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Intlab_V5.5 (0, 2008-10-06)
Intlab_V5.5\long (0, 2008-06-05)
Intlab_V5.5\long\@long (0, 2008-06-05)
Intlab_V5.5\long\@long\eq.m (321, 2005-04-06)
Intlab_V5.5\long\@long\ge.m (338, 2005-04-06)
Intlab_V5.5\long\@long\gt.m (337, 2005-04-06)
Intlab_V5.5\long\@long\le.m (338, 2005-04-06)
Intlab_V5.5\long\@long\lt.m (338, 2005-04-06)
Intlab_V5.5\long\@long\ne.m (321, 2005-04-06)
Intlab_V5.5\long\@long\isempty_.m (784, 2005-10-12)
Intlab_V5.5\long\@long\display.m (5288, 2005-04-06)
Intlab_V5.5\long\@long\mtimes.m (3730, 2006-04-02)
Intlab_V5.5\long\@long\abs.m (448, 2005-07-25)
Intlab_V5.5\long\@long\end.m (312, 2005-04-06)
Intlab_V5.5\long\@long\exp.m (1686, 2005-04-06)
Intlab_V5.5\long\@long\inf.m (1090, 2005-07-25)
Intlab_V5.5\long\@long\max.m (1732, 2005-04-06)
Intlab_V5.5\long\@long\mid.m (430, 2005-04-06)
Intlab_V5.5\long\@long\min.m (1728, 2005-04-06)
Intlab_V5.5\long\@long\rad.m (697, 2005-04-06)
Intlab_V5.5\long\@long\sup.m (1305, 2005-07-25)
Intlab_V5.5\long\@long\ldivide.m (176, 2005-11-02)
Intlab_V5.5\long\@long\mrdivide.m (4614, 2005-11-23)
Intlab_V5.5\long\@long\longshift.m (782, 2005-04-06)
Intlab_V5.5\long\@long\length.m (252, 2005-04-06)
Intlab_V5.5\long\@long\minus.m (252, 2005-04-06)
Intlab_V5.5\long\@long\times.m (304, 2005-04-06)
Intlab_V5.5\long\@long\diam.m (486, 2005-04-06)
Intlab_V5.5\long\@long\addlongerror.m (1512, 2005-11-23)
Intlab_V5.5\long\@long\subsref.m (1884, 2005-04-06)
Intlab_V5.5\long\@long\long.m (4183, 2005-04-06)
Intlab_V5.5\long\@long\plus.m (5743, 2005-04-06)
Intlab_V5.5\long\@long\subsasgn.m (1602, 2005-04-06)
Intlab_V5.5\long\@long\mpower.m (1041, 2005-04-06)
Intlab_V5.5\long\@long\long2intval.m (2466, 2005-11-23)
Intlab_V5.5\long\@long\size.m (505, 2005-04-06)
Intlab_V5.5\long\@long\uplus.m (227, 2005-04-06)
Intlab_V5.5\long\@long\numel.m (149, 2002-12-05)
Intlab_V5.5\long\@long\rdivide.m (300, 2005-11-02)
Intlab_V5.5\long\@long\long2dble.m (1217, 2005-04-06)
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A short overview how to use INTLAB (see also file FAQ): ======================================================= INTLAB was started with the introduction of an operation concept in Matlab. It is continuously developed since the first version in 19***. The only developer is Siegfried M. Rump, head of the Institue for Reliable Computing at the Hamburg University of Technology. As a reference to INTLAB please use S.M. Rump. INTLAB - INTerval LABoratory. In Tibor Csendes, editor, Developments in Reliable Computing, pages 77?104. Kluwer Academic Publishers, Dordrecht, 1999. @incollection{Ru99a, author = {Rump, {S.M.}}, title = {{INTLAB - INTerval LABoratory}}, editor = {Tibor Csendes}, booktitle = {{Developments~in~Reliable Computing}}, publisher = {Kluwer Academic Publishers}, address = {Dordrecht}, pages = {77--104}, year = {1999}, url = {http://www.ti3.tu-harburg.de/rump/} } ===> If you used INTLAB before with an older Matlab version, you had to set the system variable BLAS_VERSION: ===> Please delete this system variable. It was necessary to avoid IMKL, now with new rounding routines the Intel Math Kernel Library (IMKL), which is the default and very fast, can safely be used (thanks to Dr. Takeshi Ogita from Tokyo). The easiest way to start is create a file startup.m (or to replace the content of the existing one) in the Matlab path \toolbox\local by cd ... insert the INTLAB path ... startintlab Then the startintlab.m file is started and should do the rest. For some user-specific options you may adjust the file startintlab.m . INTLAB is successfully tested under Matlab versions 5.3, 6.0.0, 6.5.0, 6.5.1, 7.0, 7.0.1 (SP1), 7.0.4 (SP2), 7.1 (SP3), 7.2 (R2006a), 7.3 (R2006b), 7.4 (R2007a), R2007b, R2008a. The Matlab version 7.0.4 (R14) Service Pack 2, however, is not very stable, sometimes a segmentation violation may occur. There are some problems in Matlab version 7.2 (R2006a). For example, const*sparse for larger dimension can be slow: >> n=1e4, a=sparse([],[],[],n,n), tic,b=2*a,toc, tic,c=a/2,toc n = 10000 a = All zero sparse: 10000-by-10000 b = All zero sparse: 10000-by-10000 Elapsed time is 3.101008 seconds. c = All zero sparse: 10000-by-10000 Elapsed time is 0.000131 seconds. >> or, still in Matlab version 7.2 (R2006a) under Windows XP, >> m=1e6, n=1e4, I=100, J=300, A=sparse(I,J,NaN,m,n), sqrt(-1)*A the last multiplication produces an infinite loop. Matlab Version R2006b produces sometimes a core dump with segment violation. For Matlab Version 5.3 and higher on PCs and Windows, INTLAB is entirely written in Matlab. There is no system dependency. This is because the Mathworks company was so kind to add into the routine "system_dependent" a possibility to change the rounding mode (my dear thanks to Cleve). For other platforms or Matlab Version 5.3-, INTLAB is written entirely in Matlab language except one routine setround for switching rounding mode. A corresponding C-routine together with a dll-file is in directory \setround. This routine is the only portability constraint (see also function "setround"). The file setround.dll is put in a separate path "setround". If rounding does not work properly, this path is removed. So INTLAB should detect automatically whether a rounding routine is necessary. If not, for some reason, ===> and you are using Matlab Version 5.3+ on a PC under Windows, **delete** ===> the file "setround.dll" . ===> Prerequisite for INTLAB is the possibility to switch the processor ===> permanently into a specific rounding mode "downwards", "upwards" and ===> "to nearest". This is always checked when starting INTLAB under Matlab. ===> If the check fails, INTLAB will be terminated not to produce incorrect results! The progress in the different INTLAB versions can be viewed using help, for example help Version5_3 Note that '_' is used rather than a dot, otherwise the help function does not work. INTLAB supports - interval scalars, vectors and matrices, real and complex, - full and sparse matrices, - interval standard functions, real and complex, - and a number of toolboxes for intervals, gradients, hessians, slopes, polynomials, multi-precision arithmetic and more. There are some demo-routines to get acquainted with INTLAB such as demointlab demointval demogradient demohessian demoslope demopolynom demolong Call "demos" and access the INTLAB-demos. INTLAB results are verified to be correct including I/O and standard functions. Interval input is rigorous when using string constants, see help intval . Interval output is always rigorous. For details, try e.g. "help intval\display" and "demointval". You may switch your display permanently to infimum/supremum notation, or midpoint/radius, or display using "_" for uncertainties; see "help intvalinit" for more information. For example format long, x = midrad(pi,1e-14); infsup(x) midrad(x) disp_(x) produces intval x = [ 3.14159265358978, 3.14159265358***1] intval x = < 3.14159265358979, 0.00000000000002> intval x = 3.14159265358***_ Display with uncertainties represents the interval produces by subtracting and adding 1 from and to the last displayed digit of the mantissa. Note that the output is written in a way that "what you see is correct". For example, midrad(4.99993,0.0004) produces intval ans = < 4.9999, 0.0005> in "format short" and mid-rad representation. Due to non-representable real numbers this is about the best what can be done with four decimal places after the decimal point. A possible trap is for example >> Z=[1,2]+i*[-1,1] Z = 1.0000 - 1.0000i 2.0000 + 1.0000i The brackets in the definition of Z might lead to the conclusion that Z is a complex interval (rectangle) with lower left endpoint 1-i and upper right endpoint 2+i. This is not the case. The above statement is a standard Matlab statement defining a (row) vector of length 2. It cannot be an interval: Otherwise Z would be preceded in the output by "intval". Standard functions are rigorous. This includes trigonometric functions with large argument. For example, x=2^500; sin(x), sin(intval(x)) produces ans = 3.273390607896142e+150 intval ans = 0.42925739234243 the latter being correct to the last digit. For real interval input causing an exception for a real standard function, one may switch between changing to complex standard functions with or without warning or, to stay with real standard functions causing NaN result. For example, intvalinit('DisplayMidRad') intvalinit('RealStdFctsExcptnAuto'), sqrt(infsup(-3,-2)) produces ===> Complex interval stdfct used automatically for real interval input out of range (without warning) intval ans = < 0.0000 + 1.5811i, 0.1670> whereas intvalinit('RealStdFctsExcptnWarn'), sqrt(infsup(-3,-2)) produces ===> Complex interval stdfct used automatically for real interval input out of range, but with warning Warning: SQRT: Real interval input out of range changed to be complex > In c:\matlab_v5.1\toolbox\intlab\intval\@intval\sqrt.m at line 81 intval ans = < 0.0000 + 1.5811i, 0.1670> and intvalinit('RealStdFctsExcptnNaN'), sqrt(infsup(-3,-2)) gives ===> Result NaN for real interval input out of range intval ans = < NaN, NaN> All functions support vector and matrix input to minimize interpretation overhead. Pure floating point standard functions (not rigorous) are still faster; therefore those are still in INTLAB (for details, see help intvalinit). Sometimes it is useful to ignore input data out of range. This is possible using intvalinit('RealStdFctsExcptnIgnore'), sqrt(infsup(-3,4)) which gives ===> !!! Caution: Input arguments out of range are ignored !!! ===> !!! Use intvalinit('RealStdFctsExcptnOccurred') to check whether this happened !!! ===> !!! Using Brouwer's Fixed Point Theorem may yield erroneous results !!! intval ans = [ 0.0000, 2.0000] Using Brouwer's Fixed Point Theorem by checking f(X) in X is only possible if the interval vector X is completely in the range of definition of f. Consider f = inline('sqrt(x)-2'); X = infsup(-3,4); Y = f(X) which produces intval Y = [ -2.0000, 0.0000] Obviousy, Y is contained in X, but f has no real fixed point at all. You may check whether an input out of range occurred by NotDefined = intvalinit('RealStdFctsExcptnOccurred') which gives NotDefined = 1 Checking the flag 'RealStdFctsExcptnOccurred' resets it to zero. Certain data necessary for rigorous input/output and for rigorous standard functions are stored in files power10tobinaryVVV.mat and stdfctsdataVVV.mat, where VVV stands for Matlab version number. Those files must be in the search path of Matlab. They are generated automatically when starting the system the first time. Generation of the first is fast, the second file needs some 10 minutes on a 750 Mhz PC. The documentation is included in every routine. INTLAB-code, i.e. Matlab-code, is (hopefully) self-explaining. INTLAB stays to the philosophy to implement everything in Matlab. To start under Windows: - create new directory c:\matlab\toolbox\intlab and copy INTLAB and subdirectories into it; - adapt your STARTINTLAB file according to the included sample; - add a call of startintlab to your startup file. For other operating systems similar directions apply. Especially, directory names should be adapted. For a quick overview of functions, use help intval or help gradient or help slope or help polynom or help long or help utility . For some examples, try the demos. For specific help try for example help verifylss or lookfor reshape or lookfor gradientinit or lookfor 'linear systems' and alike. Certain flags like display-mode etc. are stored in global variables. Therefore ===> try to avoid a statement like "clear global" or "clear all" <=== because it clears all global variables and results in subsequent errors in INTLAB. The statements "clear" or "clear variables" do not do any harm. Try to clear only specific global variables using, for example, statements like "clear global v*". Note that all global variables used in INTLAB start with "INTLAB_". Try "who global" to list them. If a "clear global" statement is necessary for some reason, let it be followed by startintlab to restore global variables. In fact, this restores the (your) default options. Intlab uses infimum-supremum representation for real intervals, and midpoint-radius representation for complex intervals. However, this is not visible to the user. For multiplication of two real interval matrices, both of them being thick, there is a choice between - fast midpoint-radius implementation with a little overestimation issued by the Matlab command intvalinit('FastIVmult'), - slower infimum-supremum representation issued by the Matlab command intvalinit('SharpIVmult'), Needless to say that both results are verified to be correct. There is only a difference between "fast" and "sharp" - if both operands being contain intervals of nonzero diameter, and - if the inner dimension is greater than one. Otherwise, all results will be computed sharp and fast. For example, interval scalar products are always computed in sharp mode (not with long accumulator, but without overestimation due to midpoint-radius representation). Empty interval components are represented by NaN. This is because Matlab does not accept empty components []. For this, there is some ambigouity concering the function "isempty". It could mean "mathematically empty" or "empty in the Matlab sense". We define "isempty" for all INTLAB data types in the same sense as Matlab emptyness []. This is to ensure 'upward' compatability, the result of isempty(x) and isempty(intval(x)), for example, should be the same for x of type double. The result is one number, 0 or 1. Emptyness in the mathematical sense is represented by NaN or [], and is tested by "isempty_". As for similar Matlab functions like isnan or isinf, the result of "isempty" is an array of 0/1 characterizing empty components. Note that the definition of "isempty" and "isempty_" was interchanged from INTLAB version 5.1 to 5.2. For timing use for example n=500; A=midrad(rand(n),1e-12); Amid=mid(A); k=10; tic; for i=1:k, Amid*Amid; end, toc/k tic; for i=1:k, intval(Amid)*Amid; end, toc/k tic; for i=1:k, A*Amid; end, toc/k intvalinit('FastIVmult'), tic; for i=1:k, A*A; end, toc/k intvalinit('SharpIVmult'), tic; A*A; toc for the multiplication of two 500x500 interval matrices. Results on a 750 Mhz Pentium III are 0.7 sec for floating point multiplication, 1.3 sec for multiplication of point matrices (both sharp and fast), 2.4 sec for multiplication of an thick and a thin matrix (both sharp and fast), 3.3 sec for fast multiplication of interval matrices, and 379 sec for sharp multiplication of interval matrices. Default is fast multiplication. This may be changed in the startintlab file. To start, try for example the following commands: n=10; A=2*rand(n)-1; b=A*ones(n,1); X=verifylss(A,b) This solves a randomly generated 10x10 linear system with result verification, matrix entries uniformly distributed in [-1,1]. The result is a real interval vector. You may display the result in mid-rad representation by midrad(X) or intvalinit('DisplayMidrad') For a larger example, try n=1000; A=2*rand(n)-1; b=A*ones(n,1); tic; A\b; toc tic; verifylss(A,b); toc On a 750 Mhz Pentium III Laptop this takes about 2.8 seconds using floating point approximation, 20.2 seconds using fast multiplication, and 20.9 seconds using sharp multiplication. The time difference comes from interval matrix by interval vector multiplication C*X, see verifylss line 189. However, you would barely see a difference in the results because rather than the solution itself, the error with respect to an approximate solution is included. Use the routine "verifylss" to be sure to calculate verified results. Using "A\b" instead calls the built-in linear system solver, without verification (if both operands are non-interval). If the matrix or right hand side is an interval, you may use "\" safely (it calls verifylss), e.g. n=1000; A=midrad(2*rand(n)-1,1e-12); b=A*ones(n,1); tic; A\b; toc . This generates a 1000x1000 matrix with entrywise radius 10^(-12) and solves the linear system with verified bounds. The timing is 29.2 seconds . To change display format of intervals, use the built-in commands "format long", "format short e", etc. The linear system solver works for sparse s.p.d. and/or rectangular matrices as well. To generate a random sparse s.p.d. linear system try n=1000; A=sprand(n,n,1/n)+speye(n); A=A*A'; b=A*ones(n,1); and solve it (with timing) without verification by tic; x = A\b; toc and with verification by tic; X = verifylss(A,b); toc On a 750 Mhz Pentium III the timing is 0.16 seconds for built-in floating point, and 2.4 seconds with verification. Much better results are possible with reordering. Try the built-in minimum degree algorithm by p = symmmd(A); tic; X = verifylss(A(p,p),b(p)); toc . This computes a verified result in only .18 seconds on the above computer. The structure of the matrix, the Cholesky factor and the original matrix and the Cholesky factor of the reordered matrix can be displayed by spy(A), pause, spy(chol(A)), pause, spy(chol(A(p,p))) Verified solution of linear systems works for complex matrices as well. Unfortunately, there is a bug in the Cholesky decomposition of Matlab 5.2 for complex sparse matrices such that verification does not work either in that case. For nonlinear systems, automatic differentiation is used. For example, look at the function "intval\test". The code for the first function (omitting comments) is function y = test(x) y = x; c1 = typeadj( 1 , typeof(x) ); cpi = typeadj( midrad(3.14159265358979323,1e-16) , typeof(x) ); y(1) = .5*sin(x(1)*x(2)) - x(2)/(4*cpi) - x(1)/2; y(2) = (1-1/(4*cpi))*(exp(2*x(1))-exp(c1)) + ... exp(c1)*x(2)/cpi - 2*exp(c1)*x(1); return The call x=[-3;7]; y=test(x) calls "test" with input vector [-3;7] and usual floating point arithmetic. The call x=intval([-3;7]); y=test(x) uses interval arithmetic, the call x=gradientinit([-3;7]); y=test(x) uses automatic differentiation with access y.x to the function value and y.dx to the Jacobian. Finally, the call x=gradientinit(intval([-3;7])); y=test(x) evaluates "test" with automatic differentiation and interval artihmetic. The function "typeadj" is necessary for interval evaluations. It adjusts the type of constants to the type of the input argument "x". For example, exp(1) is replaced by "exp(c1)" to ensure correct rounding. The nonlinear system (Broyden's example) can be solved by X = verifynlss('test',[.6;3],'g') . The third parameter 'g' indicates function expansion by gradients; the call X = verifynlss('test',[.6;3],'s') does the same using slopes. Use of slopes does not imply uniqueness of the zero, but allows inclusion of clusters of zeros. The programs and operators support vector and matrix notation in order to diminish slow-down by interpretation. The user is encouraged to use matrix and vector notation whereever possible. See, for example, Brent's example in the function "test" in directory \intval (copy it from the end to top). The call n=200; X = verifynlss('test',10*ones(n,1),'g',1) solves Brent's example with the initial approximation 10*ones(n,1) given in his paper. The last parameter "1" tells the function "verifynlss" to display intermediate information. It shows that 10 floating point and 1 interval iteration is used together with information on the residual. The timing on an 750 Mhz Pentium III is time for n fl-pt iteration verification ------------------------------------------- 200 1.9 sec 1.3 sec 500 12.4 sec 21.6 sec and the sum of the two columns is the total computing time. Without the vector notation (see "test") this would not achievable. For sample programs, consult verifylss verifynlss verifyeig verifypoly in directories \intval and \polynom. For the homo ludens in you, try intlablogo . If you like Japan, look at intlablogo_ ... ...

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