L. Adleman and K. Manders, Reductions that lie, Proc. 20th Annual Symp. on Foundations of Computer Science (1979) 397--410.
L.M. Adleman, K. Mander and G. Miller, On taking roots in finite fields, in: Proc. 18th IEEE Symp. Foundations of Computer Science (1977) 175--178.
L.M. Adleman, K. Manders and G. Miller, On taking roots in finite fields, in: Proc. 18th Annual Symp. on Foundations of Computer Science, IEEE Computer Society, Providence, RI (1977) 175--178.
J. Albrecht, Über die Abrundungsfehler bei der Iteration für y=&sqrt;[n]x, Z. Angew. Math. Mech. 40 (1960) 191.
M. Andrews, S.F.McCormick and G.D. Taylor, Evaluation of functions on micro-processors: Square root, Comput. Math. Appl. 4 (1978) 359--367.
M. Andrews, Mathematical microprocessor software: a &sqrt;x comparison, IEEE Micro 2 (1982) 63--75.
E. Bach, A note on square roots in finite fields, IEEE Trans. Inform. Theory 36 (1990) 1494--1498.
E. Bach, Realistic analysis of some randomized algorithms, in: Proc. 19th Annual ACM Symp. on Theory of Computing (1987) 453--461.
V.A. Bailey, Prodigious calculation, Austral. J. Sci. 3 (1941) 78--80.
J.P. Ballantine, An averaging method of extracting roots, Amer. Math. Monthly 63 (1956) 249--252.
T. Barrera and P. Olsson, An integer based square root algorithm, BIT 33 (1993) 254--261.
E.H. Bateman, Solution of algebraic and transcendental equations by iteration, Math. Gaz. 37 (1953) 96--101.
R.W. Bemer, A machine method for square-root computation, Comm. ACM 1 (1958) 6--7.
Tabit ben Quarra, On the Verification of Problems of Algebra by Geometrical Proofs (c. 890); translation: P. Luckey, Berichte über die Verhandlungen der Sächs, Akad. Leipzig 93 (1941) 93--112.
L. Berg, Stabile Iterationsverfahren beliebiger Ordnung zur Berechnung von Wurzeln, Z. Angew. Math. Mech. 61 (1981) 396--399.
E. Bodewig, Über das Quadratwurzelzeihen aus kleinen Zahlen auf der Rechensmaschine, Z. Angew. Math. Mech. 29 (1949) 377--379.
R. Bombelli, L'Algebra (Venice, 1572).
A. Borodin and I. Munro, The Computational Complexity of Algebraic and Numerical Problems (Elsevier, New York, 1977) 54--76; 132--137; 148--150.
S. Breuer and G. Zwas, Polynomial iterations for root extraction, Comp. Educ. 12 (1988) 289--300.
S. Breuer, Polynomial iterations of rth order for the extraction of kth roots, J. Comput. Appl. Math. 25 (1) (1989) 61--68.
S. Breuer and G. Zwas, Computer root extraction by a priori design, Comp. Educ. 8 (1984) 305--316.
M. Cantor, Vorlesungen über Geschichte der Mathematik, 2 (Leipzig, 1900) 621--629.
R. Cazenave, Calcul de la racine nème d'un nombre sur une machine binaire, Chiffres 3 (1960) 129--135.
P.L. Chebyshev, Sur les expressions approchées de la racine carrée d'une variable par des fractions simple, in: Oeuvres, Vol. 2 (Chelsea, New York) 542--558.
P.L. Chebyshev, Sur les fractions algébriques qui représentent approximativement la racine carré d'une variable comprise entre les limites données, in: Oeuvres, Vol. 2 (Chelsea, New York) 725.
S.G. Chen and P.Y. Isieh, Fast computation of the Nth root, Comput. Math. Appl. 17 (1989) 1423--1427.
T.C. Chen, Automatic computation of exponentials, logarithms, ratios and square roots, IBM J. Res. Develop. 16 (1972) 380--388.
W.J. Cody, Double-precision square root for the CDC-3600, Comm. ACM 7 (1964) 715--718.
W.J. Cody, Jr and W. Waite, Software Manual for the Elementary Functions (Prentice-Hall, Englewood Cliffs, NJ, 1980).
D. Cowgill, Logic equations for a built-in square root method, IEEE Trans. Electronic Comput. (Corr.) 13 (1964) 156--157.
C. Davies and Peck, Extraction of roots, Sturm's theorem, square roots, cube roots, cubic equations, in: Mathematical Dictionary and Cyclopedia of Mathematical Science (Barnes, 1876) 140--145; 242; 534; 539--542.
C. Domb, On iterative solutions of algebraic equations, Proc. Cambridge Philos. Soc. 45 (1949) 237--240.
O. Dunkel, A note on the computation of arithmetic roots, Amer. Math. Monthly 34 (1927) 366--368.
O. Dunkel, A simple rule for extracting any root of a number, School Sci. Math. 18 (1918) 19--20.
O. Dunkel, A graphical representation of approximations for square roots, School Sci. Math. 18 (1918) 621--625.
A. Emch, Two hydraulic methods to extract the nth root of any number, Amer. Math. Monthly 8 (1901) 10--12.
J. Eve, Starting approximations for the iterative calculation of square roots, Comput. J. 6 (1963) 274--276.
?. Fibonacci, Liber Abbaci (1202) Chapters 14, 15; also: in: B. Boncompagni, Ed., Scritti di Leonardo Pisano (Fibonacci), Vol. 1 (Rome, 1857).
C.T. Fike, Computer Evaluation of Mathematical Functions (Prentice-Hall, Englewood Cliffs, NJ, 1968) Chapters 2, 4.
C.T. Fike, A rational approximation optimal by Moursund's criterion, Comm. ACM 10 (1967) 683--684.
C.T. Fike, Starting approximations for square root calculations on the IBM System/360, Comm. ACM 9 (1966) 297--298.
N.J. Fine, Infinite products for k-th roots, Amer. Math. Monthly 84 (1977) 629--630.
A.I. Forsythe et al., Computer Science, a First Course (Wiley, New York, 1975) Chapter 3.
J.S. Frame, Remarks on a variation of Newton's method, Amer. Math. Monthly 52 (1945) 212--214.
E. Frank, On continued fraction expansions for binomial quadratic surds II, III, Numer. Math. 4 (1962) 303--309; 5 (1963) 113--117.
E. Frank, On continued fractions and binomial quadratic surds, Numer. Math. 4 (1962) 85--95.
E. Frank, Continued fraction expansions for k-th roots, Z. Angew. Math. Mech. 60 (1980) T289--291.
W. Fraser and J.F. Hart, Minimax approximations for square root and cube root routines, in: Proc. Third Conf. of the Computing and Data Processing Society of Canada (Univ. Toronto Press, Toronto, 1962) 158--167.
J.B. Gibson, Optimal rational starting approximations, J. Approx. Theory 12 (1974) 182--189.
H.W. Gould, An iterative approximation for finding the Nth root of a number, Math. Mag. 33 (1959) 61--69.
J.C. Gower, A note on an iterative method for root extraction, Comput. J. 1 (1958) 142--143.
M.A. Grant, Approximating square roots, Math. Gaz. 66 (1982) 230--231.
H.D. Green, Square root extractor, Rev. Sci. Instr. (N.S.) 11 (1940) 262--264.
J.B.S. Haldane, The extraction of square roots, Math. Gaz. 35 (1951) 89--90.
J.F. Hart and E.W. Cheney, Computer Approximations (Wiley, New York, 1968) Chapter 6.
D.R. Hartree, Notes on iterative processes, Proc. Cambridge Philos. Soc. 45 (1949) 230--236.
R. Hashemian, Square-rooting algorithms for integer and floating point numbers, IEEE Trans. Comput. 39 (1990) 1025--1029.
S. Hitotumatu, A method of successive approximation based on the expansion of second order, Math. Japon. 7 (1962) 31--50.
P.M. Hummel and C.L. Seebeck, A generalization of Taylor's expansion, Amer. Math. Monthly 56 (1949) 243--247.
S.M. Jacob, On sequences which determine the nth root of a rational number, Proc. London Math. Soc. (2) 1 (1903) 166--174.
G. James, A rapid method of approximating arithmetic roots, Amer. Math. Monthly 31 (1924) 471--475.
W. James and P. Jarratt, The generation of square roots on a computer with rapid multiplication compared with division, Math. Comp. 19 (1965) 497--502.
M.J. Jamieson, Rapidly converging iterative formulae for finding square roots and their computational efficiencies, Comput. J. 32 (1989) 93--94.
K.C. Johnson, ALGORITHM 650: Efficient square root implementation on the 68000, ACM Trans. Math. Software 13 (1987) 138--151.
K.R. Johnson, An iterative method for approximating square roots, Math. Mag. 62 (1989) 253--259.
S. Kaplan, On finding the square root of a complex number, Math. Tables Aids Comput. 4 (1950) 177--178.
Omar Khayyam, On the Proofs of the Problems of Algebra and Muquatala, translation: H.J.J. Winter, J. Roy. Asiatic Soc. Bengal 16 (1950) 27--77.
R.F. King, On the double-precision square root routine, Comm. ACM 8 (1965) 202.
R.F. King and D.L. Phillips, The logarithmic error and Newton's method for the square root, Comm. ACM 12 (1969) 87--88.
I. Kiss, Über eine Verallgemeinerung des Newtonschen Näherungsverfahrens, Z. Angew. Math. Mech. 34 (1954) 68--69.
I. Kiss, Die theoritischen Grundlagen der Radizierung mit der Rechenmaschine, Acta Tech. Acad. Sci. Hungar. 8 (1954) 221--241.
F. Klein, Elementary Mathematics from an Advanced Standpoint, Vol. I (Dover, New York, 1939) 87--143 (translation: E.R. Hedrick and C.A. Noble).
E.G. Kogbetlianz, Computation of sinN, cosN and &sqrt;[m]N using an electronic computer, IBM J. Res. Develop. 3 (1959) 147.
M. Kuczyma and H. Swiatak, Newton-like algorithms for k'th root calculations, Ann. Polon. Math. 52 (1991) 303--312.
K.S. Kunz, Numerical Analysis (McGraw-Hill, New York, 1957) 1--37.
O.E. Lancaster, Machine method for the extraction of cube root, J. Amer. Statist. Soc. 37 (1942).
L.R. Langdon, Approximating functions for digital computers, Indust. Math. 6 (1955) 85--86.
M. Levey, The Algebra of Abu Kamil (Univ. of Wisconsin Press, Madison, WI, 1966).
L.A. Lyusternik, O.A. Chervonenkis and A.R. Yanpolskii, Handbook for Computing Elementary Functions, translation: G.J. Tee (Pergamon Press, Oxford, 1965) 10--16; 21--31; 142--148.
M. Müller, (untitled letter), Z. Angew. Math. Mech. 29 (1949) 160.
S. Majerski, Square-rooting algorithms for high-speed digital circuits, IEEE Trans. Comput. 34 (1985) 724--733.
K. Manders and L. Adleman, NP-complete decision problems for quadratic polynomials, in: Proc. 8th Annual ACM Symp. on Theory of Computing (1976) 23--29.
Y. Mansour, B. Schieber and P. Tiwari, The complexity of approximating the square root, in: 30th Annual Symp. on Foundations of Computer Science (IEEE Computer Science Press, Los Alamitos, 1989) 325--330.
G. Meinardus and G.D. Taylor, Optimal partitioning of Newton's method for calculating roots, Math. Comp. 35 (1980) 1221--1230.
G. Metze, Minimal square rooting, IEEE Trans. Electronic Comput. 14 (1965) 181--185.
M. Mignotte, Calcul des racines d-iemes dans un corps fini, C.R. Acad. Sci. Paris Ser. 1 290 (1980) 205--206.
N. Mikami et al., A new DSP-oriented algorithm for calculation of the square root using a nonlinear digital filter, IEEE Trans. Signal Process. 40 (1992) 1663--1669.
Z. Mikolajska, Remarque sur la note de A.B. Torowicz sur l'approximation des racines de nombres positifs, Ann. Polon. Math. 8 (1960) 285--289.
J. Mikusinski, Sur la méthode d'approximation de Newton, Ann. Polon. Math. 1 (1954) 184--194.
C. Moler and D. Morrison, Replacing square roots by Pythagorean sums, IBM J. Res. Develop. 27 (1983) 577--581.
P. Montuschi and M. Mezzalama, Optimal absolute error starting values for Newton-Raphson calculation of square root, Computing 46 (1991) 67--86.
P. Montuschi and L. Ciminiera, On the efficient implementation of higher radix square root algorithms, 9th IEEE Symp. on Computer Arithmetic (Santa Monica, CA, 1989) 154--161.
P. Montuschi and M. Mezzalama, Survey of square rooting algorithms, Proc. IEEE 137 (1990) 31--40.
D.G. Moursund, Optimal starting values for the Newton--Raphson calculation of &sqrt;x, Comm. ACM 10 (1967) 430--432.
R.F. Newton, A simple derivation of Hutton's formula for the computation of roots, Amer. Math. Monthly 34 (1927) 368--369.
C. Nicoletti, Su una classe di algoritmi di iterazione per l'approssimazione degli irrazionali quadratici, Rend. Circ. Math. Palermo 42 (1917) 73--79.
I. Ninomiya, Best rational starting approximations and improved Newton iteration for the square root, Math. Comp. 24 (1970) 391--404.
Benedetto of Florence, Trattato di Practicha d'Arismetica (1463) Books 13, 14, 15.
Dardi of Pisa, Aliabraa Argibra (1344); cf. W. Van Egmond, The algebra of Master Dardi of Pisa, Hist. Math. 10 (1983) 399--421.
V.G. Oklobdzija and M.D. Ercegovac, An on-line square root algorithm, IEEE Trans. Comput. 31 (1982) 70--75.
V.G. Oklobdzija and M.D. Ercegovac, An on-line square-root algorithm, IEEE Trans. Comput. 31 (1982) 70--75.
H. Peng, Algorithms for extracting square roots and cube roots, Proc. 5th IEEE Symp. on Computer Arithmetic, Ann Arbor, MI (1981) 121--126.
R. Peralta, A simple and fast probabilistic algorithm for computing square roots modulo a prime number, IEEE Trans. Inform. Theory 32 (1986) 846--847.
L. Petkovic and M.S. Petkovic, The representation of complex circular functions using Taylor series, Z. Angew. Math. Mech. 61 (1981) 661--662.
L.D. Petkovic and M.S. Petkovic, On the kth root in circular arithmetic, Computing 33 (1984) 27--35.
L.D. Petkovic, A note on the evaluation in circular arithmetic, Z. Angew. Math. Mech. 66 (1986) 371--373.
M.S. Petkovic and L.D. Petkovic, On a representation of the kth root in complex circular interval arithmetic, in: K. Nickel, Ed., Interval Mathematics 80 (Academic Press, New York, 1980) 473--479.
D.L. Phillips, Generalized logarithmic error and Newton's method for the mth root, Math. Comp. 24 (1970) 383--389.
T.J. Rolfe, On a fast integer square root algorithm, SIGNUM 22 (4) (1987) 6--11.
F.K. Rubbert, Zuschriften an der Herausgeber, Z. Angew. Math. Mech. 29 (1949) 160.
F.K. Rubbert, Zur Radizierung mit der Rechenmaschine, Z. Angew. Math. Mech. 28 (1948) 190--191.
P. Ruopp, Zur Berechnung von Quadratwurzeln, Praxis Math. 2 (1960) 295--297.
P. Ruopp, Einfache Iterationen, Praxis Math. 4 (1962) 108.
H. Rutishauser, Betrachtungen zur Quadratwurzeliteration, Monatsh. Math. 67 (1963) 452--464.
B.P. Sarkar and E.V. Krishnamurthy, Economic pseudodivision processes for obtaining square root, logarithm, and arctan, IEEE Trans. Comput. 20 (1971) 1589--1593.
C. Schmidt, Über einen Algorithmus zur Berechnung der nten Wurzel aus a, Math. Ann. 45 (1894) 301--308.
R.J. Schoof, Elliptic curves over finite fields and the computation of square roots mod p, Math. Comp. 44 (1985) 483--494.
A. Sharma, On Newton's method of approximation, Ann. Polon. Math. 6 (1959) 295--300.
P.H. Sterbenz and C.T. Fike, Optimal starting approximations for Newton's method, Math. Comp. 23 (1969) 313--318.
J. Sugai, Extraction of roots by repeated subtractions for digital computers, Comm. ACM 1 (12) (1958) 6--8.
P.N. Swarztrauber, Letter to the editor, Comm. ACM 8 (1965) 202.
G.D. Taylor, Optimal starting approximations for Newton's method, J. Approx. Theory 3 (1970) 156--163.
G.S. Taylor, Compatible hardware for division and square root, Proc. 5th IEEE Symp. on Computer Arithmetic (Ann Arbor, MI, 1981) 127--134.
J. Todd, Basic Numerical Mathematics, Vol. 1: Numerical Mathematics (Birkhäuser, Basel, 1979).
J.F. Traub, Comparison of iterative methods for the calculation of Nth roots, Comm. ACM 4 (1961) 143--145.
S.M. Turner, Square roots mod p, Amer. Math. Monthly 101 (1994) 443--449.
A.B. Turowicz, Sur l'approximation des racines de nombres positifs, Ann. Polon. Math. 8 (1960) 265--69.
J.V. Uspensky, Note on the computation of roots, Amer. Math. Monthly 34 (1927) 130--134.
B.L. Van der Waerden, A History of Algebra (Springer, New York, 1985).
A. Vogel, Praktische Berechnung der Wurzeln, Praxis Math. 1 (1959) 238--240.
W.G. Wadey, Two square-root approximations, Comm. ACM 1 (11) (1958) 13--14.
J.S. Walther, A unified algorithm for elementary functions, Proc. AFIPS 1971 Spring Joint Computer Conf. (1971) 379--385.
G.W. Ward, Successive approximations to &sqrt;[n]a, Math. Gaz. 17 (1933) 52; 157.
G. Wertheim, Die Berechnung der irrationalem Quadratwurzeln und die Erfindung der Kettenbrüche, Z. Math. Phys. 42 (1898) S149--160.
M.W. Wilson, Optimal starting approximations for generating square root for slow or no divide, Comm. ACM 13 (1970) 559--560.
E. Wingler, An infinite product expansion for the square root function, Amer. Math. Monthly 97 (1990) 836--839.
A.K. Yeyios, On two sequences of algorithms for approximating square roots, J. Comput. Appl. Math. 40 (1992) 63--72.
J. Yohe, Interval bounds for square roots and cube roots, Computing 11 (1973) 51--53.