List of commands using functions

add_func f1 f2 f3

Puts the sum of the functions f1 and f2 in the function f3. These functions must be of the same type but don't need to have the same x-range.

comp_func f1 f2 f3

Computes the composition of the real functions f1 and f2 and puts the result in f3, i.e

$$f_3 \ = \ f_1\circ f_2 .$$

The functions $f_1$, $f_2$, $f_3$ must be real and have the same precision. Here the functions f1, f2 need not be true functions. They can be formulas, with x as variable. These formulas must be understood by the expression evaluator. For example

comp_func f sin(x)+x g

will compute the composition of the function f and of the function sin(x)+x.

All the functions that appear in this command (if they are not formulas) must be real and have the same precision.

const_func f c

Fills the real function f with the constant expression c which can be a number or a numerical expression. For complex functions, the syntax is different :

const_func f cr ci

will fill f with the complex constant cr + i.ci.

convol f1 f2 f n

Computes the convolution of the functions f1, f2, and puts the result in f, using FFTs on 2*n points. Here n must be a power of 2. The functions must have the same precision and their types must be consistent. For example, if f is real anf f2 complex, then f must be complex.

The smallest of the xranges of f1, f2 will be divided in n-1 intervals. The other x-range will also divided in intervals of the same length. Only the values of f1, f2 on the end points of these small intervals will be used.

copy_func f1 f2

This command copies the function f1 in f2. The two functions must be of the same type. This is not a real copy, since f1 and f2 need not have the same x-range.

def_bes_par xx n1 n2

Defines an object xx containing some parameters used to compute order n1 Bessel transforms of functions, using integrals on n2 points (cf. 3.3.2).

def_four tr f n

Computes the parameters needed to obtain the Fourier transform of the function f, using FFT's with n points (n must be a power of 2). These parameters are stored in the object tr. To obtain the Fourier transform of f, the command trans_four must be used (cf. 3.3.1).

desc_func f

Prints a description of the function f. The minimal value of the x-range is put in the (local) variable xmin of the expression evaluator, and the maximum in xmax. To find the extrema of the function, use the commands min_func and max_func.

diff f1 f2

Computes the derivative of the function f1 and puts the result in f2. The two functions must be of the same type.

fft f1 f2 n

Computes the FFT of f1 on n points and puts the result in f2.Here n must be a power of 2.

div_func f1 f2 f3

Puts the quotient of the functions f1 and f2 in the function f3. These functions must be of the same type but don't need to have the same x-range.

fill_func f xxx

Fills the real function f using the formula xxx. This formula can be the name of a numerical function recognized by the expression evaluator or an expression with x as variable. For example the two instructions

fill_func f sin

and

fill_func f sin(x)

will give the same result, but the first runs faster. The second way must be used for more complicated formulas, for example

fill_func f sin(x)+x

fix_func_R f n x

Sets the value of the n-th point of the real function f to x. Here x is a numerical expression that can be parsed by the expression evaluator.

fix_func_C f n x1 x2

Sets the value of the n-th point of the complex function f to x1 + i.x2. Here x1,x2 are numerical expressions that can be parsed by the expression evaluator.

fix_xrange xr x1 x2

If xr is the name of a x-range of type 1 (equally spaced points) in double precision, this sets the first point to x1 and the step to x2 (these two last parameters are numbers or numerical expressions).

If xr is the name of a x-range of type 0 in double precision, the expression x1 is converted to an integer n, and the n-th value of the x-range is set to x2 (if n is a valid index). It is also possible to omit the last two parameters. In this case the program will wait for the input of all the points of the x-range (or read them if this instruction is inside a program).

fix_xrange_f xr x1 x2

If xr is the name of a x-range of type 2 (equally spaced points) in simple precision, this sets the first point to x1 and the step to x2 (these two last parameters are numbers or numerical expressions).

If xr is the name of a x-range of type 1 in simple precision, the expression x1 is converted to an integer n, and the n-th value of the x-range is set to x2 (if n is a valid index). It is also possible to omit the last two parameters. In this case the program will wait for the input of all the points of the x-range (or read them if this instruction is inside a program).

function f xr

Defines a real double precision function f with double precision x-range xr. This x-range must have been defined previously.

function_f f xr

Defines a real simple precision function f with simple precision x-range xr. This x-range must have been defined previously.

function_C f xr

Defines a complex double precision function f with double precision x-range xr. This x-range must have been defined previously.

function_fC f xr

Defines a complex simple precision function f with simple precision x-range xr. This x-range must have been defined previously.

imag fC fR

Puts the imaginary part of the complex function fC in the real function fR. Here the two functions must be of the same precision, but the x-ranges need not be the same.

imag_fix fR fC

Fills the imaginary part of the complex function fC with the real function fR. Here the two functions must be of the same precision, but the x-ranges need not be the same.

integ f1 f2

Computes the integral of the function f1 and puts the result in f2. The two functions must be of the same type. The integral has value 0 at the first point of f1.

inv_func f1 f2

Computes the generalized inverse of the function f1 and puts the result in the function f2. Both functions must be real of the same precision. The generalized inverse is defined as follows : suppose f1 is defined in the interval $\lbrack a,b\rbrack$. Then the value of f2 at $x$ will be the measure of the set of points in $\lbrack a,b\rbrack$where f1 is greater than or equal to $x$. So if $a=0$ and f1is decreasing, f2 will be the inverse of f1 (at least on theinterval $\lbrack {\bf f1}(b),{\bf f1}(a)\rbrack$).

load_create_f f xr ccc

This command will create the real simple precision function f with x-range xr from the data contained in the file ccc of the result directory. Such a file has the same structure as the files produced by the command save_func, i.e its lines are pairs consisting of real numbers . The first term in a pair represents the variable x and the other f(x). The order of the x's must be increasing. The x-range xr will contain exactly the x's of the data file. There is a test of the existence of xr and f, i.e. if one on them already exists, the command will be rejected.

load_create_d f xr ccc

The same as the preceeding command, but with a double precision real function.

load_create_C f xr ccc

This command will create the complex simple precision function f with x-range xr from the data contained in the file ccc of the result directory. Such a file has the same structure as the files produced by the command save_func, i.e its lines are triples consisting of real numbers . The first term in a triple represents the variable x and the others f(x) (the real and imaginary parts). The order of the x's must be increasing. The x-range xr will contain exactly the x's of the data file. There is a test of the existence of xr and f, i.e. if one on them already exists, the command will be rejected.

load_create_dC f xr ccc

The same as the preceeding command, but with a double precision complex function.

load_funct f xxx

This commands fills the function f with the data contained in the file xxx of the result directory. Such a file has the same structure as the files produced by the command save_func, i.e its lines are pairs consisting of two real values if the function is a real one, triples if the function is complex. The first term in a pair (or triple) represents the variable x and the other(s) f(x). The order of the x's must be increasing. If the values of the x of the data file are not the same as those of the x-range of f, interpolations will be done.

max_func f

If f is a real function this command computes the (first) maximum of f. The value of $x$ is put in the variable xmax_func of the expression evaluator and the corresponding value of $f(x)$ in the variable fmax_func. If f is complex, the maximal modulus is computed. The value of $x$ is again in the variable xmax_func, the real part of $f(x)$ in fmax_func_r, its imaginary part in fmax_func_i and the modulus in fmax_func_R.

min_func f

If f is a real function this command computes the (first) minimum of f. The value of $x$ is put in the variable xmin_func of the expression evaluator and the corresponding value of $f(x)$ in the variable fmin_func. If f is complex, the minimal modulus is computed. The value of $x$ is again in the variable xmin_func, the real part of $f(x)$ in fmin_func_r, its imaginary part in fmin_func_i and the modulus in fmin_func_R.

mul_func f1 f2 f3

Puts the product of the functions f1 and f2 in the function f3. These functions must be of the same type but don't need to have the same x-range.

real fC fR

Puts the real part of the complex function fC in the real function fR. Here the two functions must be of the same precision, but the x-ranges need not be the same.

real_fix fR fC

Fills the imaginary part of the complex function fC with the real function fR. Here the two functions must be of the same precision, but the x-ranges need not be the same.

rmul f1 x f2

Computes the product of the real function f1 by the value of the numerical expression x and puts the result in f2, which must be of the same type as f1 (but the x-ranges need not be the same). For complex functions the syntax is different :

rmul f1 xr xi f2

will multiply the function f1 by xr+i.xi and put the result in f2.

save_func f fic

Saves the function f in the file fic of the result directory. If f is a real function, this file will contain lines consisting of two numbers : the $x_i$ and $f(x_i)$. If f is a complex function it contains lines consisting of three numbers : the $x_i$, $Re(f(x_i))$ and $Im(f(x_i))$.

sub_func f1 f2 f3

Puts the difference of the functions f1 and f2 in the function f3. These functions must be of the same type but don't need to have the same x-range.

trans_bessel tr par f

Computes the Bessel transform of a real function, knowing its Fourier transform tr, the parameters of the Bessel transform par. The result is stored in the function f (see 3.3.2 and the command def_bes_par).

trans_four tr f

Computes the Fourier transform of a function using the data contained in tr (cf. 3.3.1 and the command def_four). The result is put in the complex function f.

val_func f x

Computes the value of the function f at x and prints the result. Here x is a numerical expression that will be parsed by the expression evaluator. Moreover, for a real function the variable func of the expression evaluator will contain the value of f at x, and for a complex one, the variables func_r and func_i will contain respectively the real part and the imaginary part of the value of f at x.

xrange xr n1 n2

Defines a x-range xr in double precision of type n1 with n2 points. The type must be 0 or 1.

xrange_f xr n1 n2

Defines a x-range xr in simple precision of type n1 with n2 points. The type must be 0 or 1.