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Interval versions of some procedures for the simultaneous estimation of complex polynomial zeros
.
3.
s
-2
4.
(5.1)
l t=
s
i
)
.
5=
=
v
M. MONSIAND M. A. WOLFE
The pdure
mss
with
m, mss, and mws have been 101using tit&mm for nx%ngth complex Wls ms ad PI on a VAX U-785 caqm&m. Numerical aumdtsare presented for six examples in Tables l-6, in rr,
Is,
rss, mss,
UpT,
(1W) ( - aso) @So)
( - LO) ( - 108)
wo W) ao)
ih
wn ( - Lo) (OS 1)
(0; -1) (- w (- 1s- 2)
TABLE& 1331
(o,O) (-ii-l)
(-10s -10)
(OS-5)
We (oa
8
9 10
TABLE4.
[36]
0
(GO)
1 2 3 4 5 6 ? 8 9 l0
( - 1W)
(0,101 (0, -10) (10110) (10, - lc$ ( - 10,lO) ( - 10, - 10)
mo) (OSB
TABLE5
[30]
Pa aad P!! mspectidy to satisfy a given stopping givextinitial data Then the rat&s in Tables 7-10 and other
TABLEi7 Cputime(srpx)
~*oo W4)
5-3(4) 5*4(4)
TABLE9 Cputime(se@ -A
l&7(4) =2(3) 22.213) 22.1(3) fW3) 7*5(3)
1
2 3 4 5 6
683) 5*30 9.1(2) 9.1(2) 4.1(3) 4*3(3)
TABLE 10 Cputime(set) 15.4(3) 11.0(2) 19.7(2) 19.7(2) 7*3(3) 7JK3)
5*4(2) 582) 6.0(l) $.0(l) 3*2(2) 3*3(2)
resuks which are not reported in this paper suggest that IS < IPS,
IT
IPT c IPSS 6 IPRSS,
that of the
IRSS < IPRSS,
ISS < IPSS, an
IPS a wss.
208
M. MONSIAND M. A. WOLFE
wm, the prwxhm mim with rk=3 (VbO) is likely to be the most efficient for bounding the simple zems of a complex polynomial. REFERENCES 1 0. Abed& Iteratkmmethods for finding all zeros of a polynomial simultaneously, n!wh. cbmp. 27:33%3u (lQ73). htmbcth to Zntmal Cbmputations, Acacbmic, 2 G. Ah&&l and J. II-, NewYork, 1983. 3 G. Al&d and J. Herzbqpr, On the convergencespeed of some algorithmsfor the shhwous appro;rdmation of polynomial zeros, SLAM1. NW. Anal. lu37-243 (lQ74). 4 D. Brams and K. P. Hadeler,Sim~eous inchuion of the zeros of a polynomial, NunMIr.M&. 2&l&16!5 (m73). and App-, McGraw-EmICom 5 R v. ChurcIbi&GnRpkx v&&k Th 1mQ. 6 A. j. Cole and R Morrison, ‘Riplex: A system for interval arithmetic, s&mm-e antf Eajmhwe 12:341-350 (1982). 7 L W. E&h, A mod&d Newton method for polynomials, Comm. Assoc. cbf#q&. Mb&. 10=101-108(1967). 8 M. R Famer and G. Mzou, A classof iteration functions for improviig, simm, ato the zeros of polynomials, BZT 95:250-258 0. 9 I. Gargmtini, Phrabl squue mot iterations,in ZntrmolMdhemcrtics(K. Nickel Ed.), bcture Notes in Computer Science 29, Springer, Heidelberg, 1975, pp. m-204. 10 I. Gaqptini, Pam&l Laguem iterations: The complex case, Numer. Math. 28:317-323 (1976). 11 I. Gargmtini, Fbther applications of circuh s@unetic: ScMder-like algorithms with error bounds for finding zeros of polynomials, SZAM1. Namer. Ana& l&497-510 (1978). 12 I. Gaqpntini, An ap#cation of interval mathematics:A polynomial solver with degree four convergence, Fw ZntmaWwchte 81(7):x5-25 (1981). 13 I. -tini and P. bnrici, Circuk arithmetic and the determination of polynomial Z~MS,NW. Math. 18905-320 (1912). 14 G. Glatx, Newton-Algorithmen zur Restimmung von Polynomwbzeln unter Verwendung bmplexer bisarithmetik, in Znti Mathematim (K. Nihl, Ed.), Lecture Notes in Computer Science 29, Springer, Heidelberg, 1975, pp. 205-214. 15 E. Hansen, M. Patrick,and J. Rusnak, Some modificationsof Laguerre’smethod, BZT 17&B-417 (lQ77). 16 PObhi, AppEced(LndChnpukrfsoMiCompkx Anal@s, Vol. 1, Wiley, New York, lQ74. 17 0. IcemeI, Eh Gemwchiw zur Berechnmg der Nullstellen von Polynoznen, Nlcmer. M&k 8:~294 (19$@ and P. S engen von Polynom-Nullsteilen,in
19
20 21 22 23 24 25
26 27 28
29
30
31
32
33 34 35 36
2~~ ~U~~~ (K. MCkeLEd.), Lecture Notes in ~orn~ter Science29* springer,lwd~, 1@75, p3*223-228. R. Morrison,A. js Cole, P, Raihy, MeA.Wolfe, and J. M. Shearer*Egperienoein usbg a hi& level language which supports inter\tuttarithmetic, in ~~~~~ 6th CI 8q%qX&~&*. u* ~~~ ~~~ (T. R. N. Rao and P. I(;onrenrp, Ms.), Aiarhls~ libnmah, lQs3, pp. l-li3. G. V. ~V~O~C ad M.S. P&~vic, On the cxmqeme ofa modifiedme&d for simultaneous finding of plynomial zeros, Cafe 30:171-178 (1983). M. MO& and MO A. WoEe, An a&ritbm for the simultan~ fnclusionof re& polynomial zeros~Appt. Mu&. Cbmput., to appe;ur, A. Neumaier, An intervalversion of the secant method, BH’ 24:366-372 (X9&4). ALNeumaier, Intervaliterationfor zeros of systemsof equations, BI11‘ 25956-273 (1985). J. M. Ortega and W. C. Rheinbok#t,IterativeSulutim of llkmhem &uatiom in SmeraZVaniablarr, Academic,New York,1970. M. S. Petkovic, On the generalizationof some algo&hms for the simul~eous appmximation of polynomial roots, in Zntewal IbiathW 1980 (K. Nickel, Ed,), Academic, New York,lQ80, pp. 461-471. M. S. Petkovic, Qn a geaerabation of the root iterationsfor polynomialcomplex zeros in circuhr intervalarithmetic, Cbmpu~~ 27:37-55 (1981). Me S*Petkovic, On an iterativemethod for simultaneousinch&on of polynomial complex zeros, J. Compuh Appl. Math. 851-56 (1982). M. S. Petkovic and G. V. Mihmovic, A note on some ~p~ve~~~ of the A&. s~~t~~ methods for determination of ~l~orn~ zeros, f. ~~* Iu&h. ~~~ (1983). M. S. Petkovic and IL.V. Stoops, On the ~~~~~ method of the second order for f&ding ~~~~ complex zeros in circular srithmetic, ~~ Z~~~~~~~~ ~3):~-~ (lQ85). M. 5. Petkotiilc md L. V’ StefanoviG On a second order method for the simultaneous inchsion of pc@nomialcomplex zeros in rectanguhr 33rithmeticp ~~~~ 3&24Q-26l(lQ86). MI S.Petkovic and L. V. Stefmov& On some ~p~ve~~ of square roat iteration for ~~0~~ complex zeros9 I. ~~~. MzdADZ. ~~. 1513-25 (lQS@* M. S. Petkovic and I, V. Stef~o~c, On some items ~~0~ for the ~~~~ ~rn~~on of rn~~ abmplex~~~0~ zeF1)s;, BIT 27:Pll-122 (lag* J. Rokne, A~to~~~ errorboundsfor simple zeros of ~~~~ ~~~~, ~~* A~~ ~~~1~1-1~ (1973). J. Rokne and P. bmcaster*Complexinterval~~e~~, urn* A~ 141:111-1112 (1971). J. Rokne ad P. Lancaster,Algorithm8rE), compk?xinterval arithmetic,~~~* 1. l&83--85 (lQ75). iter&iorbmethod bFOfphe De ‘&bg ad Y. Wu, *me m and their convergenwB~~~~ 3&75-87 (lQ87).
3.
s
-2
4.
(5.1)
l t=
s
i
)
.
5=
=
v
M. MONSIAND M. A. WOLFE
The pdure
mss
with
m, mss, and mws have been 101using tit&mm for nx%ngth complex Wls ms ad PI on a VAX U-785 caqm&m. Numerical aumdtsare presented for six examples in Tables l-6, in rr,
Is,
rss, mss,
UpT,
(1W) ( - aso) @So)
( - LO) ( - 108)
wo W) ao)
ih
wn ( - Lo) (OS 1)
(0; -1) (- w (- 1s- 2)
TABLE& 1331
(o,O) (-ii-l)
(-10s -10)
(OS-5)
We (oa
8
9 10
TABLE4.
[36]
0
(GO)
1 2 3 4 5 6 ? 8 9 l0
( - 1W)
(0,101 (0, -10) (10110) (10, - lc$ ( - 10,lO) ( - 10, - 10)
mo) (OSB
TABLE5
[30]
Pa aad P!! mspectidy to satisfy a given stopping givextinitial data Then the rat&s in Tables 7-10 and other
TABLEi7 Cputime(srpx)
~*oo W4)
5-3(4) 5*4(4)
TABLE9 Cputime(se@ -A
l&7(4) =2(3) 22.213) 22.1(3) fW3) 7*5(3)
1
2 3 4 5 6
683) 5*30 9.1(2) 9.1(2) 4.1(3) 4*3(3)
TABLE 10 Cputime(set) 15.4(3) 11.0(2) 19.7(2) 19.7(2) 7*3(3) 7JK3)
5*4(2) 582) 6.0(l) $.0(l) 3*2(2) 3*3(2)
resuks which are not reported in this paper suggest that IS < IPS,
IT
IPT c IPSS 6 IPRSS,
that of the
IRSS < IPRSS,
ISS < IPSS, an
IPS a wss.
208
M. MONSIAND M. A. WOLFE
wm, the prwxhm mim with rk=3 (VbO) is likely to be the most efficient for bounding the simple zems of a complex polynomial. REFERENCES 1 0. Abed& Iteratkmmethods for finding all zeros of a polynomial simultaneously, n!wh. cbmp. 27:33%3u (lQ73). htmbcth to Zntmal Cbmputations, Acacbmic, 2 G. Ah&&l and J. II-, NewYork, 1983. 3 G. Al&d and J. Herzbqpr, On the convergencespeed of some algorithmsfor the shhwous appro;rdmation of polynomial zeros, SLAM1. NW. Anal. lu37-243 (lQ74). 4 D. Brams and K. P. Hadeler,Sim~eous inchuion of the zeros of a polynomial, NunMIr.M&. 2&l&16!5 (m73). and App-, McGraw-EmICom 5 R v. ChurcIbi&GnRpkx v&&k Th 1mQ. 6 A. j. Cole and R Morrison, ‘Riplex: A system for interval arithmetic, s&mm-e antf Eajmhwe 12:341-350 (1982). 7 L W. E&h, A mod&d Newton method for polynomials, Comm. Assoc. cbf#q&. Mb&. 10=101-108(1967). 8 M. R Famer and G. Mzou, A classof iteration functions for improviig, simm, ato the zeros of polynomials, BZT 95:250-258 0. 9 I. Gargmtini, Phrabl squue mot iterations,in ZntrmolMdhemcrtics(K. Nickel Ed.), bcture Notes in Computer Science 29, Springer, Heidelberg, 1975, pp. m-204. 10 I. Gaqptini, Pam&l Laguem iterations: The complex case, Numer. Math. 28:317-323 (1976). 11 I. Gargmtini, Fbther applications of circuh s@unetic: ScMder-like algorithms with error bounds for finding zeros of polynomials, SZAM1. Namer. Ana& l&497-510 (1978). 12 I. Gaqpntini, An ap#cation of interval mathematics:A polynomial solver with degree four convergence, Fw ZntmaWwchte 81(7):x5-25 (1981). 13 I. -tini and P. bnrici, Circuk arithmetic and the determination of polynomial Z~MS,NW. Math. 18905-320 (1912). 14 G. Glatx, Newton-Algorithmen zur Restimmung von Polynomwbzeln unter Verwendung bmplexer bisarithmetik, in Znti Mathematim (K. Nihl, Ed.), Lecture Notes in Computer Science 29, Springer, Heidelberg, 1975, pp. 205-214. 15 E. Hansen, M. Patrick,and J. Rusnak, Some modificationsof Laguerre’smethod, BZT 17&B-417 (lQ77). 16 PObhi, AppEced(LndChnpukrfsoMiCompkx Anal@s, Vol. 1, Wiley, New York, lQ74. 17 0. IcemeI, Eh Gemwchiw zur Berechnmg der Nullstellen von Polynoznen, Nlcmer. M&k 8:~294 (19$@ and P. S engen von Polynom-Nullsteilen,in
19
20 21 22 23 24 25
26 27 28
29
30
31
32
33 34 35 36
2~~ ~U~~~ (K. MCkeLEd.), Lecture Notes in ~orn~ter Science29* springer,lwd~, 1@75, p3*223-228. R. Morrison,A. js Cole, P, Raihy, MeA.Wolfe, and J. M. Shearer*Egperienoein usbg a hi& level language which supports inter\tuttarithmetic, in ~~~~~ 6th CI 8q%qX&~&*. u* ~~~ ~~~ (T. R. N. Rao and P. I(;onrenrp, Ms.), Aiarhls~ libnmah, lQs3, pp. l-li3. G. V. ~V~O~C ad M.S. P&~vic, On the cxmqeme ofa modifiedme&d for simultaneous finding of plynomial zeros, Cafe 30:171-178 (1983). M. MO& and MO A. WoEe, An a&ritbm for the simultan~ fnclusionof re& polynomial zeros~Appt. Mu&. Cbmput., to appe;ur, A. Neumaier, An intervalversion of the secant method, BH’ 24:366-372 (X9&4). ALNeumaier, Intervaliterationfor zeros of systemsof equations, BI11‘ 25956-273 (1985). J. M. Ortega and W. C. Rheinbok#t,IterativeSulutim of llkmhem &uatiom in SmeraZVaniablarr, Academic,New York,1970. M. S. Petkovic, On the generalizationof some algo&hms for the simul~eous appmximation of polynomial roots, in Zntewal IbiathW 1980 (K. Nickel, Ed,), Academic, New York,lQ80, pp. 461-471. M. S. Petkovic, Qn a geaerabation of the root iterationsfor polynomialcomplex zeros in circuhr intervalarithmetic, Cbmpu~~ 27:37-55 (1981). Me S*Petkovic, On an iterativemethod for simultaneousinch&on of polynomial complex zeros, J. Compuh Appl. Math. 851-56 (1982). M. S. Petkovic and G. V. Mihmovic, A note on some ~p~ve~~~ of the A&. s~~t~~ methods for determination of ~l~orn~ zeros, f. ~~* Iu&h. ~~~ (1983). M. S. Petkovic and IL.V. Stoops, On the ~~~~~ method of the second order for f&ding ~~~~ complex zeros in circular srithmetic, ~~ Z~~~~~~~~ ~3):~-~ (lQ85). M. 5. Petkotiilc md L. V’ StefanoviG On a second order method for the simultaneous inchsion of pc@nomialcomplex zeros in rectanguhr 33rithmeticp ~~~~ 3&24Q-26l(lQ86). MI S.Petkovic and L. V. Stefmov& On some ~p~ve~~ of square roat iteration for ~~0~~ complex zeros9 I. ~~~. MzdADZ. ~~. 1513-25 (lQS@* M. S. Petkovic and I, V. Stef~o~c, On some items ~~0~ for the ~~~~ ~rn~~on of rn~~ abmplex~~~0~ zeF1)s;, BIT 27:Pll-122 (lag* J. Rokne, A~to~~~ errorboundsfor simple zeros of ~~~~ ~~~~, ~~* A~~ ~~~1~1-1~ (1973). J. Rokne and P. bmcaster*Complexinterval~~e~~, urn* A~ 141:111-1112 (1971). J. Rokne ad P. Lancaster,Algorithm8rE), compk?xinterval arithmetic,~~~* 1. l&83--85 (lQ75). iter&iorbmethod bFOfphe De ‘&bg ad Y. Wu, *me m and their convergenwB~~~~ 3&75-87 (lQ87).