1941 R. J. Schmidt in London, Edinb. & Dublin Philos. Mag. 7th Ser. 32 370 Denote the nth approximation to the value of any unknown x_r by x_r⁽ⁿ⁾. To solve these [simultaneous] equations by the method of successive approximations, we assume approximations x₂⁽⁰⁾, x₃⁽⁰⁾,…x_m⁽⁰⁾ to the unknowns x₂, x₃,..x_m. Making use of these values we then use the first equation to find x₁⁽¹⁾, the first approximation to x₁. As the approximations will not always converge to the value of the unknowns, we shall, in future, call the numbers x_s^(r) iterates.

1956 F. B. Hildebrand Introd. Numerical Anal. x. 449 The error t_k in the kth iterate will be approximately the square of that in the preceding iterate, and will be of opposite sign.

1968 L. Fox & D. F. Mayers Computing Methods for Scientists & Engineers i. 10 In the days of machine saturation it is tempting to take 25 iterates even though 250 are really necessary.

1968 N. Bourbaki Theory of Sets iii. 186 The mapping fⁿ is called the nth iterate of the mapping f.

1658 Sir T. Browne Hydriotaphia: Urne-buriall iii. 40 The cemiteriall Cels..were filled with draughts of Scripture Stories..iterately affecting the pourtraits of Enoch, Lazarus, Jonas, and the Vision of Ezechiel.

?1548 tr. J. Calvin Faythfvl Treat. Sacrament sig. Cv^v Neither doe they thynke that it ought to be so often iterated and repeted, after that we haue ones receyued Christe.

1594 W. West Symbolæogr.: 2nd Pt. §175 Amongst heretikes are numbered Anabaptists, which wickedly yterate holy baptisme.

1650 E. Ashmole tr. A. Dee Fasciculus Chemicus 78 The dregs being cast away, iterate the Sublimation of the most white Dust by it self.

1682 R. Boyle Contin. New Exper. Physico-mech.: 2nd Pt. vii. iv Having wiped and cleansed away the soot, I iterated the experiment.

1744 R. North & M. North Life Sir D. North & Rev. J. North 264 He found that, by often iterating, his Thoughts lost of their Force.

1862 W. F. Hook Lives Archbishops Canterbury II. 642 That cannot be said to be iterated, which is not known to have been done before.

1533 W. Tyndale Supper of Lord in Wks. (Parker Soc.) III. 245 I am here compelled to inculk and iterate it with so many words.

1597 R. Hooker Of Lawes Eccl. Politie v. xxxvii. 74 This is the very cause why we iterate the Psalmes oftner then any other part of Scripture.

1611 Bible (King James) Ecclus. xli. 23 Iterating and speaking againe that which thou hast heard. View more context for this quotation

1661 S. Morgan Sphere of Gentry ii. i. 5 You must not itterate or name one Colour twice in the blazon of one Coat.

1858 Sat. Rev. 20 Nov. 500/2 Scientific research iterates and reiterates one moral..the greatness of little things.

1863 C. C. Clarke Shakespeare-characters v. 134 She iterates..to all the charges crowding in against him, ‘My husband!’

1660 J. Lloyd Treat. Episcopacy 70 Our Saviour iterated their ordinary into the pastoral extraordinary and ordinary offices.

1953 A. S. Householder Princ. Numerical Anal. ii. 45 Since the ‘approximation’ x₀ with which one may start an iteration does not necessarily need to be close, it is sometimes advantageous to..start with an arbitrary x₀, perhaps x₀ = 0, and iterate until the approach is sufficiently close.

1957 L. Fox Numerical Solution Two-point Boundary Probl. iv. 81 As an example..of an uncommon but convenient iterative procedure consider the solution of the differential equation y″ − (1/x⁴ − 2/x³)y + 1/x² = 0... We iterate according to the scheme y_{n + 1} = x²(1 + x²y^″/ _n)/(1 − 2x), with y₀ = 0.

a1593 C. Marlowe Tragicall Hist. Faustus (1604) sig. C2 The iterating of these lines brings golde.

a1626 F. Bacon Elements Common Lawes (1630) 82 The doubling or iterating of that and no more..is reputed nugation.

1644 K. Digby Two Treat. ii. Concl. 452 The iterating of those actes, which brought it from ignorance to knowledge.