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Xnumbers Tutorial

280

Example 4 - Optimization of Integral function

The method of optimization on-site used by Xnumbers macros and by the Excel Solver is very

flexible. This is expecially useful when the optimization function has a complicated form.

Assume, for example, to have to maximize the following integral respect to the parameter α

∫

−

)

(

The function integr returns the integral in

the cell B5. The integration limits and the

parameter α are contained in the cells B2,

B3 and B4 respectively.

Start the optimization macro that you like,

setting the objective cell B5 and changing

the cell B4

Here, we have choose three different algorithms. They seem work very fine for this problem

Algorithm

α (max)

error

Divide-conquer

2.210360464

9.8E-09

Downhill-Simplex

2.210360527

5.4E-08

Excel Solver

2.210361438

9.6E-07

Another example. Find the maximum of the following function for 0 < ω < 2

∫

)

sin(

( )

xdx

Here we use the Excel Solver for finding the

value of the cell D2 (the variable ω)

maximizing the cell E2 (the integral),

subjected to the constraints 0 < ω < 2 .

The final result w = 0.74201929194068

has an high precision, better 1E-8.

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Xnumbers Tutorial

281

How to call Xnumbers functions from VBA

There are two ways for calling Xnumbers functions from other user VBA macros

• by reference method

• by Application.Run method

Both methods have advantages and drawbacks. Let's see.

Reference method

Prepare a new worksheet called, for example test.xls

First of all , after we have installed

and loaded Xnumbers.xla in Excel,

turn on the VBA editor by shift+F11

and add a new module.

Our VBA editor should look like this.

In particular we must see the

XNUMBERS (xnumbers.xla) project

and our new project VBAProject

(test.xls)

Now we add a reference link in our

VBAProject to XNUMBERS project

From the menu tool/ references...

Switch on the box of XNUMBERS library and click OK

Note that a references object has been added to our project

From now on all functions (except the functions declared private) of XNUMBERS are available

in our macros. Of course all functions that you use in worksheet are public. But not all the

functions that we can use in a worksheet can be called from VBA and vice versa.

The link that we have added is permanent. This means that when we load our test.xls file, the

Xnumbers.xla will also be automatically loaded.

If you want to remove the link to Xnumbers.xla simply return to menu tools/references and

deselect the box

Now prepare the following simple macro that reads two numbers from the sheet and returns the

division in multiprecision

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Xnumbers Tutorial

282

Sub Test()

Dim a, b, c, DgtMax

DgtMax = Range("B2")

a = Range("B3")

b = Range("B4")

c = XNUMBERS.xdiv(a, b,

DgtMax)

Range("B5") = "'" & c

End Sub

Note that the above code works also if we write the compact statement

c = xdiv(a, b, DgtMax)

The XNUMBERS prefix is not necessary; it is useful only - at the developing time - to generate

automatically the list of the functions when we write the dot "." after XNUMBERS.

The result of the Test macro will be:

Of course the x-functions can be nested as any other function.

For example assume now to have to compute the expression = a * Ln(b)

The code will be

Sub Test1()

Dim a, b, c, DgtMax

DgtMax = Range("B2")

a = Range("B3")

b = Range("B4")

c = xmult(a, xln(b, DgtMax), DgtMax)

Range("B5") = "'" & c

End Sub

The output of the Test1 macro will be

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Xnumbers Tutorial

283

As we have seen calling x-function using the reference method is very easy and straightforward

because it works as we have "virtually" copied all the Xnumbers.xla code in our project

The main drawback is that the link set in our workbook is static. This mean that if we move or

install the Xnumbers.xla addin in another directory, also the link must be removed and then

rebuilt.

Application.Run method

A metod for calling external functions that always works, independently where you have

installed the addin, uses the Excel Application.Run method. Using this method we do not need

to add any reference link. We have only to load the addin in Excel.

Therefore our VBA environment looks like as simple as possible and the code of the macro

Test will modified as the following

Sub test()

Dim a, b, c, DgtMax

DgtMax = Range("B2")

a = Range("B3")

b = Range("B4")

With Application

c = .Run("xdiv", a, b,

DgtMax)

End With

Range("B5") = "'" & c

End Sub

The first argument of the "Run" method is the name of the function that we want to call, that is

"xdiv" in our case (note the quotes around the name).

The other arguments are, in sequence, the parameters that we pass to the function xdiv.

The "Run" method returns in the variable c the result of the xdiv function.

This method is both simple and efficient. The only drawback is that it cannot accept nested

functions. Thus the Test1 macro becomes a bit more tricky.

Sub test1()

Dim a, b, c, DgtMax

DgtMax = Range("B2")

a = Range("B3")

b = Range("B4")

With Application

c = .Run("xLn", b, DgtMax)

c = .Run("xmult", a, c, DgtMax)

End With

Range("B5") = "'" & c

End Sub

The final result, of course, is the same.

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Xnumbers Tutorial

284

References & Resources

"MSIEVE: A Library for Factoring Large Integers", Jason Papadopoulos , v1.17, March 2007

"LAPACK -- Linear Algebra PACKage 3.0", Updated: May 31, 2000

"EISPACK Guide, Matrix Eigensystem Routines”, Smith B. T. at al. 1976

"Numerical Analysis" F. Sheid, McGraw-Hill Book Company, New-York, 1968

"Handbook for Autom. Computation - Linear Algebra", vol II, Wilkinson, Martin, Peterson, 1970

"Matrix Analysis and Applied Linear Algebra", C. D. Mayer, Siam, 2000

"Linear Algebra", J. Hefferon, Saint Michael’s College, Colchester, Vermont, 2001,

"Numerical Methods that usually work", F. S. Acton, The Mathematica Association of America, 1990

"Analysis Nunerical Methods", E. Isaacson, H. B. Keller, Wiles & Sons, 1966

"Calculo Numérico", Neide M. B. Franco, 2002

"Metodos Numericos" Sergio R. De Freitas, 2000

"Numerical Mathematics in Scientific Computation", G. Dahlquist, Å. Björck, vol II.

"Numerical Recipes in FORTRAN 77- The Art of Scientific Computing - 1986-1992 by Cambridge

"Computation of Special Functions", by Shanjie Zhang and Jianming Jin - John Wiley and Sons, Inc

"Lanczos Implementation of the Gamma Function" by Paul Godfrey, Intersil Corp, 2001

"clsMathParser - A Class for Math Expressions Evaluation in Visual Basic", Leonardo Volpi , Michael

Ruder, Thomas Zeutschler, Lieven Dossche, . 3.2 Jan. 2003, by .Volpi

"clsMathParserC -A Class for Complex Math Expressions Evaluation in Visual Basic", Arnaud de

Grammont, Leonardo Volpi, v. 3.2 , Jan. 2003

"F F T (Fast Fourier Transform)", Paul Bourke, June 1993, http://astronomy.swin.edu.au

"2 Dimensional FFT" , Paul Bourke, June 1998, http://astronomy.swin.edu.au

“Solutions Numeriques des Equations Algebriques”, E., Durand, Tome I, Masson,Paris ,1960.

”A modified Newton method for Polynomials” , W.,Ehrilich, Comm., ACM, 1967, 10, 107-108.

“Ein Gesamschrittverfahren zur Berechnung der Nullstellen eines Polynoms”, O., Kerner, Num.Math.,

1966, 8, 290-294.

“Iteration methods for finding all the zeros of a polynomial simultaneously”, O. Aberth, Math. Comp. ,1973,

27, 339-344.

“The Ehrlich-Aberth Method for the nonsymmetric tridiagonal eigenvalue problem”, D. A. Bini, L.

Gemignani, F, Tisseur, AMS subject classifications. 65F15

“Progress on the implementetion of Aberth’s method”, D. A. Bini, G. Fiorentino, , 2002, The FRISCO

Consortiunm (LTR 21.024)

“A Method for Finding Real and Complex Roots of Real Polynomials With Multiple Roots”, by C. Bond, July

10, 2003

“A Robust Strategy for Finding All Real and Complex Roots of Real Polynomials”, by C. Bond, April 24,

2003

"Nonlinear regression", Gordon K. Smyth Vol. 3, pp 1405/1411, in Encyclopedia of Environmetrics (ISBN

0471 899976), Edited by John Wiley & Sons, Ltd, Chichester, 2002

"Optimization", Gordon K. Smyth Vol. 3, pp 1481/1487, in Encyclopedia of Environmetrics (ISBN 0471

899976), Edited by John Wiley & Sons, Ltd, Chichester, 2002

"Process Modeling", The National Institute of Standards and Technology (NIST) website for Statistical

Reference Datasets, (http://www.itl.nist.gov/div898/handbook/pmd/pmd)

"Metodos numericos con Matlab"; J. M Mathewss et al.; Prentice Hall

"Genetic and Nelder-Mead", E. Chelouan, et al, EJOR 148(2003) 335-348

"Convergence properties", J.C. Lagarias, et al, SIAM J Optim. 9(1), 112-147

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Xnumbers Tutorial

285

"Optimization for Engineering Systems", Ralph W. Pike, 2001, Louisiana State University

(http://www.mpri.lsu.edu/bookindex)

"Zeros of Orthogonal Polynomials" , Leonardo Volpi, Foxes Team

(http://digilander.libero.it/foxes/Documents)

"How to tabulate the Legendre’s polynomials coefficients" Leonardo Volpi, Foxes Team

(http://digilander.libero.it/foxes/Documents)

“Advanced Excel for scientific data analysis", Robert de Levie, 2004, Oxford University Press

"Microsoft Excel 2000 and 2003 Faults, Problems, Workarounds and Fixes", David A. Heiser, web site

http://www.daheiser.info/excel/frontpage.html

"NIST/SEMATECH e-Handbook of Statistical Methods", January 26, 2005

(http://www.itl.nist.gov/div898/handbook)

Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-01, Japan

(http://momonga.t.u-tokyo.ac.jp/~ooura)

"A Comparison of three high-precision quadrature schemes", David H. Bailey and Xiaoye S. Li, Copyright

2003, University of California (http://repositories.cdlib.org/lbnl/LBNL-53652)

"Asymptotic Bit Cost of Quadrature Formulas Obtained by Variable Transformation" P. Favati, Appl. Math.

"Gli algoritmi della crittografia a chiave pubblica", Giovanni Fraterno, settembre 2000.

http://digilander.libero.it/crittazione

Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. "Integer Relation

Detection." §2.2 in Experimental Mathematics in Action. Wellesley, MA: A K Peters, pp. 29-31, 2006.

Bailey, D. H. and Broadhurst, D. J. "Parallel Integer Relation Detection: Techniques and Applications."

Math. Comput. 70, 1719-1736, 2001.

Bailey, D. H. and Ferguson, H. R. P. "Numerical Results on Relations Between Numerical Constants Using

a New Algorithm." Math. Comput. 53, 649-656, 1989.

Bailey, D. and Plouffe, S. "Recognizing Numerical Constants."

http://www.cecm.sfu.ca/organics/papers/bailey/.

Borwein, J. M. and Lisonek, P. "Applications of Integer Relation Algorithms." To appear in Disc. Math.

http://www.cecm.sfu.ca/preprints/1997pp.html.

Weisstein, Eric W. "Integer Relation." From MathWorld--A Wolfram Web Resource.

http://mathworld.wolfram.com/IntegerRelation.html

Helaman R. P. Ferguson and David H. Bailey , "A Polynomial Time, Numerically Stable Integer Relation

Algorithm" RNR Technical Report RNR-91-032, July 14, 1992

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Xnumbers Tutorial

286

Credits

Software developed

Xnumbers contains code developed by the following authors that kindly contributed to this

collection.

Luis Isaac Ramos Garcia

Orthogonal Polynomials and Downhill Simplex Nelder-Mead routine

Olgierd Zieba

Cubic Spline and documentation

Alfredo Álvarez Valdivia

Robust Method fitting routines

Michael J. Kozluk

Log Relative Error function, Linear regression debugging, and

documentation (StRD benchmark)

Ton Jeursen

Format function, Xnumber debugging; Xnumber for Excel95,

xBaseChange - Multiprecision BaseChange

xCalc - multiprecision conversion function

Richard Huxtable

ChangeBase and Prime functions

Michael Ruder

MathParser improvement and debugging

Thomas Zeutschler

MathParser improvement and debugging

Lieven Dossche

Class MathParser development

Arnaud de Grammont

Complex MathParser developement

Rodrigo Farinha

MathParser improvement and debugging

Vladimir Zakharov

Installation and initialization improvement

Hans Günter

Extension of exponential integral for negative x

David A. Heiser

Weilford's algorithm for statistic standard deviation

John Jones

Modular arithmetic and Xnumber converter parser

Software translated

Xnumbers contains VB code translated from the following packages:

Takuya OOURA

DE-Quadrature (Numerical Automatic Integrator) Package

Shanjie Zhang - Jianming Jin

FORTRAN routines for computation of Special Functions

Software used

This documentation was developed thanks to:

Microsoft Help Workshop 4

Compiler for Help-on line

Microsoft Word and Equation Editor

RTF main document and formulas

Winplot ( Peanut Software

by Richard Parris) Math plots and graphical elaboration

Derive 5 (Texas Instrument)

Math graph and symbolic elaboration

MathGV 3.1 (by Greg VanMullem)

Math graphs and bitmap elaboration

Msieve 1.17

by Jason Papadopoulos. Large number factoring

Many thanks to everybody that have contributed to this document and made it more intelligible.

I am especially grateful for invaluable help on many occasions to Robert de Levie.

Xnumbers Tutorial

287

Analytical Index

Analytical index

Absolute; 28

Adams; 211; 212

Adams-Bashfort-Moulton; 211

Addition; 23

Airy functions; 252

AiryA; 252

AiryAD; 252

AiryB; 252

AiryBD; 252

Aitken; 158; 183

Arccos; 62

Arcsine; 62

Arctan; 62

Arithmetic Mean; 40

Base conversion; 240

baseChange; 240

Bessel functions; 249

BesselSphJ; 255

Beta function; 247

Bisection; 231

Bivariate Polynomial; 88

Brouncker-Pell Equation; 120

Cebychev; 78

Change sign; 28

Check digits; 32

Check Integer; 112

Check odd/even; 112

Check Prime; 109

Cholesky; 131

Coefficients of Orthogonal Polynomials; 96

Combinations; 39

Combinations function; 248

Complement of right angle; 63

Complex absolute; 99

Complex Addition; 98

Complex ArcCos; 102

Complex ArcSin; 102

Complex ArcTan; 102

Complex Complementary Error Function; 104

Complex conjugate; 101

Complex Cos; 102

Complex digamma; 103

Complex Division; 98

Complex Error Function; 104

Complex Exp; 101

Complex Exponential Integral; 103

Complex Expression Evaluation; 260

Complex Function Integration (Romberg

method); 156

Complex Gamma Function; 104

Complex Hyperbolic Cosine; 103

Complex Hyperbolic Sine; 102

Complex Hyperbolic Tan; 103

Complex inverse; 101

Complex Inverse Hyperbolic Cos; 103

Complex Inverse Hyperbolic Sin; 103

Complex Inverse Hyperbolic Tan; 103

Complex Log; 101

Complex Logarithm Gamma Function; 104

Complex Multiplication; 98

Complex negative; 101

Complex power; 100

Complex Quadratic Equation; 105

Complex Roots; 100

Complex Series Evaluation; 184

Complex Sin; 102

Complex Subtraction; 98

Complex Tangent; 102

Complex Zeta Function; 104

Constant e; 59

Constant Ln(10); 59

Constant Ln(2); 59

Constant pi; 62

Continued Fraction; 108

Continued Fraction of Square Root; 109

Convert extended into double; 34

Convol; 189

corrector; 211

Corrector; 212; 214

Cos; 61

Cosine Integral Ci(x); 249

CosIntegral; 249

cplxabs; 99

cplxacos; 102

cplxacosh; 103

cplxadd; 98

cplxasin; 102

cplxasinh; 103

cplxatan; 102

cplxatanh; 103

cplxconj; 101

cplxcos; 102

cplxcosh; 103

cplxdigamma; 103

cplxdiv; 98

cplxei; 103

cplxEquation2; 105

cplxerf; 104

cplxerfc; 104

cplxeval; 260

cplxExp; 101

cplxgamma; 104

cplxgammaln; 104

Xnumbers Tutorial

288

cplxintegr; 156

cplxinv; 101

cplxLn; 101

cplxmult; 98

cplxneg; 101

cplxpolar; 99

cplxpow; 100

cplxrect; 99

cplxroot; 100

cplxserie; 184

cplxsin; 102

cplxsinh; 102

cplxsub; 98

cplxtan; 102

cplxtanh; 103

cplxzeta; 104

Crout; 131

cspline_coeff; 199

cspline_eval; 197

cspline_interp; 197

cspline_pre; 197

Cubic Spline 2nd derivatives; 197

Cubic Spline Coefficients; 199

cvBaseDec; 240

cvBinDec; 240

cvDecBase; 240

cvDecBin; 240

Data Conditioning; 53

Data Integration (Newton Cotes); 157

dBel; 240

Decibel; 240

Decimal part; 29

DFSP; 140

DFSP_INV; 140

DFT; 137; 142

DFT_INV; 139

Diff1; 234

Diff2; 235

digamma; 247

Digamma function; 247

Digit_Max; 22; 35

Digits count; 31

Digits sum; 33

DigitsAllDiff; 32

Diophantine; 119

Diophantine Equation; 119

DiophEqu; 119

Discrete 2D Fourier Transform; 141

Discrete Convolution; 189

Discrete Fourier Inverse Transform; 139

Discrete Fourier Spectrum; 140

Discrete Fourier Transform; 137

Division; 24

Double Data integration; 179

Double Exponential; 152

Double Integral; 168

Double integration function; 174

Double Series; 185

Downhill; 268

DPOLYN; 74

Elliptic Integrals; 252

errfun; 244

Error Function Erf(x); 244

Euler; 211

Euler constant γ; 60

Euler-Mascheroni Constant; 245

Euler's Totient function; 121

exp_integr; 244

expn_integr; 244

Exponential; 58

Exponential any base; 58

Exponential integral Ei(x); 244

Exponential integral En(x); 244

Extended Number Check; 32

Factor; 116

Factorial; 39

Factorial with double-step; 39

Factorize; 113

Factorize function; 116

FFT; 137; 142

FFT_INV; 139

FFT2D; 141

FFT2D_INV; 141

Fibonacci numbers; 250

First Derivative; 234

Flip; 33

Format Extended Number; 32

Fourier; 165

Fourier_cos; 164

Fourier_cos; 166

Fourier_sin; 164

fract; 107

Fract_Interp; 195

Fract_Interp_Coef; 195

FractCont; 108

FractContSqr; 109

Fresnel cosine Integral; 250

Fresnel sine Integral; 250

Fresnel_cos; 250

Fresnel_sin; 250

Function Integration (Double Exponential

method); 152

Function Integration (mixed method); 154

Function Integration (Newton-Cotes); 159

Function Integration (Romberg method); 151

Gamma F-factor; 247

Gamma function); 245

Gamma quotient; 246

GCD; 106

Geometric Mean; 40

Grad; 235

Gradient; 235

Greatest Common Divisor; 106

Hermite; 78

Hessian; 236

Hessian matrix; 236

Hyperbolic Arc Cosine; 60

Hyperbolic Arc Sine; 59

Xnumbers Tutorial

289

Hyperbolic Arc Tangent; 60

Hyperbolic Cosine; 60

Hyperbolic Sine; 59

Hyperbolic Tangent; 60

Hypergeometric function; 251

Hypgeom; 251

I BesselSphY; 255

IElliptic1; 252

IElliptic2; 252

Incomplete Beta function; 248

Incomplete Gamma function; 248

Infinite integral; 177

Infinite Integration of oscillating functions; 165

Integer Division; 24

Integer part; 29

Integer polynomial; 86

Integer relation; 122

Integer Remainder; 25

Integer roots; 70

integr; 166

Integr; 154

Integr_2D; 174

Integr_fcos; 162

Integr_fsin; 162

Integr_nc; 159

Integr_ro; 151

Integr2D; 168

Integr3D; 171

Integral function; 147

Integral of sine-cosine power; 254

Integral_Inf; 177

Integration of oscillating functions (Filon

formulas); 162

Integration of oscillating functions (Fourier

transform); 164

IntegrData; 147

IntegrData2D; 179

IntegrDataC; 157

Interp_Mesh; 200

Interpolation with continue fraction; 195

Interpolation with Cubic Spline; 197

IntPowCos; 254

IntPowSin; 254

Inverse; 24

Inverse 2D Discrete Fourier Transform; 141

Inverse Discrete Fourier Spectrum; 140

Jacobian; 236

Jacobian matrix; 236

Kummer confluent hypergeometric functions;

253

Kummer1; 253

Kummer2; 253

LCM; 106

Least Common Multiple; 106

Legendre; 78

Linear Regression - Coefficients; 50

Linear Regression - Standard Deviation of

Estimate; 45

Linear Regression Coefficients; 43

Linear Regression Covariance Matrix; 47

Linear Regression Evaluation; 49

Linear Regression Formulas; 46

Linear Regression Min-Max; 56

Linear Regression Statistics; 48

Linear Regression with Robust Method; 55

Log Gamma function; 246

Log Relative Error; 242

Logarithm in any base; 58

Logarithm natural (Napier’s); 58

Macro for Double Integration; 168

Macro for Multiprecision Matrix operations; 135

Macro for Triple Integration; 171

Macro Integer relation finder; 126

Macro Mesh Fill; 201

Macro Poly Regression; 51

Macro Regression; 51

Macro Sampler; 145

Macros for optimization on site; 267

Macros X-Edit; 35

Math expression strings; 263

MathParser; 265

matrix; 135

Matrix Addition; 129

Matrix Determinant; 129

Matrix Inverse; 129

Matrix LL

decomposition; 131

Matrix LU decomposition; 131

Matrix Modulus; 130

Matrix Multiplication; 129

Matrix Power; 130

Matrix Subtraction; 129

Maximum Common Divisor; 106

MCD; 106

MCM; 106

Mesh Interpolation 2D; 200

Minimum Common Multiple; 106

mjkLRE; 242

Modular Addition; 110

Modular Division; 110

Modular Multiplication; 110

modular power; 110

Modular Power; 110

Modular Subtraction; 110

Multiple roots; 72

Multiplication; 24

Multiprecision Base Conversion; 241

Multiprecision Excel Formula Evaluation; 261

Multiprecision Expression Evaluation; 256

Next Prime; 109

NextPrime; 109

Non Linear Equation Solving; 238

N-Root; 28

Numbers comparison; 31

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