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270

For functions having more than one argument, the successive arguments are separated by

commas "," or alternatively, by semicolon ";" if your system has comma as decimal separator.

max(a;b) root(x;y) comb(n;k) beta(x;y) (only if comma "," is decimal sep.)

max(a,b) root(x,y) comb(n,k) beta(x,y) (only if point "." is decimal sep.)

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List of basic functions and operators

The following table lists all functions and operators recongize in a math expression string

Use: R = Real, C = Complex, M = Multiprecision

The functions with M can be evaluated either in standard or multiprecision

The functions with C alone must be used with cplxeval only

The functions with R alone must be used with xeval, xevall setting DgtMax = 0 (standard

precision)

Function

Use

Description

Note

R C M subtraction

R C M factorial

5!=120 (the same as fact)

R M

percentage

35% = 0.35 , 100+35% =103.5

R C M multiplication

R C M division

35/4 = 8.75

R C M integer division

35\4 = 8

R C M raise to power

3^1.8 = 7.22467405584208

| |

R C M absolute value

|-5|=5 (the same as abs)

R C M addition

R M

less than

return 1 (true) 0 (false)

R M

equal or less than

returns 1 (true) 0 (false)

R M

not equal

returns 1 (true) 0 (false)

R M

equal

returns 1 (true) 0 (false)

R M

greater than

returns 1 (true) 0 (false)

R M

equal or greater than

returns 1 (true) 0 (false)

abs(x)

R C M absolute value

abs(-5)= 5

acos(x)

R C M inverse cosine

argument -1 ≤ x ≤ 1

acosh(x)

R C M inverse hyperbolic cosine

argument x ≥ 1

acot(x)

R M

inverse cotangent

acoth(x)

R M

inverse hyperbolic cotangent

argument x<-1 or x>1

acsc(x)

R M

inverse cosecant

acsch(x)

R M

inverse hyperbolic cosecant

alog(z)

complex exponential with base 10

10^z

and(a,b)

logic and

returns 0 (false) if a=0 or b=0

arg(z)

polar angle of complex number

-pi < a <= pi

asec(x)

R M

inverse secant

asech(x)

R M

inverse hyperbolic secant

argument 0 ≤ x ≤ 1

asin(x)

R C M inverse sine

argument -1 ≤ x ≤ 1

asinh(x)

R C M inverse hyperbolic sine

atanh(x)

R C M inverse hyperbolic tangent

argument -1 < x < 1

atn(x), atan(x)

R C M inverse tangent

beta(x,y)

R C

beta

argument x>0 y>0

betaI(x,a,b)

Beta Incomplete function

x >0 , a >0 , b >0

cbr(x)

cube root

cbr(2) =1.25992104989487, also 2^(1/3)

clip(x,a,b)

Clipping function

return a if x<a , return b if x>b, otherwise

return x.

comb(n,k)

R C M combinations

comb(6,3) = 20

conj(x)

conjugate

cos(x)

R C M cosine

argument in radians

cosh(x)

R C M hyperbolic cosine

cot(x)

R M

cotangent

argument (in radians) x≠ k*π with k = 0,

2…

coth(x)

R M

hyperbolic cotangent

argument x>2

csc(x)

R M

cosecant

argument (in radians) x≠ k*π with k = 0,

2…

csch(x)

R M

hyperbolic cosecant

argument x>0

dec(x)

R M

decimal part

dec(-3.8) = -0.8

deg(x)

R M

degree sess. conversion

conv. degree (60) into current unit of

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angle

digamma(x)

R C

digamma

argument x>0

Ei(x)

R C

exponential integral function

argument x>0

Ein(x,n)

Exponential integral of n order

x >0 , n = 1, 2, 3…

Elli1(x)

Elliptic integral of 1st kind

∀ φ , 0 < k < 1

Elli2(x)

Elliptic integral of 2st kind

∀ φ , 0 < k < 1

erf(x)

R C

error Gauss's function

argument x>0

erfc(x)

R C

error Gauss's function for complex

value

R M

Euler-Mascheroni constant

Gamma constant 0.577..

exp(x)

R C M exponential

exp(1) = 2.71828182845905

fact(x)

R M

factorial

argument x >0

fix(x)

R M

integer part

fix(-3.8) = 3

FresnelC(x)

Fresnel's cosine integral

∀ x

FresnelS(x)

Fresnel's sine integral

∀ x

gamma(x)

R C

gamma

argument x>0

gammaln(x)

R C

logarithm gamma

argument x>0

gcd(a,b)

greatest common divisor

The same as mcd

grad(x)

degree cent. conversion

conv. degree (100) into current unit of

angle

HypGeom(x,a,b,c)

Hypergeometric function

-1 < x <1 a,b >0 c ≠ 0, −1, −2…

im(z)

immaginary part of complex number

int(x)

integer part

int(-3.8) = 4

integral(f,z,a,b)

Def. integral of complex function f(z) Integral('1/z^2','z',1-i,1+i)

inv(x)

inverse of a number

1/x

lcm(a,b)

R M

lowest common multiple

The same as mcm

ln(x), log(x)

R C M logarithm natural

argument x>0

max(a,b,...)

R M

maximum

max(-3.6, 0.5, 0.7, 1.2, 0.99) = 1.2

min(a,b,...)

R M

minimum

min(13.5, 24, 1.6, 25.3) = 1.6

mcd(a,b,...)

R M

maximun common divisor

mcm(4346,174) = 2

mcm(a,b,...)

R M

minimun common multiple

mcm(1440,378,1560,72,1650) =

21621600

mod(a,b)

R M

modulus

mod(29, 6) = 5 mod(-29, 6) = 1

nand(a,b)

logic nand

returns 1 (true) if a=1 or b=1

neg(z)

opposit of complex

-z

nor(a,b)

logic nor

returns 1 (true) only if a=0 and b=0

not(a)

logic not

returns 0 (false) if a ≠ 0 , else 1

nxor(a,b)

logic exclusive-nor

returns 1 (true) only if a=b

or(a,b)

logic or

returns 0 (false) only if a=0 and b=0

R M

Pi greek

3.141….

PolyCh(x,n)

Chebycev's polynomials

∀x , orthog. for -1 ≤ x ≤1

PolyHe(x,n)

Hermite's polynomials

∀ x , orthog. for −∞ ≤ x ≤ +∞

PolyLa(x,n)

Laguerre's polynomials

∀ x , orthog. for 0 ≤ x ≤1

PolyLe(x,n)

Legendre's polynomials

∀ x , orthog. for -1 ≤ x ≤1

rad(x)

R M

radians conversion

converts radians into current unit of angle

re(x)

real part of complex number

rnd(x)

random

returns a random number between x and

root(x,n)

R C M n-th root

argument x ≥ 0 (the same as x^(1/n)

round(x,d)

R M

round a number with d decimal

round(1.35712, 2) = 1.36

sec(x)

R M

secant

argument (in radians) x≠ k*π/2 with k =

2…

sech(x)

R C M hyperbolic secant

argument x>1

serie(….)

serie expansion of complex value z

Serie('z/n','n',1,10,'z',1+3i)

sgn(x)

R C

sign

returns 1 if x >0 , 0 if x=0, -1 if x<0

sin(x)

R C M sin

argument in radians

sinh(x)

R C M hyperbolic sine

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sqr(x)

R C M square root

sqr(2) =1.4142135623731, also 2^(1/2)

tan(x)

R C M tangent

argument (in radians) x≠ k*π/2 with k =

2…

tanh(x)

R C M hyperbolic tangent

xor(a,b)

logic exclusive-or

returns 1 (true) only if a ≠ b

BesselI(x,n)

Bessel's function of 1st kind, nth

order, mod.

x >0 , n = 0,1, 2, 3…

BesselJ(x,n)

Bessel's function of 1st kind, nth

order

x ≥0 , n = 0, 1, 2, 3…

BesselK(x,n)

Bessel's function of 2nd kind, nth

order, mod.

x >0 , n = 0,1, 2, 3…

BesselY(x,n)

Bessel's function of 2nd kind, nth

order

x ≥0 , n = 0,1, 2, 3…

J0(x)

Bessel's function of 1st kind

x ≥0

Y0(x)

Bessel's function of 2nd kind

x ≥0

I0(x)

Bessel's function of 1st kind, modified x >0

K0(x)

Bessel's function of 2nd kind,

modified

x >0

Si(x)

Sine integral

∀ x

Ci(x)

Cosine integral

x >0

zeta(x)

Riemman's zeta function

argument x<-1 or x>1

AiryA(x)

Airy function Ai(x)

∀x, example cbr(2) = 1.2599, cbr(-2) = -

1.2599

AiryB(x)

Airy function Bi(x)

argument (in radian) x≠ k*π/2 with k =

2…

Mean(a,b,...)

R M

Arithmetic mean

mean(8,9,12,9,7,10) = 9.1666

Meanq(a,b,...)

R M

Quadratic mean

meanq(8,9,12,9,7,10) = 9.300

Meang(a,b,...)

R M

Arithmetic mean

meang(8,9,12,9,7,10) = 9.035

Var(a,b,...)

R M

Variance

var(1,2,3,4,5,6,7) = 4.6666

Varp(a,b,...)

R M

Variance pop.

varp(1,2,3,4,5,6,7) = 4

Stdev(a,b,...)

R M

Standard deviation

Stdev(1,2,3,4,5,6,7) = 2.1602

Stdevp(a,b,...)

R M

Standard deviation pop.

Stdevp(1,2,3,4,5,6,7) = 2

Step(x,a)

Haveside's step function

Returns 1 if x ≥ a , 0 otherwise

DSBeta(x, a, b, [j])

Beta distribution (j=1 cumulative)

0 < x < 1 , a > 0, b > 0,

DSBinomial(k, n, p, [j])

Binomial distribution (j=1 cumulative) k integer, n integer , 0 < p < 1

DSCauchy(x, m, s, n, [j])

Cauchy (generalized) distribution (j=1

cumul.)

n integer , s > 0

DSChi(x, r, [j])

Chi distribution (j=1 cumulative)

r integer, x > 0

DSErlang(x, k, l, [j]))

Erlang distribution (j=1 cumulative)

k integer, x > 0

DSGamma(x, k, l, [j]))

Gamma distribution (j=1 cumulative)

x > 0, k > 0, l > 0

DSLevy(x, l, [j]))

Levy distribution (j=1 cumulative)

x > 0, l > 0

DSLogNormal(x, m, s, [j]))

Log-normal distribution (j=1

cumulative)

x > 0, m ≥ 0, s > 0

DSLogistic(x, m, s, [j]))

Logistic distribution (j=1 cumulative)

x > 0, m ≥ 0, s > 0

DSMaxwell(x, a, [j]))

Maxwell-Boltzman distribution (j=1

cumulative)

x > 0, a > 0

DSMises(x, k, [j]))

Von Mises distribution (j=1

cumulative)

k > 0, -π < x < π

DSNormal(x, m, s, [j]))

Normal distribution (j=1 cumulative)

s > 0

DSPoisson(k, z, [j]))

Poisson distribution (j=1 cumulative)

k integer, z > 0

DSRayleigh(x, s, [j]))

Rayleigh distribution (j=1 cumulative)

x > 0, s > 0

DSRice(x, v, s, [j]))

Rice distribution (j=1 cumulative)

x > 0, v ≥ 0 , s > 0

DSStudent(t, v, [j]))

Student distribution (j=1 cumulative)

v integer degree of freedom

DSWeibull(x, k, l, [j]))

Weibull distribution (j=1 cumulative)

x > 0, k integer, l > 0

Symbol "!" is the same as "Fact", symbol "\" is the integer division, symbols “|x|” is the same as Abs(x)

Logical function and operators returns 1 (true) or 0 (false)

Note: the arguments separator of the functions changes automatically from "," to ";" if your system decimal separator is

comma ",".

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Function Optimization

Macros for optimization on site

These macros has been ideated for performing the optimization task directly on the worksheet.

This means that you can define any function that you want simply using the standard Excel

built-in functions.

Objective function. For example: if you want to search the minimum of the bivariate function

(

)

(

)

100

( , , )

−

f x x y

insert in the cell E4 the formula "=(B4-0.51)^2+(C4-0.35)^2", where the cells B4 and C4

contain the current values of the variables x and y respectively. Changing the values of B4 e/o

C4 the function value E4 also changes consequently.

For optimization, you can choose two different algorithms

Downhill-Simplex

The Nelder–Mead downhill simplex algorithm is a popular derivative-free

optimization method. Although there are no theoretical results on the

convergence of this algorithm, it works very well on a wide range of

practical problems. It is a good choice when a one-off solution is wanted

with minimum programming effort. It can also be used to minimize

functions that are not differentiable, or we cannot differentiate.

It shows a very robust behavior and converges for a very large set of

starting points. In our experience is the best general purpose algorithm,

solid as a rock, it's a "jack" for all trades.

For mono and

multivariate

functions

Divide-Conquer 1D

For monovariable function only, it is an high robust derivative free

algorithm. It is simply a modified version of the bisection algorithm

Adapt for every function, smooth or discontinue.

It converges for very large segments. Starting point not necessary

For monovariable

function only. It

needs the segment

where the max or

min is located

Example assume to have to minimize the following function for x > 0

cos(4 )

sin(3 )

( )

x e

f x

−

The Downhill-Simplex of Nelder and Maid routine appears by the courtesy of Luis Isaac Ramos Garcia

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We try to search the minimum in the range 0 < x < 10

Choose a cell for the variable x , example B6, and insert the function

= SIN(3*B6)*EXP(- 2*B6) + COS(4*B6)*EXP(-B6)

in a cell that you like, for example C6.

After this, add the constrain values into another range, for example B3:C3

The values of the variables at the start are not important

Select the cell of the function C6 and start the macro "1D divide and conquer", filling the input

field as shown

Stopping limit. Set the maximum evaluation points allowed.

Max/Min. The radio buttons switches between the minimization and maximization algorithm

The "Downhill-Simplex" macro is similar except that:

• The constrain box is optional.

• It accepts up to 9 variables (range form 1 to 9 cells)

• The algorithm starts from the point that you give in the variable cells. If the constrain

box is present, the algorithm starts from a random point inside the box

Let's see how it works with some examples

The following examples are extracted from "Optimization and Nonlinear Fitting" , Foxes Team, Nov. 2004

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276

Example 1 - Rosenbrock's parabolic valley

This family of test functions is well known to be a minimizing problem of high difficult

(

)

(

)

( , , )

m y x

f x x y

+ −

= ⋅ ⋅ −

The parameter "m" tunes the difficult: high value means high difficult in minimum searching.

The reason is that the minimum is located in a large flat region with a very low slope. The

following 3D plot shows the Rosenbrock's parabolic valley for m = 100

The following contour plot is obtained for m = 10

The function is always positive except in the point (1, 1) where it is 0. it is simple to

demonstrate it, taking the gradient

(

)

(

)











−

⋅ − − =

−

⋅ +

⇒

∇ =

2 0

2 1 1 2

my x

m y

m x

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277

From the second equation, we get

(

)

y x

my x

⇒

−

Substituting in the first equation, we have

(

)

2 0

2 1 1 2

⇒

− =

⇒

⋅ − − =

−

⋅ +

m x

So the only extreme is the point (1, 1) that is the absolute minimum of the function

To find numerically the minimum, let's arrange a similar sheet.

We can insert the function and the parameters as we like

Select the cell D4 - containing the objective function - and start the macro "Downhill-Simplex".

The macro fills automatically the variables-field with the cells related to the objective function.

But, In that case, the cell A4 contains the parameter m that the macro must not change. So

insert only the range B4:C4 int the variables field.

The cells B4:C4 will change for minimizing the

objective function in the cell D4

Leave empty the constraint input box and press "Run"

Starting from the point (0, 0) we obtain the following good results

Algorithm

error

time

Simplex

2.16E-13

2 sec

100

Simplex

4.19E-13

2 sec

where the error is calculated as |x-1|+|y-1|

Example 2 - Constrained minimization

Example: assume to have to minimize the following function

( , , )

−

f x x y

with the ranges constrains

0.5

2 , 0

≤ ≤

≤

≤ ≤

The Excel arrangement can be like the following

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278

Compare with the exact solution x = 1.5, y = 0.5

Note that the function has a free minimum at x = 1, y = 1

Repeat the example living empty the constrains box input, for finding those free extremes.

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279

Example 3 - Nonlinear Regression with Absolute Sum

This example explains how to perform a nonlinear regression with an objective function

different from the "Least Squared". In this example we adopt the "Absolute Sum".

We choose the exponential model

a e

f xak

− ⋅

( , , , )= = ⋅

The goal of the regression is to find the best couple of parameters (a, k) that minimizes the sum

of the absolute errors between the regression model and the given data set.

∑

−

( , , , )|

f x ak

The objective function AS depends only by parameter a, k. Giving in input this function to our

optimization algorithm we hope to solve the regression problem

A possible arrangement of the worksheet may be:

We hope that changing the parameters "a" and "k" int the cells E2 and F3, the objective

function (yellow cell) goes to its minimum value. Note that the objective function depends

indirectly by the parameters a and k.

0.2

0.4

0.6

0.8

1.2

0.2

0.4

0.6

0.8

1.2

The starting condition is the following, where y

indicates the given data and y* is the

regression plot (a flat line at the beginning)

Start the Downhil-Simplex and insert the

appropriate ranges: objective function = G3

and variables = E3:F3

Starting form the point (1, 0) you will see the cells changing quickly until the macro stops itself

leaving the following "best" fitting parameters and the values of the regression y*

Best fitting parameters

-2

The plot of the y* function and the samples

y are shown in the graph. As we can see

the regression fits perfectly the given

dataset.

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