Nash - Compact Numerical Methods for Computers (523163), страница 55
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Root-findingExample 12.5. Minimum of a function of one variableExample 13.1. Grid and linear searchExample 13.2. A test of root-finding algorithmsExample 13.3. Actuarial calculationsExample 14.1. Using the Nelder-Mead simplex procedure (algorithm 19)Example 15.1. Illustration of the variable metric algorithm 21Example 16.1. Conjugate gradients minimisationExample 17.1.
Marquardt’s minimisation of a nonlinear sum of squaresExample 18.1. Optimal operation of a public lottery256202023242844455262779192100108117125131138144144144145146156164165180196204216228List of examplesExample 18.2.ExampleExampleExampleExample18.3.19.1.19.2.19.3.Example 19.4.Market equilibrium and the nonlinear equations that resultMagnetic rootsSolution of Fröberg’s differential equation (example 2.2)Surveying-data fittingConjugate gradients for inverse iteration and Rayleighquotient minimisationNegative definite property of Fröberg’s matrix257231232238240249251Previous Home NextAppendix 4FILES ON THE SOFTWARE DISKETTEThe files on the diskette fall into several categories.
For the new user of the diskette,we strongly recommend looking at the fileREADME.CNMwhich contains notes of any errors or additions to material in either the book or thediskette. This can be displayed by issuing a commandTYPE(drive:)README.CNMwhere drive: is the disk drive specification for the location of the README.CNMfile. The file may also be printed, or viewed with a text editor.The algorithms (without comments) are in the files which follow. OnlyALG03A.PAS has not appeared on the pages of the book.ALG01.PASALG02.PASALG03.PASALG03A.PASALG04.PASALG05.PASALG06.PASALG07.PASALG08.PASALG09.PASALG10.PASALG11.PASALG12.PASALG13.PASALG14.PASALG15.PASALG16.PASALG17.PASALG18.PASALG19.PASALG20.PASALG21.PASALG22.PAS258Files on the software diskette259ALG23.PASALG24.PASALG25.PASALG26.PASALG27.PASThe following files are driver programs to run examples of use of the algorithms.DR102.PASDR03.PASDR03A.PASDR04.PASDR0506.PASDR0708.PASDR09.PASDR10.PASDR13.PASDR14.PASDR15.PASDR1617.PASDR1618.PASDR1920.PASDR21.PASDR22.PASDR23.PASDR24II.PASDR24LE.PASDR24LS.PASDR25.PASDR26.PASDR27.PAS-algorithms 1 and 2, svd and least-squares solution-algorithm 3, columnwise Givens’ reduction-algorithm 3a, row-wise Givens’ reduction-algorithm 4, Givens’ reduction, svd and least-squares solution-algorithms 4 and 5, Gauss elimination and back-substitutionto solve linear equations-algorithms 7 and 8, Choleski decomposition and back-substitution to solve linear equations of a special form-algorithm 9, to invert a symmetric, positive-definite matrix-algorithm 10, to find eigensolutions of matrices via inverseiteration using Gauss elimination-algorithm 13, eigensolutions of a symmetric matrix via the svd-algorithm 14, eigensolutions of a symmetric matrix via a cyclicJacobi method--algorithm 15, solution of a generalised matrix eigenproblemvia two applications of the Jacobi method-algorithms 16 and 17, grid search and one-dimensional minimisation--algorithms 16 and 18, grid search and one-dimensional rootfinding-algorithms 19 and 20, Nelder-Mead function minimiser andaxial search for lower points in the multivariate space-algorithm 21, variable metric function minimiser-algorithm 22, conjugate gradients function minimiser-algorithm 23, modified Marquardt nonlinear least-squaresmethod-algorithm 24, applied to finding eigensolutions of a symmetricmatrix by inverse iteration-algorithm 24, applied to finding solutions of linear equations--algorithm 24, applied to solving least-squares problems-algorithm 25, solutions of a generalised symmetric eigenproblem by conjugate gradients minimisation of the Rayleighquotient-algorithms 26, 11, and 12, to find eigensolutions of a generalcomplex matrix.
standardise the eigenvectors and computeresiduals-algorithm 27, Hooke and Jeeves function minimiser260Compact numerical methods for computersThe following support codes are needed to execute the driver programs:CALCEPS.PAS --to compute the machine precision for the Turbo Pascal computing environment in which the program is compiledCONSTYPE.DEF ---a set of constant and type specifications common to the codesCUBEFN.PAS ---a cubic test function of one variable with minimum at 0.81650FNMIN.PAS ---a main program to run function minimisation proceduresGENEVRES.PAS ---residuals of a generalised eigenvalue problemGETOBSN.PAS ---a procedure to read a single observation for several variables(one row of a data matrix)HTANFN.PAS---the hyperbolic tangent, example 13.2JJACF.PAS---Jaffrelot’s autocorrelation problem, example 14.1MATCOPY.PAS ---to copy a matrixMATMUL.PAS ---to multiply two matricesMATRIXIN.PAS ---to create or read in matricesPSVDRES.PAS---to print singular-value decomposition results---real valued test function of x for [1D] minimisation and rootQUADFN.PASfindingRAYQUO.PAS---to compute the Rayleigh quotient for a generalised eigenvalueproblem---to compute residuals for linear equations and least-squaresRESIDS.PASproblems---to set up and compute function and derivative information forROSEN.PASthe Rosenbrock banana-shaped valley test problemSPENDFN.PAS ---the expenditure example, illustrated in example 12.5 andexample 13.1STARTUP.PAS ---code to read the names of and open console image and/orconsole control files for driver programs.
This common codesegment is not a complete procedure, so cannot be included inTurbo Pascal 5.0 programs.---to compute various tests of a singular-value decompositionSVDTST.PASTDSTAMP.PAS ---to provide a time and date stamp for output (files). This codemakes calls to the operating system and is useful only for MSDOS computing environments. In Turbo Pascal 5.0, there areutility functions which avoid the DOS call.VECTORIN.PAS ---to create or read in a vectorThe following files provide control information and data to the driver programs.Their names can be provided in response to the questionFile for input of control data ([cr] for keyboard)?Be sure to include the filename extension (.CNM).
The nomenclature follows that forthe DR*.PAS files. In some cases additional examples have been provided. For thesefiles a brief description is provided in the following list of control files.Files on the software diskette261EX0102.CNMEX03.CNMEX03A.CNMEX04.CNMEX0506.CNMEX0506S.CNM --- a set of equations with a singular coefficient matrixEX0708.CNMEX09.CNMEX10.CNMEX13.CNMEX14.CNMEX15.CNMEX1617.CNMEX1618.CNMEX19.CNMEX1920.CNMEX1920J.CNM --- data for the Jaffrelot problem (JJACF.PAS), example 14.1EX21.CNMEX22.CNMEX23.CNMEX24II.CNMEX24LE.CNMEX24LS.CNMEX24LS1.CNM --- data for example 19.2EX25.CNMEX26.CNMEX26A.CNMEX27J.CNM --- data for the Jaffrelot problem (JJACF.PAS), example 14.1.EX27R.CNM --- console control file for the regular test problem, the Rosenbrock test function (ROSEN.PAS)If the driver programs have been loaded and compiled to saved executable (.COM)files, then we can execute these programs by typing their names, e.g.
DR0102. Theuser must then enter command information from the keyboard. This is not difficult,but it is sometimes useful to be able to issue such commands from a file. Such aBATch command file (.BAT extension) is commonly used in MS-DOS systems. In thedriver programs we have included compiler directives to make this even easier to useby allowing command input to come from a file. A batch file EXAMPLE.BAT whichcould run drivers for algorithms 1 through 6 would have the formrem EXAMPLE.BATrem runs Nash Algorithms 1 through 6 automaticallyDR0102<DR0102X.DR03A<DR03AX.DR03<DR03X.DR04<DR04X.DR0506<DR0506X.262Compact numerical methods for computersThe files which end in an ‘X.’ contain information to control the drivers, in fact, theycontain the names of the EX*.CNM control files.
This facility is provided to allowfor very rapid testing of all the codes at once (the technical term for this is ‘regressiontesting’). Note that console image files having names of the form OUT0102 arecreated, which correspond in form to the driver names, i.e. DR0102.PAS. Thecommand line files present on the disk are:DR0102X.DR09X.DR1618X.DR24LEX.DR03AX.DR10X.DR19X.DR24LSX.DR03X.DR13X.DR21X.DR25X.DR04X.DR14X.DR22X.DR26X.DR0506X.DR15X.DR23X.DR27X.DR0708X.DR1617X.DR24IIX.Users may wish to note that there are a number of deficiencies with version 3.01 a ofTurbo Pascal. I have experienced some difficulty in halting programs with theControl-C or Control-Break keystrokes, in particular when the program is waitingfor input.
In some instances, attempts to halt the program seem to interfere with thefiles on disk, and the ‘working’ algorithm file has been over-written! On someoccasions, the leftmost characters entered from the keyboard are erased by READinstructions. From the point of view of a software developer, the absence of a facilityto compile under command of a BATch command file is a nuisance. Despite thesefaults, the system is relatively easy to use.
Many of the faults of Turbo Pascal 3.01ahave been addressed in later versions of the product. We anticipate that a diskette ofthe present codes adapted for version 5.0 of Turbo Pascal will be available about thetime the book is published. Turbo Pascal 5.0 is, however, a much ‘larger’ system interms of memory requirements.Previous Home NextBIBLIOGRAPHYABKAMOWITZ M and STEGUN I A 1965 Handbook of Mathematical Functions with Formulas, Graphs andMathematical Tables (New York: Dover)ACTON F S 1970 Numerical Methods that Work (New York: Harper and Row)BARD Y 1967 Nonlinear Parameter Estimation and Programming (New York: IBM New York ScientificCenter)---1970 Comparison of gradient methods for the solution of nonlinear parameter estimation problemsSIAM J.
Numer. Anal. 7 157-86--1974 Nonlinear Parameter Estimation (New York/London: Academic)BATES D M and WATTS D G 1980 Relative curvature measures of nonlinearity J. R. Stat. Soc. B 42 1-25---1981a A relative offset orthogonality convergence criterion for nonlinear least squares Technometrics23 179-83---1988 Nonlinear Least Squares (New York: Wiley)BAUER F L and REINSCH C 1971 Inversion of positive definite matrices by the Gauss-Jordan method inlinear algebra Handbook for Automatic Computation vol 2, eds J H Wilkinson and C Reinsch (Berlin:Springer) contribution l/3 (1971)BEALE E M L 1972 A derivation of conjugate gradients Numerical Methods for Nonlinear Optimization ed.F A Lootsma (London: Academic)BELSLEY D A, KUH E and WELSCH R E 1980 Regression Diagnostics: Identifying Influential Data andSources of Collinearity (New York/Toronto: Wiley)BIGGS M C 1975 Some recent matrix updating methods for minimising sums of squared terms HatfieldPolytechnic, Numerical Optimization Centre, Technical Report 67BOOKER T H 1985 Singular value decomposition using a Jacobi algorithm with an unbounded angle ofrotation PhD Thesis (Washington, DC: The American University)BOWDLER H J, MARTIN R S, PETERS G and WILKINSON J H 1966 Solution of real and complex systems oflinear equations Numer.
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