numerical analysis includes many sub-disciplines. Some of the major ones are:

Izračunavanje vrednosti funkcija

Interpolation: We have observed the temperature to vary from 20 degrees Celsius at 1:00 to 14 degrees at 3:00. A linear interpolation of this data would conclude that it was 17 degrees at 2:00 and 18.5 degrees at 1:30pm. Extrapolation: If the gross domestic product of a country has been growing an average of 5% per year and was 100 billion dollars last year, we might extrapolate that it will be 105 billion dollars this year. A line through 20 points Regression: In linear regression, given n points, we compute a line that passes as close as possible to those n points. How much for a glass of lemonade? Optimization: Say you sell lemonade at a lemonade stand, and notice that at $1, you can sell 197 glasses of lemonade per day, and that for each increase of $0.01, you will sell one glass of lemonade less per day. If you could charge $1.485, you would maximize your profit, but due to the constraint of having to charge a whole cent amount, charging $1.48 or $1.49 per glass will both yield the maximum income of $220.52 per day. Wind direction in blue, true trajectory in black, Euler method in red. Differential equation: If you set up 100 fans to blow air from one end of the room to the other and then you drop a feather into the wind, what happens? The feather will follow the air currents, which may be very complex. One approximation is to measure the speed at which the air is blowing near the feather every second, and advance the simulated feather as if it were moving in a straight line at that same speed for one second, before measuring the wind speed again. This is called the Euler method for solving an ordinary differential equation.

One of the simplest problems is the evaluation of a function at a given point. The most straightforward approach, of just plugging in the number in the formula is sometimes not very efficient. For polynomials, a better approach is using the Horner scheme, since it reduces the necessary number of multiplications and additions. Generally, it is important to estimate and control round-off errors arising from the use of floating point arithmetic.

Interpolacija, ekstrapolacija, i regresija

Interpolation solves the following problem: given the value of some unknown function at a number of points, what value does that function have at some other point between the given points?

Extrapolation is very similar to interpolation, except that now we want to find the value of the unknown function at a point which is outside the given points.

Regression is also similar, but it takes into account that the data is imprecise. Given some points, and a measurement of the value of some function at these points (with an error), we want to determine the unknown function. The least squares-method is one popular way to achieve this.

Rešavanje jednačina i sistema jednačina

Another fundamental problem is computing the solution of some given equation. Two cases are commonly distinguished, depending on whether the equation is linear or not. For instance, the equation $2x+5=3$ is linear while $2x^{2}+5=3$ is not.

Much effort has been put in the development of methods for solving systems of linear equations. Standard direct methods, i.e., methods that use some matrix decomposition are Gaussian elimination, LU decomposition, Cholesky decomposition for symmetric (or hermitian) and positive-definite matrix, and QR decomposition for non-square matrices. Iterative methods such as the Jacobi method, Gauss–Seidel method, successive over-relaxation and conjugate gradient method are usually preferred for large systems. General iterative methods can be developed using a matrix splitting.

Root-finding algorithms are used to solve nonlinear equations (they are so named since a root of a function is an argument for which the function yields zero). If the function is differentiable and the derivative is known, then Newton's method is a popular choice. Linearization is another technique for solving nonlinear equations.

Rešavanje svojstvene vrednositi ili singularne vrednosti problema

Several important problems can be phrased in terms of eigenvalue decompositions or singular value decompositions. For instance, the spectral image compression algorithm is based on the singular value decomposition. The corresponding tool in statistics is called principal component analysis.

Optimizacija

Optimization problems ask for the point at which a given function is maximized (or minimized). Often, the point also has to satisfy some constraints.

The field of optimization is further split in several subfields, depending on the form of the objective function and the constraint. For instance, linear programming deals with the case that both the objective function and the constraints are linear. A famous method in linear programming is the simplex method.

The method of Lagrange multipliers can be used to reduce optimization problems with constraints to unconstrained optimization problems.

Diferencijalne jednačine

Numerička analiza se isto tako bavi izračunavanjem (na aproksimativan način) rešenja diferencijalnih jednačina, običnih diferencijalnih jednačina i parcijalnih diferencijalnih jednačina.

Partial differential equations are solved by first discretizing the equation, bringing it into a finite-dimensional subspace. This can be done by a finite element method, a finite difference method, or (particularly in engineering) a finite volume method. The theoretical justification of these methods often involves theorems from functional analysis. This reduces the problem to the solution of an algebraic equation.

Softver

Since the late twentieth century, most algorithms are implemented in a variety of programming languages. The Netlib repository contains various collections of software routines for numerical problems, mostly in Fortran and C. Commercial products implementing many different numerical algorithms include the IMSL and NAG libraries; a free alternative is the GNU Scientific Library.

There are several popular numerical computing applications such as MATLAB, TK Solver, S-PLUS, LabVIEW, and IDL as well as free and open source alternatives such as FreeMat, Scilab, GNU Octave (similar to Matlab), IT++ (a C++ library), R (similar to S-PLUS) and certain variants of Python. Performance varies widely: while vector and matrix operations are usually fast, scalar loops may vary in speed by more than an order of magnitude.

Many computer algebra systems such as Mathematica also benefit from the availability of arbitrary precision arithmetic which can provide more accurate results.

Also, any spreadsheet software can be used to solve simple problems relating to numerical analysis.

Numeričko integriranje

Površina ispod funkcije f(x) (označene plavom) aproksimira se površinom trapeza ispod po dijelovima linearne aproksimacije (označene crvenom).

Numerical integration, in some instances also known as numerical quadrature, asks for the value of a definite integral. Popular methods use one of the Newton–Cotes formulas (like the midpoint rule or Simpson's rule) or Gaussian quadrature. These methods rely on a "divide and conquer" strategy, whereby an integral on a relatively large set is broken down into integrals on smaller sets. In higher dimensions, where these methods become prohibitively expensive in terms of computational effort, one may use Monte Carlo or quasi-Monte Carlo methods (see Monte Carlo integration), or, in modestly large dimensions, the method of sparse grids.

Jedan od najčešćih problema s kojima se susrećemo u numeričkoj analizi je računanje vrijednosti određenog integrala $\int _{a}^{b}f(x)\,dx$.

Dvije osnovne metode numeričke integracije su proširena trapezna formula i proširena Simpsonova formula.

Kod proširene trapezne formule, interval integracije [a,b] podijeli se u n podintervala uz slijedeću oznaku: a=x₀<x₁<...<x_n=b. U svim se točkama razdiobe izračunaju vrijednosti podintegralne funkcije y_i=f(x_i), te se nad svakim podintegralom formira trapez spajanjem točaka T_i(x_i,y_i) i T_i+1(x_i+1,y_i+1). Tim se trapezom, čija je površina jednaka P_i=(x_i+1-x_i)(y_i+y_i+1)/2, aproksimira stvarna površina ispod funkcije f(x) na tom intervalu. Uz uobičajen postupak ekvidistantne razdiobe, tj razdiobe intervala na n jednakih podintervala (kod kojeg je x_i+1-x_i=(b-a)/n ), te zbrajanjem površina trapeza konstruiranih nad svim intervalima razdiobe dobijamo trapeznu formulu:

$\int _{a}^{b}f(x)\,dx\,\approx \,{\frac {b-a}{2n}}\cdot (y_{0}+2y_{1}+2y_{2}+\ldots +2y_{n-1}+y_{n})$

Ocjena greške ove numeričke aproksimacije dana je s:

$E(f)=\max _{\xi \in [a,b]}{\frac {(b-a)^{3}}{12n^{2}}}|f''(\xi )|.$

Površina ispod funkcije f(x) (označene plavom) aproksimira se površinom ispod parabole koja interpolira funkciju u tri zadane točke (označene crvenom).

Proširena Simpsonova formula, kao i trapezna formula počinje razdiobom intervala [a,b] na n, ne nužno, jednakih podintervala. No ovoga puta se na svaka dva podintervala, odnosno kroz točke T_i-1(x_i-1,y_i-1), T_i(x_i,y_i) i T_i+1(x_i+1,y_i+1) povlači jedinstveno određena kvadratna funkcija (parabola). Zbog toga kod provođenja Simpsonove formule imamo dodatni zahtjev da je broj podintervala n paran. Računanjem površina ispod tako kontruiranih parabola, te njihovim zbrajanjem dobijamo proširenu Simpsonovu formulu:

$\int _{a}^{b}f(x)\,dx\approx {\frac {b-a}{3n}}{\bigg [}y_{0}+4y_{1}+2y_{2}+4y_{3}y+2y_{4}+\cdots +4y_{n-1}+y_{n}{\bigg ]}.$

Ocjena greške proširene Simpsonove formule dana je izrazom:

$E(f)=\max _{\xi \in [a,b]}{\frac {(b-a)^{5}}{180n^{4}}}|f^{(4)}(\xi )|.$

Kako u pravilu parabola bolje aprokisimira nasumične funkcije od pravca, Simpsonova formula u pravilu daje točniji rezultat od trapezne formule.

Numeričko rješavanje diferencijalnih jednadžbi

U numeričku analizu spadaju i metode kojima se traži numeričko aproksimativno rješenje "Cauchyjevog problema"; diferencijalne jednadžbe s zadanim početnim uvjetom. Razvijene su metode za numeričko rješavanje običnih, ali i parcijalnih diferencijalnih jednadžbi. Dvije osnovne metode su Eulerova metoda, i familija Runge-Kutta metoda.

Vidite još

• Analiza algoritama
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• Spisak tema numeričke analize
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Literatura

• Golub, Gene H. and Charles F. Van Loan (1986). Matrix Computations, Third Edition (Johns Hopkins University Press, ISBN 0-8018-5413-X). 
• Higham, Nicholas J. (1996). Accuracy and Stability of Numerical Algorithms (Society for Industrial and Applied Mathematics, ISBN 0-89871-355-2). 
• Hildebrand, F. B. (1974). Introduction to Numerical Analysis (2nd edition izd.). McGraw-Hill. ISBN 0-07-028761-9. 
• Leader, Jeffery J. (2004). Numerical Analysis and Scientific Computation. Addison Wesley. ISBN 0-201-73499-0. 
• Wilkinson, J.H. (1965). The Algebraic Eigenvalue Problem (Clarendon Press). 
• Kahan, W. (1972). "A survey of error-analysis," in Info. Processing 71 (Proc. IFIP Congress 71 in Ljubljana), vol. 2, pp. 1214–39, North-Holland Publishing, Amsterdam.  (examples of the importance of accurate arithmetic).
• Lloyd N. Trefethen (2006). "Numerical analysis", 20 pages. In: Timothy Gowers and June Barrow-Green (editors), Princeton Companion of Mathematics, Princeton University Press.

Vanjske veze

• Numerische Mathematik, volumes 1-66, Springer, 1959-1994 (searchable; pages are images). (en) (de)
• Numerische Mathematik at SpringerLink, volumes 1-112, Springer, 1959–2009
• SIAM Journal on Numerical Analysis, volumes 1-47, SIAM, 1964–2009
• Numerical Recipes, William H. Press (free, downloadable previous editions)
• First Steps in Numerical Analysis (internet archivearchived), R.J.Hosking, S.Joe, D.C.Joyce, and J.C.Turner
• CSEP (Computational Science Education Project), U.S. Department of Energy
• Numerical Methods, Stuart Dalziel University of Cambridge
• Lectures on Numerical Analysis, Dennis Deturck and Herbert S. Wilf University of Pennsylvania
• Numerical methods, John D. Fenton University of Karlsruhe
• Numerical Methods for Science, Technology, Engineering and Mathematics, Autar Kaw University of South Florida
• Numerical Analysis Project, John H. Mathews California State University, Fullerton
• Numerical Methods - Online Course, Aaron Naiman Jerusalem College of Technology
• Numerical Methods for Physicists, Anthony O’Hare Oxford University
• Lectures in Numerical Analysis , R. Radok Mahidol University
• Introduction to Numerical Analysis for Engineering, Henrik Schmidt Massachusetts Institute of Technology
• Numerical Methods for time-dependent Partial Differential Equations, J.W. Haverkort, based on a course by P.A. Zegeling Utrecht_University
• Numerical Analysis for Engineering, D. W. Harder University of Waterloo