fulltext

Derivation of e-beam emittances based on undulator radiation

G. DATTOLI(1), G. K. VOYKOV(1) (*), A. H. LUMPKIN(2) and B. X. YANG(2) (1) ENEA, Dipartimento Innovazione, Divisione Fisica Applicata

Centro Ricerche Frascati - C.P. 65, 00044 Frascati, Roma, Italy (2) Argonne National Laboratory - Argonne, IL 60439, USA (ricevuto il 21 Giugno 1996; approvato il 30 Ottobre 1996)

Summary. — We present a method to extract the e-beam emittances from undulator

brightness data. The analysis is based on the Newton method and on a stochastic procedure for the solution of a non-linear system of algebraic equations. The algorithm is shown to be stable and its reliability is tested by performing the analysis with the data available from the advanced photon source.

PACS 41.85 – Beam optics.

1. – Introduction

It has been elsewhere noted that one of the challenges of the third generation of synchrotron facilities, with low natural emittances, is the characterization and the monitoring of the particle beams [1]. A well-known method is that of inferring the

e-beam qualities from the undulator brightness [2-8]. The method is somewhat

complicated and needs a careful analysis to be applied unambiguously. This paper is devoted to such an analysis and to a discussion of the intrinsic limitations of this procedure.

The analysis developed in the forthcoming sections is based on the following elements:

1) We derive the undulator brightness for a given set of emittance values, by means of a numerical code capable of including the effect of the betatron motion, the e-beam Twiss parameters and emittances, energy spread etc. 2) The data from the code are used as input data to construct a model function

which characterizes the brightness in terms of the e-beam emittances (1).

(*) ENEA Guest.

(1_{) We focus our attention on the emittances and not on the energy spread.}

3) It is assumed that the peak brightness of a given harmonic can be characterized by a suitable linear combination of functions whose variables are the horizontal and vertical emittances (ex and ey, respectively).

4) The coefficients of the linear combination are evaluated from the data. 5) The procedure is finally inverted to get the emittances from the model function.

The inversion procedure is performed by using two different techniques, the Newton method for a system of non-linear equations and a second algorithm which exploits a stochastic procedure for the solution of non-linear algebraic systems [9].

The most important result of the present analysis will be that it is not possible to obtain a one-to-one correspondence between brightness and emittances, but it will be shown that a region of emittances values can be connected to one value of the perturbed brightness.

We will also prove that the procedure we have developed does not induce any error due to the model and that the error is entirely due to the precision of the brightness measurement.

2. – Description of the numerical code

The code we have developed (see also ref. [10]) is based on a numerical integration of the Lienard-Wiechert integral [11] and on the inclusion of the initial e-beam phase-space distribution by means of a Monte Carlo sampling of the mean electron brightness as a function of the frequency. The initial phase-space values (x0, x 80, y0, y 80) are obtained, by using a Neumann procedure [9], from the distribution function

.

/

´

y

z

where peAh4 eh is the e-beam emittance and (ah, bh, gh) are the e-beam Twiss

parameters.

The brightness radiated by an accelerated charged particle is given by [11] d2 J dV dv 4 e2_v2 4 p2c3

N



k

l

N

where n is a unit vector determining the direction of observation, r and b specify the particle position and velocity, respectively. The integration of (2) proceeds as follows.

We introduce the quantities

.

`

/

`

´

k

l

k

l

TABLEI. – Principal symbols list.

B0f peak on-axis magnetic field

luf undulator period

N f number of undulator periods Lu4 Nluf length of the undulator

K 4 eB0lu 2 pm0c2

f undulator parameter g f e-beam relativistic factor b * 4 g

pKlu

and then we write the electron equations of motion, which take into account the dependence on the transverse coordinates of the undulator field. The latter is assumed to be linearly polarized along the vertical direction (see table I and fig. 1 for the specifications of the symbols and of the axis convention),

.

`

/

`

´

g

h

k

g

h

g

h

l

g

h

k

g

h

g

h

g

h

l

g

h

k

g

h

l

g

h

g

h

The differential equations (4) are the Lorentz force equations for an electron moving in

Fig. 2. – a) First-harmonic brightness vs. frequency(parameter of table IIa)). Case ax4 ay4 0 ,

bx4 by4 b * : a) analytical approximation; b) numerical analysis. Case ax4 ay4 1 , bx4 by4

b * O10: c) analytical approximations; d) numerical analysis. b) Third-harmonic brightness vs. frequency (parameters of table IIb)). Case ax4 ay4 0 , bx4 by4 b * : a) analytical

approximation; b) numerical analysis. c) Third-harmonic brightness vs. frequency(parameters of table IIc)). Case ax4 ay4 0 .5 , bx4 by4 b * O5 : a) analytical approximation; b) numerical

analysis.

a magnetic field with components

.

`

/

`

´

g

h

g

h

m

g

h

n

g

h

g

h

g

h

The magnetic field has been expanded up to the lowest order in the transverse coordinates and we leave d, namely the sextupolar term, unspecified for the moment.

TABLEII. – Parameters of the analytical and numerical direct-problem examples. a) g 41.4Q104_{, e} x4 7 Q 1027cm rad , ey4 8 Q 1028cm rad lu4 5 cm , N 420 , K 41.48 b) g 440 , ex4 ey4 7 .8 Q 1026cm rad , lu4 5 cm , N 420 , K 41.41 c) g 440 , ex4 ey4 3 .1 Q 1024cm rad , lu4 5 cm , N 420 , K 41.41

The initial conditions of the system (4) are provided by

.

`

/

`

´

N

N

N

The solution is evaluated in the interval [ 0 , Lu] and the brightness is expressed as d2 J dv dV 4 e2 4 p2_c

g

h

!

Examples of undulator brightness calculated by using the above procedure are given in fig. 2.

3. – Description of the model and results

The code we have described in the previous section has been exploited to derive the Advanced Photon Source (APS) brightness [1], undulator and beam parameters are listed in table III.

To test the sensitivity of the brightness to the beam emittance, we have assumed

exOeyA 10 and the following interval of variation (the emittances are expressed in

cm Q rad ):

exmin4 6 Q 1027, emaxx 4 9 Q 1027,

TABLEIII. – Advanced photon source beam and undulator parameters. Undulator parameter K

Undulator period lu( cm )

Number of period Electron energy (GeV) Energy spread Twiss parameters: ax4 ay bx(cm) by(cm) On-axis observer 0.22 1.8 158 7 2 Q 1023 0 1420 1010

TABLEIV. – Direct problem approximation functions.

j Wj xj 1 2 3 4 5 1 ex1 ey (ex1 ey)2 ex2 ey (ex2 ey)2 1 eex eex1 ey eey eex2 ey

The peak brightness is assumed to be reproduced by the two functions P(ex, ey) 4

!

!

and Wj, xj are specified in table IV.

The values of the coefficients ciand diare calculated by using the data points given

in table V.

Once the coefficients ]ci, di( are specified, the functions P(ex, ey) and Q(ex, ey)

are inverted to obtain (ex, ey). The situation is illustrated in table V and in fig. 3.

The inversion is performed by using the Newton method for the solution of non-linear equations and a method called intersection. The second method, which will be more carefully discussed in the concluding section, consists in finding the values of (ex, ey) providing the brightness with the required tolerance, by means of a stochastic

procedure.

The model provides the emittances with a precision of 1% (in most cases less). It is, however, evident that the model is pathological around some points (those denoted by J and C). For these points the model and the direct calculation do not provide the same results. We must remark that these points are external, in the sense that they have not been utilized to specify the (cj, dj) coefficients. If they are included in the data set

points and the interval of emittances is reduced, we find the result of fig. 4, which provides a close agreement between model and data points. It is therefore clear that the functions P and Q are good test functions.

A more intuitive, but also more quantitative, idea of the correspondence between emittances and brightness is given by fig. 5, in which we have reported the most

TABLEV. – Numerical data of the solution for the APS inverse problem. Radiated energy (photonsOsOrad) 31023 Sample emittance (cmOrad) exQ 1027 eyQ 1028 Reproduced emittance (cmOrad) Newton’s method exQ 1027 eyQ 1028 intersection Model points E 4 .52 60.88 8.83_8.99 8.84 9.00 8.74 8.91 D 5 .00 60.90 7.56_7.84 7.57 7.82 7.58 7.92 K1 5 .23 60.92 7.59 7.32 7.55 7.31 7.58 7.29 K2 5 .35 60.90 7.62 7.26 7.65 7.27 7.58 7.26 B 6 .12 61.01 6.65 7.14 6.69 7.23 6.76 7.26 A 6 .21 61.03 5.85 6.02 5.78 5.94 6.59 7.26 Test points J 5 .19 60.94 8.10_6.83 7.52_7.32 7.42_7.42 K 5 .29 60.93 7.61_7.29 7.60_7.29 7.58_7.29 C 5 .33 60.93 7.09_7.81 7.63_7.28 7.58_7.26

probable emittances, obtained with the method of intersections. We have chosen the

values ( B0) of the brightness given with a given precision, and we have considered the equations

P(ex, ey) 4 B0, Q(ex, ey) 4 B0. (10)

By using a stochastic procedure we have found the values of (ex, ey) satisfying (10) in

the required interval of precision. We have then represented the most probable emittance values, by plotting the density of solutions (i.e. the number of solutions in the subregion specified by the chosen Dex, Deygrid). It is evident from the figures that the

number of possible solutions increases where the uncertainty on B0 increases. We conclude this section by providing an idea of the brightness sensitivity to the emittance values. In fig. 6 we show the first-harmonic peak brightness for different values of the emittances vs. the number of particles used for the simulation. Variations of 7% in emittances correspond to about 1 or 2% brightness variation. It is however worth stressing that for the parameters of table III, the second harmonic appears more sensitive.

Fig. 3. – a) Emittance vs. radiated energy (parameters in table III). First-harmonic frequency region: exOeyB 10 . Continuous line: sample emittance; dashed line: emittances reproduced by

Newton’s method; dotted line: reproduction by the intersection method. b) The same as in a). The region of the points J, K, C.

Fig. 4. – Emittances vs. radiated energies (parameters in table III). First-harmonic frequency region: exOeyB 10 . Restriction of the region of emittances, higher number of the model points.

Fig. 5. – a) Density of the solutions in the region of emittances (parameters in table III). First-harmonic frequency region: exOeyB 10 . Test point K. Intersection method with

perturbation DeOe41% of the test energy. b) The same as in a), perturbation 2%. c) The same as in a), perturbation 5%.

Fig. 6. – Energy vs. number of electron histories of the beam simulation (parameters in table III). First-harmonic frequency region: exOeyB 10 ; introduced emittances (cm rad): a) ex4 7 Q 1027,

4. – Problem definition and numerical solution

In this section we will analyze in a more rigorous way the problem of deriving the beam emittance from undulator brightness data.

The linear undulator brightness depends on three principal groups of parameters concerning the undulator, the electron beam and the observer. We have to deal with the case when all these parameters are fixed except the electron beam emittances.

We suppose that the region D % R2 _{to which the values of the emittance vectors} belong is known:

D 4 ](ex, ey) Neminx G exG emaxx , eminy G eyG emaxy ( .

(11)

Let f be the mapping of D onto the interval E % R of the radiated energy. This mapping is not explicitly known but we are able to calculate the value of the energy for each vector of the region D by solving the direct problem of the undulator brightness.

The numerical solution of the direct problem is performed by the code MDK1F [6] which evaluates the energy produced by a sampled electron beam passing through the undulator. The values of the energy e  E and the emittances ex and ey are randomly

distributed around some mean values. We suppose that there are not two different values of the energy which correspond to the same emittance vector, i.e. that there exists a well-defined functional relation between the emittance of the beam and the undulator brightness. In this case the interval of the energies is well determined and

E 4f(D).

We define the inverse problem of the undulator radiation to be evaluation of the emittance vector u 4 (ex, ey) given the region D to which u belongs and one value e0of the radiated energy, e0E .

This inverse problem is in general close to the class of ill-conditioned problems. We will face cases in which a small deviation of the energy corresponds to a significant change of the emittance. As a consequence, we have to propose a numerical solution which does not increase the physical instability of the inverse problem. This instability can be expressed by the proposition that the function f : D KE is not one-to-one, i.e. the same value of the emitted energy corresponds to different emittance vectors. Our procedure distinguishes two cases. In the first the relation V f (u1) 2f(u2) V Ee implies V(u1) 2 (u2) V Ed, where e and d are the error tolerances of the energy and the emittance for some norms in E and D. The second case, which is also individuated by the procedure, occurs when the same energy is obtained from significantly different emittances.

This situation is linked to the main question whether the linear undulator radiation is a tool for solving the inverse emittance problem. This question lies out of our considerations but it is clear that we need a more rigorous definition of the notion of inverse problem and of its solution.

Suppose that f is a surjective mapping of the set of emittances D onto the interval of energies E. Let (El)l  Lbe a partition of E, i.e. (El)l  L is a family of subintervals of E

such that E 4

0

l  LEland ElO Em4 ¯ for l c m . For any u, u8 of D the statement “there

exists l  L such that f (u)  El and f (u8) El” is the statement of an equivalence

relation R. Then the mapping

F»DNRK L

(12)

is bijective. The inverse mapping

F21_{»L KDN}

R

(13)

exists and is bijective too.

The definition of the inverse problem of the undulator brightness is restricted to the mapping F21 _{which puts in correspondence the energy intervals of the family} (El)l  L and the equivalence classes of DNR which are the subsets of the set of

emittances D. For any l  L we take El4 [el2 Del, el1 Del], where elis the measured

or calculated mean energy value and Del some error tolerance estimation. Finally, if

the solution class F21_{(l) f [ (d(l) ]}

mod R contains the values of the emittance which are too different, we can restate the problem by putting E 4El, D 4 [ (d(l) ]mod R and

f : [ (d(l) ]mod RK El.

In our considerations the elements of E and D are random variables. The numerical code which solves the direct problem provides an error tolerance De by using the sampled variance of the energy value e  E . This energy value is a function of the effects produced by the electron beam passing through the undulator. The initial position r and velocity b of each one of N electrons modeling the beam are sampled for given Twiss parameters and given introduced emittance vector u0D . The mean values of r and b correspond to each number N of electrons. Using their sampled variances we obtain the sampled emittance u 4u(N). By definition u(N) cu0for any finite N.

We assume that the number N1is big enough and that N1, N2, R is an increasing sequence of numbers of electrons in the sampling of the same beam. Then we can consider the elements of the corresponding family

(u–1, E1), (u–2, E2) , R (14)

as independent random variables. The entire family expresses the influence of the perturbation of the emittance data on the brightness values and can be used as an indirect way to introduce into the model the error tolerance of the emittance data basis. Another way to have a family like (14) is to sample the introduced emittance u0within the region D. If u0( 1 ), u0( 2 ), R , is such sequence, then the model basis should be

(

0 , E(u( 1 )0 , N1)

)

(

)

where E is again a set of intervals determined by the tolerance of the calculated or measured value of the energy and N14 N24 R 4 N is the number of the beam-sampling electrons.

The first step of the numerical procedure is to approximate the function f : D KE into the region D. This approximation is then applied to solve the inverse problem for given test values e  E of the energy. Finally, within the class of candidate solutions we select one or more values according to criteria which interpret the physical peculiarities of the problem and the properties of the applied numerical methods.

We approximate the function f : D KE by the functional polynomial P(u) 4

!

over a set (ui, ei), uiD , eiE , i 41, 2, R, m solving the following least-square

problem [12] usually denoted as Ac Be: “given a real m3n matrix A of rank rGnGm and given a real m-vector e, find a real n-vector c minimizing the Euclidean length of

Ac 2e”, where c4 (c1, R , cn)T, e 4 (e1, R , em) and the elements of the matrix A are

given by

aij4 Wj(ui) , i 41, R, m; j41, R, n

(17)

and Wj are some (not necessarily linear) functions defined and assumed differentiable

in D.

We use the singular value decomposition (SVD) [13] method for analyzing matrices and questions involving matrices both for the direct and inverse problems.

The SVD of an m 3n real matrix A is a factorization of the form

A 4USVT_, (18)

where U is an m 3m orthogonal matrix, V is an n3n orthogonal matrix and S is an

m 3n diagonal matrix with elements sj , jfsjF 0 , j 4 1 , R , n , named singular

values of A.

The rank of the matrix A is expressed by the number of non-zero singular values of the diagonal matrix S. An m 3n matrix A with mFn is said to be of full rank if Rank (A) 4n or rank-deficient if Rank (A) En. In practice the small changes in a rank-deficient matrix can make all its singular values non-zero and hence create a matrix which is of full rank. This fact is used to generate the family (14); the sampled values ui, i 41, R, m are distributed close to the values of the introduced emittance u0.

The SVD provides information for the condition number of the matrix A Cond (A) 4 smax

smin ; (19) since sminG VAcV VcV G smax, (20)

it is clear that the condition number is a measure of the amplification of the errors in the model data. An orthogonal matrix has a condition number 1.

We replace the problem Ac Be by the problem A 8 c8Be, where A84AH and c 4Hc8. The matrix H is chosen in order to make the condition number of A small. For a non-singular matrix A there exists a matrix H such that Cond (AH) 41. Such a matrix results from the Householder triangularization of the matrix A.

Introducing the approximate size of the uncertainty in e we use a left multiplication of A8 and e by a diagonal matrix G with elements gii4 1 ODei, i 41, R, m. Here DeiD 0

is the standard deviation of the i-th component of the energy vector e.

Suppose that the least-square problem Ac Be is solved and a vector c of coefficients of the polynomials P(u) is selected.

We repeat the same procedure for another Q-approximation, Q(u) 4

!

of the function f : D KE.

Then for a given test energy e0E with error tolerance De0we consider a non-linear problem

G(e0) u 40 (22)

for the unknown emittance vector u 4 (ex, ey). Here G(e0) is a mapping of D into D and

G(e0) u 4 [p(e0, u), q(e0, u) ]T, (23) where p(e0, u) 4

!

!

We approximate the solution of (22) by applying two different methods.

Let Dp be the subset of the set D of emittances which contains all solutions of

eq. (24) when the energy e0 runs the interval [e02 De0, e01 De0].

If Dq is the corresponding set for eq. (25), then we consider the intersection I 4

DpO Dq as a solution of the perturbed problem (22). Providing this intersection with

some structure we could observe the density of the solution vector u within I. The intersection I corresponds to one class (mod R) described above in the definition of the reduced inverse problem. In the examples we refer to this method as “the intersection”. For an approximation of the solution within a neighborhood of the model energies

ei, i 41, R, m we apply Newton’s method [14]

uk 11_{4 u}k_{2 [G 8 (u}k) ]21_Guk

, _{k 40, 1 , R}

(26)

where G8(u) denotes the Jacobian matrix and k is the iteration number. We solve the linear system

G 8(u)k

₎

y 42Guk

(27)

and add the correction y 4uk 11_{2 u}k _{to u}k_{. If e}

0E is the test energy and De0 its standard deviation, then we select the initial approximation value of u to be ui, where i

is the index of the nearest value ei of the model energy to e0. The iterative procedure terminates for an approximation uk _{such that}

[p(e0, uk)21 q(e0, uk)2]1 O2Oe0G De0. (28)

The linear system (27) is solved as a full-rank problem applying SVD for a condition number check at each iteration.

vectors ui, i 41, 2, R, m and functions Wj, xj, j 41, R, n must be selected in the P

and Q approximations of the function f : D KE.

Note first that for ex4 ey the function f is nearly linear. This function is smooth

enough for a constant ratio exOey. Let the functions W and x be such that the

approximation of f at the model points of u has an uncertainty less than the error tolerance of the energy for these values. Then for a sufficient density of the model vectors u in the region D we observe a local convergence of Newton’s method. The numerical procedure solving the inverse problem is stable for the relative deviation DelOel of the energy up to some percent. The error tolerances of the energy in our

examples correspond to the maximum ( 3 s) sampled deviation. Their values are usually too high. For the inverse-problem calculation we take the deviation of the approximated energies as more realistic values for Del.

5. – Low-energy, exB ey inverse-problem calculation and concluding remarks

The next example treats the case exOey near to 1. The same procedure (the same

functions W and x) has been used for a solution of the inverse problem. For ex4 ey a

solution can be found with less computational effort by using the almost linear dependence of the energy on the emittances [15].

The undulator and beam parameters are given in table VI. Figure 7 shows the dependence of the emittance on the energy for the frequence of the first harmonic; table VII contains the numerical results.

The energy value is taken here to be the brightness value (in photonsOrad) at the maximum of the first harmonic peak.

We see that the procedure provides a good reproduction of the emittances for the model points and that the relative deviation at the test point is less than 1%. Figure 8 shows the dependence of the solution region dimensions on the perturbation DelOel.

The increase in the diameter of the classes of the solution follows that of the physical behavior.

For this example the linearity of the relation between the energy and the emittance allows a less rigorous selection of the initial approximation point for Newton’s method.

TABLEVI. – Low-energy, exB ey, beam and undulator parameters.

Undulator parameter K Undulator period lu(cm) Number of periods k2 4.4 50 Electron energy (MeV)

Energy spread Twiss parameters: ax4 ay bx(cm) by(cm) 30 4 .3 Q 1023 0 110 73.6 On-axis observer

Fig. 7. – Emittance vs. radiated energy (parameters in table VI). First-harmonic frequency region, exB ey. Continuous line: ex; dotted line: ey.

TABLEVII. – Numerical data of the solution for the low-energy, exB ey, inverse problem.

Radiated energy (photonsOsOrad) 3104 Sample emittance (cmOrad) exQ 1024 eyQ 1024 Reproduced emittance (cmOrad) Newton’s method exQ 1024 eyQ 1024 intersection Model points A 1 .428 60.050 0.9581 0.9523 0.9585 0.9526 0.9640 0.9582 C 1 .408 60.051 1.001 0.9946 1.001 0.9937 0.9995 0.9935 D 1 .388 60.052 1.043 1.037 1.043 1.037 1.047 1.041 E 1 .367 60.053 1.086_1.079 1.087_1.080 1.082_1.076 F 1 .348 60.053 1.128_1.122 1.128_1.122 1.130_1.123 B 1 .328 60.054 1.171_1.164 1.171_1.164 1.165_1.158 Test point T 1 .378 60.052 1.065 1.059 1.065 1.059 1.059 1.052

It is obvious that the inverse problem defined in this paper can be solved in a different way. We selected a method which seems to answer the natural questions about the practical application of a linear undulator radiation as a tool for an electron beam diagnostics.

Fig. 8. – a) Density of the solutions in the region of emittances (parameters in table VI). First-harmonic frequency region: exOeyB 1 . Test point T of table VII. Intersection method with

perturbation DeOe40.1%. b) The same as in a), perturbation 1%.

The presented procedure

1) realizes a one-to-one correspondence between the energy intervals and the subregions of the two-dimensional region of emittances;

2) for any energy value provides a point of convergence within the set of emittances.

We stress again that the diameter of the subregion of solutions can coincide with that of the entire region and that the point of convergence is not unique in general. Fortunately, the functional relation between the energy and the emittance is not very complicated. Using appropriate approximations of this relation and a sufficient number of model points within the region of emittances we can expect a reliable evaluation of the emittances corresponding to a given energy interval.

In particular, the examples confirm that the points of convergence of Newton’s method belong to the subregions of the most probable solutions given by the intersection method.

R E F E R E N C E S

[1] LUMPKIN A., YANG B., CHUNG J., DEJUS R., VOYKOV G. and DATTOLI G., Preliminary calculations on the determination of APS particle-beam parameters based on undulator radiation, in Proceedings of the Particle Accelerator Conference, Dallas (Texas) May 5-9, 1995, Vol. IV (American Physical Society) 1995, p. 2598.

[2] XU LAI B. and CERRINA F., Nucl. Instrum. Methods A, 243 (1986) 337.

[3] XULAIB. and VICCAROP. J., APS Undulator and Wiggler Source: Monte Carlo Simulation, NLOPAOTB-1 (1992).

[4] ELLEAUME P., J. Phys. (Paris), 44 (1983) PC1-333.

[6] CIOCCI F., DATTOLI G., GIANNESSI L. and VOYKOV G., Numerical and analytical computations of undulatorOWiggler radiation, in Proceedings of the 1994 EPAC, Vol. 2 (World Scientific, London)), 1994, p. 1241.

[7] CAIZ. et al., Rev. Sci. Instrum., 66 (1995) 1859.

[8] TARAZONA E. and ELLAUMEP., Rev. Sci. Instrum., 66 (1995) 1974.

[9] SOBOL F. M., The Numerical Methods Monte Carlo (Nauka, Moscow) 1975. [10] DATTOLI G., VOYKOV G. K. and CARPANESE M., Phys. Rev. E, 52 (1995) 6809. [11] JACKSON J. D., Classical Electrodynamics (Wiley, New York) 1975.

[12] LAWSON C. L. and HANSON R. J., Solving Least Square Problems (Prentice-Hall, Inc., Englewood Cliffs, N.J.) 1974.

[13] FORSYTHE G., MALCOLM M. and MOLER C., Computer Methods for Mathematical Computations (Prentice-Hall, Inc., Englewood Cliffs, N.J.) 1977.

[14] ORTEGA J. M., Numerical Analysis (Academic Press, New York) 1972. [15] DATTOLI G. and VOYKOV G., Nuovo Cimento B, 111 (1996) 743.