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A method of determining all the zeros of a generalized polynomial with respect to an arbitrary Chebyshev system
U.S.S.R. Comput.Maths.Math.Phys.,Vo1.27,No.l,pp.9-13,1987 Printed in Great Britain
0
0041-5553/87 $10.00+0.00 1988 Pergamon Press plc
A METHOD OF DETERMINING ALL THE ZEROS OF A GENERALIZED POLYNOMIAL WITH RESPECT TO AN ARBITRARY CHEBYSHEV SYSTEM* KH. I. SEMERDZHIYEV
and S.G. TAMBUROV
An iterational method for simultaneously finding all the zeros of a certain generalized polynomial with respect to an arbitrary Chebyshev system, if the multiplicities of the zeros are given, is considered. The method is more general than the ones previously described, which relate only to algebraic, trigonometric, and exponential polynomials. Its quadratic convergence is established, and the method is realized on a computer. Suppose
where
we are given the generalized
is a Chebyshev {cph(z)}?_,',,
system
polynomial
in the interval
[a,b]. The fundamental
properties
of
Chebyshev systems have been presented fairly fully in /l, 2/. Without loss of generality, we will assume that the coefficient in front of (Pi is equal to unity. The problem of finding zeros of polynomial (1) in the interval [a,b] can be formulated quite simply, but difficulties arise when trying to solve it; these are due mainly to the non-linearity of polynomial (1). Methods of simultaneously obtaining all the zeros of polynomial (1) have been developed in /3-6/, for the case when the system [~p,(s)}~'_~ is algebraic, trigonometric, or exponential. A method of the same type for polynomial (1) was obtained in /7/ in the case of an arbitrary Chebyshev system ('pI(z))fYO,but when all the zeros z,,...,z~ are simple. It turns out that if the functions
(cpa(s)}~=, are fairly continuous,
can be generalized to the case when the multiplicities nomial (1) are arbitrary: Z&a,=N. We will introduce
the following
a,,...,&
the method of zeros
described
z~,...,z,
in /7/
of poly-
notation: cpo(s).. .'PN(d 'co(4 '. .cP.&l)
%‘(d . . .TN M~/~;;;;+
(4
:;;;.:fjpff
,
&~).::&z~) c+pqz,)
. . .cp($qz,)
~,;1~~~:I_,=~~~~,~~~~~~~~;~~~,.
The following iteration method was proposed (1) in the case when ai=...=c&,=l: IA+11 51
in /4/ for determining
=z* [*‘-P~(z~~~) [Q~‘(z:~~)]-‘,
i=l,
2,. . . ,N,
all the zeros of polynomial
k=O, 1,. . . ,
where Q,(z) is a generalized polynomial with respect to a specified Chebyshev system of basis numbers while the functions, the coefficient of which in front of TN(X) is unity, [kl [kl are used as the zeros. The initial approximations z:"',...,z~',generally speaking, zi:e'df;%.-ent arbitrary numbers in the interval [a,b]. Specific computing schemes were obtained in /4-S/, and their convergence was proved only for the case of algebraic, trigonometric, and This method Here we describe a computing scheme in the general case. exponential systems. is easier to realize on a computer than the methods described in /3, 5/. Consider the following iterational process: 27,1,16-21,1987 *Zh.vych~sl.Mat.mat.F~z., 9
10
(2) where
To find the derivative
Qp)(r)
it is sufficient
to differentiate
a+ times the first row of
the determinant in the numerator of Qdx). If we We will transform (2) toa form that is more convenient for proving the theorem. subtract the root z, from both sides of Eq.(Z) and use the fact that P$-' (z,)=O when i=l,2, . ..*m.
we obtain N Xi~~+"_z,~x~~~-z*_(_~)~[p~-~~ (I~"')-P'"~-"(zi)][~~)(x:l')]-'.
Applying
the theorem
on finite
11+11 -_z~=(~~*‘_~~) +t
[&(_~)“+“~$d($‘)
cpp” (Lp)
(s{Xl_ zi) I_
we can convert
increments,
. . . I$$
this equation
[Qy)
(2p)
/ py
j
(3)
($‘)I-+
(@I)
------;;-------I---i----
I
to the form
[#)
(z\k]Jl-’
,
‘0’
I
where giml=(zirkl, z<), i=l, 2,. .., m.
Rh(z)=QI(r)detB,,
we will convert the determinant in the numerator on the right-hand side of (3) as follows. We multiply the first N columns by --a~,...,-UN-~ respectively, and add them to the (Nfl)-th column, which, as a result, is converted to the form
~,,(2~l)_PN(Z~~),...,(P >-*I (SF.“‘)-pglrlf The expression
in braces
in (3) can then be written @i)(@I). . . ,gj;
90 ----
--------
(z!“l) 1 p!jp) (zy) ---,_________________
; ,.......
I’,
where
we
have
once
p&. %-1)
again used the equations
apply the theorem on finite written in the form
increments
[Pi”)
in the form
-
py
(#I)
N . . . . . . . ...!
; , I
) 1’.
(ql)
P, (zy’) - p, (ZI)
i
I pii4
Bk
(I’
_
pb%-1)(21)
.._.____.....
fi . . . . . . . . . .
cRk
@i) (=I rU)l-1,
(4)
..
@)
-
p,
(Q
(pl) m
_
p(c$n-~) (% ) m
P$'(z,)=O, i=& 2,...,m, r=O, I,...,&-1. We again
to the last column in (4), after which it can be
(x;:“‘-g:“‘), P/(&f’) (x:kl-Z,),...
P’,““(g) (x:*’ -z,), . . . , P,‘(~!z) (x7-z,), . . . PC?’ (b2,“)(xdkl) I', where qr[k'~(xll'l, &P) , and hence it follows that 1=1,2,. . , oh. Hence, (2) can be represented in the form
?)~[k'~(z~'*', ZC), and
w @i[kl ) 1, p(%+l) ($1) Q’ (zp) . . . (PN-* N ________-_____,---------------, /
%
@{bl _
@l~(xcml, 2,). i=l,2 1. . . , m,
@I)
PN’@I) (zf@’ - 21)
PgJ (cg, (zy -II) 1I__ __...p.~;;“;;“c,Iri...s”) !
m1mm
......
[R(P’) 6 (+!q-‘, I
(5)
11
Theorem.
Moreover,
OCcc1,O
Suppose
suppose
and
cp, @+l) (z), i=l, 2,...,m
the derivatives
Iv!” (x) I<&
Vz=[a, bl,
-0,
(a,+l).
1,. . . , max
and constants
M.,
exist such that
s=O, 1,. . . , N,
I.z(Crn
Suppose,
at the same time, that the following
inequalities
are satisfied:
(7)
Then, if the following
inequalities
are satisfied
IO,_ &I Gcq, 1x1
then for each
k=O,l,..., the following
for the initial
inequalities
LhI_z,(
of order a, of a certain
(rPr(4)Eo8 which
agrees with the polynomial
i=l, 2,...,m.
polynomial
C=O,
the function
with respect
apart from a non-zero
P,(z) mo?.
(9)
with a left and
generalized
D
(8)
hold:
Note. If c belongs to a fairly small interval is increasing and L(c)>O. In fact, the determinant
is a derivative
approximations:
i-l, Z,...,m,
L(C)
to the system
factor
. .t (P&L1
!A, . . ., I/,
I
al,.
. .,am
1
(since the system (mA(z))'~' is Chebyshev, and z,fzJ when ~fj). But the polynomial al,...,a, respectively, and, consequently, Py zeros z~,...,z~ with multiplicities
P,(Z)
has (ZJZO, i=l,
2, ...( m. It follows from the continuity of all the determinants that .L(c)>O. k=O, inequality (9) is We will prove the theorem by mathematical induction. When identical with (8). Suppose (9) is satisfied for any natural k. Then, for the denominator on the right-hand the numerator
Bearing
in
side of
(5) we have
I@~'($' ) I"L(c)>O.
(5) we will use Hadamard's
the inequalities
(6) in mind,
(7) and
the absolute
value of
inequality
and also the estimates
PP'(xIk'), i=l,Z,. . ..m. and also inequalities
To estimate
obtained
from them
r=O, 1I...,max (a,+l),
1
*
(91, we obtain
The theorem is proved, Note that in the special case when CI,=...=CL,=~, we obtain from (2) the method described In the more special case when cpk(x)=x*,k=O,l,...,N, the determinants in (2) are in /7/. Vandermonde determinants, and from (2) we obtain the method proposed in /6/ $+I =,lkl - ~~($1) 1
fi (rI"J_ $I)]-I, [j=l j#i
i=1,2,...,N,
k-=0,1,....
12
Inequality (7) shows that the convergence of the method (2) is local. But, condition (7) is obtained using Hadamard's inequality. Ve will give some numerical examples which show that method (2) also converges for a non-local choice of initial approximations. Table 1
Table
Table
2
3
We have used the following abbreviated notation in the tables: if the number 1 is encountered when writing a given number n times in succession, it is denoted by (nil). The calculations were carried out on an ES1020 computer. Example
1.
The polynomial P&)=x’-15x’-14xJ+36sz+24z-32
aa=4 with multiplicities has zeros z,=l, z,=-2, and The numerical results are shown in Table 1,
a,=&
aa=
and
us=1
respectively.
Example 2. The coefficients a,, a, and a, of the polynomial Es(x)~ao-fa,e'i/5+a2ex'2+erare chosen so that they have the zeros z,=--1 and &=2 with multiplicities of ccl=2 and a,=1 respectively. The numerical results of several successive iterations are shown in Table 2. Exam+?
are given
3.
The numerical
results
for the trigonometric
polynomial
in Table 3.
Example 4. For the system of basis functions (1,x2,sinsx, e-',(i+z')-') , we constructed the generalized polynomial P,(x)=aO+a,zZ+az sin3zfase-‘+- (l+z*)-‘, the coefficients of which are determined so that they have simple roots x,=-0.5, x2=1, x,=1.5, x,=3. The results of several successive iterations are shown in Table 4. For comparison with Newton's classical method , modified to determine multiple roots of the polynomial P,(x)
XiI*+"= x:*'- aiP,(x:“) we initial
results
[PN’(z:“)]-‘,
for
i=l,
2,. . , m, k=O, 1,. . ,
approximations. methods
which
(10)
1 obtained
of simultaneously compared with
finding in
the roots are found individually.
proposed
in /9/, Table 4
R
-0.65 -0.746 -0.633 -0.531 -0.502 -0.5(4*0)9
1.75 1.693 1.612 1.534 1.501
2.70 2.763 2.354 2.947 2994
13
Table 5
$1
WI
R
=1
-30.333 -29.222 -19.536 4
-13.104 -8.653 -6.069 -4.283
2.923
-2.154 -2.006 -2.(4*0)1 -1.(5*9)5 -1.(4*9)6 Ip;"'"
I fi
5.215 4.632 4.245 :z: $;*w&
-3.179
-2.547 -2.224 -2.063 -2.028 -2.nO9 -2.003 -2.001 -2.(3+0)1
-i(S.O)l -1.(3.9)8 yu;"'" -iO33 -2.(3*0)3 -2.(7*0)1 -2.(7.0)1 -2.(7.0)1 -2.(7=0)1
4.&O) ::[::*i] j:&;:;] 4.(15:0, 4.(15-O) 4.(15eO) ye&*;{ . l
REFERENCES 1. 2. 3. 4.
5.
6. 7. 8. 9.
CAR.LIN S. and STADDEN
V., Chebyshev Systems and their Application in Analysis and Statistics, Nauka, Moscow, 1976. SCHUMAKER L.L., Spline functions: Basic theory. N.Y.: J. Wiley & Sons, 1981. SEMERDZHIYEV Kh.I., A method for simultaneously obtaining all the roots of an algebraic equation if their multiplicity is given, Dokl. BAN, 35, 8, 1057-1060, 1982. ANGELOVA E.D. and SEMEPDZHIYEV Kh.I., Methods for the simultaneous approximate determination of the roots of algebraic, trigonometric and exponetial equations, Zh. Vych. Mat. mat. Fin., 22, 1, 218-223, 1982. Methods of simultaneously obtaining all the roots of MAKRELOV I.V. and SEMERDZHIYEVKh.I., algebraic, trigonometric and exponential equations, Zh. vych. Nat. mat. Fiz., 24, 10, 1443-1453, 1984. DOCHEV K. and BYRNEV P.O., Some modifications of Newton's method for the approximate solution of algebraic equations, Zh. vych. Mat. mat. Fiz., 4, 5, 915-920, 1964. MAKRELOV I.V. and SEMERDZHIYEV Kh.I., Dochev's method for a generalized polynomial with respect to an arbitrary Chebyshev system, Dokl. BAN 38, 10, 1263-1266, 1985. On the convergence of two methods for the simultaneous MAKRELOV I. and SEMERDZHIYEVKh., finding of all roots of exponential equations IMA J. Numer. Analys. 5, 2, 191-200, 1985. of the SEMERDZHIYEV Kh.1. and TAMBUROV S.G., A method of determining the multiplicities zeros of algebraic polynomials, Dokl. BAN, 37, 9, 1143-1145, 1984.
Translated
by R.C.G.
0
0041-5553/87 $10.00+0.00 1988 Pergamon Press plc
A METHOD OF DETERMINING ALL THE ZEROS OF A GENERALIZED POLYNOMIAL WITH RESPECT TO AN ARBITRARY CHEBYSHEV SYSTEM* KH. I. SEMERDZHIYEV
and S.G. TAMBUROV
An iterational method for simultaneously finding all the zeros of a certain generalized polynomial with respect to an arbitrary Chebyshev system, if the multiplicities of the zeros are given, is considered. The method is more general than the ones previously described, which relate only to algebraic, trigonometric, and exponential polynomials. Its quadratic convergence is established, and the method is realized on a computer. Suppose
where
we are given the generalized
is a Chebyshev {cph(z)}?_,',,
system
polynomial
in the interval
[a,b]. The fundamental
properties
of
Chebyshev systems have been presented fairly fully in /l, 2/. Without loss of generality, we will assume that the coefficient in front of (Pi is equal to unity. The problem of finding zeros of polynomial (1) in the interval [a,b] can be formulated quite simply, but difficulties arise when trying to solve it; these are due mainly to the non-linearity of polynomial (1). Methods of simultaneously obtaining all the zeros of polynomial (1) have been developed in /3-6/, for the case when the system [~p,(s)}~'_~ is algebraic, trigonometric, or exponential. A method of the same type for polynomial (1) was obtained in /7/ in the case of an arbitrary Chebyshev system ('pI(z))fYO,but when all the zeros z,,...,z~ are simple. It turns out that if the functions
(cpa(s)}~=, are fairly continuous,
can be generalized to the case when the multiplicities nomial (1) are arbitrary: Z&a,=N. We will introduce
the following
a,,...,&
the method of zeros
described
z~,...,z,
in /7/
of poly-
notation: cpo(s).. .'PN(d 'co(4 '. .cP.&l)
%‘(d . . .TN M~/~;;;;+
(4
:;;;.:fjpff
,
&~).::&z~) c+pqz,)
. . .cp($qz,)
~,;1~~~:I_,=~~~~,~~~~~~~~;~~~,.
The following iteration method was proposed (1) in the case when ai=...=c&,=l: IA+11 51
in /4/ for determining
=z* [*‘-P~(z~~~) [Q~‘(z:~~)]-‘,
i=l,
2,. . . ,N,
all the zeros of polynomial
k=O, 1,. . . ,
where Q,(z) is a generalized polynomial with respect to a specified Chebyshev system of basis numbers while the functions, the coefficient of which in front of TN(X) is unity, [kl [kl are used as the zeros. The initial approximations z:"',...,z~',generally speaking, zi:e'df;%.-ent arbitrary numbers in the interval [a,b]. Specific computing schemes were obtained in /4-S/, and their convergence was proved only for the case of algebraic, trigonometric, and This method Here we describe a computing scheme in the general case. exponential systems. is easier to realize on a computer than the methods described in /3, 5/. Consider the following iterational process: 27,1,16-21,1987 *Zh.vych~sl.Mat.mat.F~z., 9
10
(2) where
To find the derivative
Qp)(r)
it is sufficient
to differentiate
a+ times the first row of
the determinant in the numerator of Qdx). If we We will transform (2) toa form that is more convenient for proving the theorem. subtract the root z, from both sides of Eq.(Z) and use the fact that P$-' (z,)=O when i=l,2, . ..*m.
we obtain N Xi~~+"_z,~x~~~-z*_(_~)~[p~-~~ (I~"')-P'"~-"(zi)][~~)(x:l')]-'.
Applying
the theorem
on finite
11+11 -_z~=(~~*‘_~~) +t
[&(_~)“+“~$d($‘)
cpp” (Lp)
(s{Xl_ zi) I_
we can convert
increments,
. . . I$$
this equation
[Qy)
(2p)
/ py
j
(3)
($‘)I-+
(@I)
------;;-------I---i----
I
to the form
[#)
(z\k]Jl-’
,
‘0’
I
where giml=(zirkl, z<), i=l, 2,. .., m.
Rh(z)=QI(r)detB,,
we will convert the determinant in the numerator on the right-hand side of (3) as follows. We multiply the first N columns by --a~,...,-UN-~ respectively, and add them to the (Nfl)-th column, which, as a result, is converted to the form
~,,(2~l)_PN(Z~~),...,(P >-*I (SF.“‘)-pglrlf The expression
in braces
in (3) can then be written @i)(@I). . . ,gj;
90 ----
--------
(z!“l) 1 p!jp) (zy) ---,_________________
; ,.......
I’,
where
we
have
once
p&. %-1)
again used the equations
apply the theorem on finite written in the form
increments
[Pi”)
in the form
-
py
(#I)
N . . . . . . . ...!
; , I
) 1’.
(ql)
P, (zy’) - p, (ZI)
i
I pii4
Bk
(I’
_
pb%-1)(21)
.._.____.....
fi . . . . . . . . . .
cRk
@i) (=I rU)l-1,
(4)
..
@)
-
p,
(Q
(pl) m
_
p(c$n-~) (% ) m
P$'(z,)=O, i=& 2,...,m, r=O, I,...,&-1. We again
to the last column in (4), after which it can be
(x;:“‘-g:“‘), P/(&f’) (x:kl-Z,),...
P’,““(g) (x:*’ -z,), . . . , P,‘(~!z) (x7-z,), . . . PC?’ (b2,“)(xdkl) I', where qr[k'~(xll'l, &P) , and hence it follows that 1=1,2,. . , oh. Hence, (2) can be represented in the form
?)~[k'~(z~'*', ZC), and
w @i[kl ) 1, p(%+l) ($1) Q’ (zp) . . . (PN-* N ________-_____,---------------, /
%
@{bl _
@l~(xcml, 2,). i=l,2 1. . . , m,
@I)
PN’@I) (zf@’ - 21)
PgJ (cg, (zy -II) 1I__ __...p.~;;“;;“c,Iri...s”) !
m1mm
......
[R(P’) 6 (+!q-‘, I
(5)
11
Theorem.
Moreover,
OCcc1,O
Suppose
suppose
and
cp, @+l) (z), i=l, 2,...,m
the derivatives
Iv!” (x) I<&
Vz=[a, bl,
-0,
(a,+l).
1,. . . , max
and constants
M.,
exist such that
s=O, 1,. . . , N,
I.z(Crn
Suppose,
at the same time, that the following
inequalities
are satisfied:
(7)
Then, if the following
inequalities
are satisfied
IO,_ &I Gcq, 1x1
then for each
k=O,l,..., the following
for the initial
inequalities
LhI_z,(
of order a, of a certain
(rPr(4)Eo8 which
agrees with the polynomial
i=l, 2,...,m.
polynomial
C=O,
the function
with respect
apart from a non-zero
P,(z) mo?.
(9)
with a left and
generalized
D
(8)
hold:
Note. If c belongs to a fairly small interval is increasing and L(c)>O. In fact, the determinant
is a derivative
approximations:
i-l, Z,...,m,
L(C)
to the system
factor
. .t (P&L1
!A, . . ., I/,
I
al,.
. .,am
1
(since the system (mA(z))'~' is Chebyshev, and z,fzJ when ~fj). But the polynomial al,...,a, respectively, and, consequently, Py zeros z~,...,z~ with multiplicities
P,(Z)
has (ZJZO, i=l,
2, ...( m. It follows from the continuity of all the determinants that .L(c)>O. k=O, inequality (9) is We will prove the theorem by mathematical induction. When identical with (8). Suppose (9) is satisfied for any natural k. Then, for the denominator on the right-hand the numerator
Bearing
in
side of
(5) we have
I@~'($' ) I"L(c)>O.
(5) we will use Hadamard's
the inequalities
(6) in mind,
(7) and
the absolute
value of
inequality
and also the estimates
PP'(xIk'), i=l,Z,. . ..m. and also inequalities
To estimate
obtained
from them
r=O, 1I...,max (a,+l),
1
*
(91, we obtain
The theorem is proved, Note that in the special case when CI,=...=CL,=~, we obtain from (2) the method described In the more special case when cpk(x)=x*,k=O,l,...,N, the determinants in (2) are in /7/. Vandermonde determinants, and from (2) we obtain the method proposed in /6/ $+I =,lkl - ~~($1) 1
fi (rI"J_ $I)]-I, [j=l j#i
i=1,2,...,N,
k-=0,1,....
12
Inequality (7) shows that the convergence of the method (2) is local. But, condition (7) is obtained using Hadamard's inequality. Ve will give some numerical examples which show that method (2) also converges for a non-local choice of initial approximations. Table 1
Table
Table
2
3
We have used the following abbreviated notation in the tables: if the number 1 is encountered when writing a given number n times in succession, it is denoted by (nil). The calculations were carried out on an ES1020 computer. Example
1.
The polynomial P&)=x’-15x’-14xJ+36sz+24z-32
aa=4 with multiplicities has zeros z,=l, z,=-2, and The numerical results are shown in Table 1,
a,=&
aa=
and
us=1
respectively.
Example 2. The coefficients a,, a, and a, of the polynomial Es(x)~ao-fa,e'i/5+a2ex'2+erare chosen so that they have the zeros z,=--1 and &=2 with multiplicities of ccl=2 and a,=1 respectively. The numerical results of several successive iterations are shown in Table 2. Exam+?
are given
3.
The numerical
results
for the trigonometric
polynomial
in Table 3.
Example 4. For the system of basis functions (1,x2,sinsx, e-',(i+z')-') , we constructed the generalized polynomial P,(x)=aO+a,zZ+az sin3zfase-‘+- (l+z*)-‘, the coefficients of which are determined so that they have simple roots x,=-0.5, x2=1, x,=1.5, x,=3. The results of several successive iterations are shown in Table 4. For comparison with Newton's classical method , modified to determine multiple roots of the polynomial P,(x)
XiI*+"= x:*'- aiP,(x:“) we initial
results
[PN’(z:“)]-‘,
for
i=l,
2,. . , m, k=O, 1,. . ,
approximations. methods
which
(10)
1 obtained
of simultaneously compared with
finding in
the roots are found individually.
proposed
in /9/, Table 4
R
-0.65 -0.746 -0.633 -0.531 -0.502 -0.5(4*0)9
1.75 1.693 1.612 1.534 1.501
2.70 2.763 2.354 2.947 2994
13
Table 5
$1
WI
R
=1
-30.333 -29.222 -19.536 4
-13.104 -8.653 -6.069 -4.283
2.923
-2.154 -2.006 -2.(4*0)1 -1.(5*9)5 -1.(4*9)6 Ip;"'"
I fi
5.215 4.632 4.245 :z: $;*w&
-3.179
-2.547 -2.224 -2.063 -2.028 -2.nO9 -2.003 -2.001 -2.(3+0)1
-i(S.O)l -1.(3.9)8 yu;"'" -iO33 -2.(3*0)3 -2.(7*0)1 -2.(7.0)1 -2.(7.0)1 -2.(7=0)1
4.&O) ::[::*i] j:&;:;] 4.(15:0, 4.(15-O) 4.(15eO) ye&*;{ . l
REFERENCES 1. 2. 3. 4.
5.
6. 7. 8. 9.
CAR.LIN S. and STADDEN
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Translated
by R.C.G.