Merge pull request #1470 from j-kieffer/acb_theta

new acb_theta module

CMakeLists.txt

@@ -205,7 +205,7 @@ set(_BUILD_DIRS
     acb_mat         acb_poly        acb_calc        acb_hypgeom
     arb_fmpz_poly   arb_fpwrap
     acb_dft         acb_elliptic    acb_modular     acb_dirichlet
-    dirichlet       bernoulli       hypgeom
+    acb_theta       dirichlet       bernoulli       hypgeom
 
     gr              gr_generic      gr_vec          gr_mat
     gr_poly         gr_mpoly        gr_special

Makefile.in

@@ -179,7 +179,7 @@ HEADER_DIRS :=                                                              \
         acb_mat         acb_poly        acb_calc        acb_hypgeom         \
         arb_fmpz_poly   arb_fpwrap                                          \
         acb_dft         acb_elliptic    acb_modular     acb_dirichlet       \
-        dirichlet       bernoulli       hypgeom                             \
+        acb_theta       dirichlet       bernoulli       hypgeom             \
                                                                             \
         gr              gr_generic      gr_vec          gr_mat              \
         gr_poly         gr_mpoly        gr_special                          \

doc/source/arb_mat.rst

@@ -403,12 +403,6 @@ Vector arithmetic
 
     The underscore methods do not allow aliasing between *res* and *v*.
 
-.. function:: void arb_mat_bilinear_form(arb_t x, const arb_mat_t A, arb_srcptr v1, arb_srcptr v2, slong prec)
-
-    Sets *res* to the product `v_1^T A v_2`, where `v_1` and `v_2` are seen as
-    column vectors. The lengths of the vectors must match the dimensions of
-    *A*.
-
 Gaussian elimination and solving
 -------------------------------------------------------------------------------
 
@@ -804,18 +798,17 @@ LLL reduction
 
 .. function:: void arb_mat_spd_lll_reduce(fmpz_mat_t U, const arb_mat_t A, slong prec)
 
-    Given a symmetric positive definite matrix *A*, compute a unimodular
-    transformation *U* such that *U^T A U* is close to being LLL-reduced. If
+    Given a symmetric positive definite matrix *A*, sets *U* to an invertible
+    matrix such that `U^T A U` is close to being LLL-reduced. If
     :func:`arb_mat_spd_get_fmpz_mat` succeeds at the chosen precision, we call
     :func:`fmpz_lll`, and otherwise set *U* to the identity matrix. The
     warnings of :func:`arf_get_fmpz` apply.
 
 .. function:: int arb_mat_spd_is_lll_reduced(const arb_mat_t A, slong tol_exp, slong prec)
 
-    Returns nonzero iff *A* is LLL-reduced with a tolerance of `\varepsilon =
-    2^{tol\_exp}`. This means the following. First, the error radius on
-    each entry of *A* must be at most `\varepsilon/16`. Then we consider the
-    matrix whose entries are `2^{\mathit{prec}}(1 + \varepsilon)^{i + j}
-    A_{i,j}` rounded to integers: it must be positive definite and pass
-    :func:`fmpz_mat_is_reduced` with default parameters. The warnings of
-    :func:`arf_get_fmpz` apply.
+    Given a symmetric positive definite matrix *A*, returns nonzero iff *A* is
+    certainly LLL-reduced with a tolerance of `\varepsilon = 2^{tol\_exp}`,
+    meaning that it satisfies the inequalities `|\mu_{j,k}|\leq \eta +
+    \varepsilon` and `(\delta - \varepsilon) \lVert b_{k-1}^*\rVert^2 \leq
+    \lVert b_k^*\rVert^2 + \mu_{k,k-1}^2 \lVert b_{k-1}^*\rVert^2` (with the
+    usual notation) for the default parameters `\eta = 0.51`, `\delta = 0.99`.

doc/source/index.rst

@@ -202,6 +202,7 @@ Real and complex numbers
    arb_hypgeom.rst
    acb_elliptic.rst
    acb_modular.rst
+   acb_theta.rst
    acb_dirichlet.rst
    bernoulli.rst
    hypgeom.rst

doc/source/index_arb.rst

@@ -102,6 +102,7 @@
        arb_hypgeom.rst
        acb_elliptic.rst
        acb_modular.rst
+       acb_theta.rst
        dirichlet.rst
        acb_dirichlet.rst
        bernoulli.rst

doc/source/references.rst

@@ -49,6 +49,8 @@ References
 
 .. [Bog2012] \I. Bogaert, B. Michiels and J. Fostier, "O(1) computation of Legendre polynomials and Gauss-Legendre nodes and weights for parallel computing", SIAM Journal on Scientific Computing 34:3 (2012), C83-C101
 
+.. [Bol1887] \O. Bolza, "Darstellung der rationalen ganzen Invarianten der Binärform sechsten Grades durch die Nullwerthe der zugehörigen Theta-Functionen", Math. Ann. 30:4 (1887), 478--495. https://doi.org/10.1007/BF01444091
+
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 .. [Bor2000] \P. Borwein, "An Efficient Algorithm for the Riemann Zeta Function", Constructive experimental and nonlinear analysis, CMS Conference Proc. 27 (2000) 29-34, http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf
@@ -63,6 +65,10 @@ References
 
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+.. [CFG2017] \F. Cléry, C. Faber, and G. van der Geer. "Covariants of binary sextics and vector-valued Siegel modular forms of genus two", Math. Ann. 369 (2017), 1649--1669. https://doi.org/10.1007/s00208-016-1510-2
+
+.. [CFG2019] \F. Cléry, C. Faber, and G. van der Geer. "Covariants of binary sextics and modular forms of degree 2 with character", Math. Comp. 88 (2019), 2423--2441. https://doi.org/10.1090/mcom/3412
+
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 .. [CP2005] \R. Crandall and C. Pomerance, *Prime Numbers: A Computational Perspective*, second edition, Springer (2005).
@@ -83,6 +89,8 @@ References
 
 .. [CraPom2005] \Richard Crandall and Carl Pomerance: Prime numbers: a computational perspective. 2005.
 
+.. [DHBHS2004] \B. Deconinck, M. Heil, A. Bobenko, M.  van Hoeij, and M. Schmies, "Computing Riemann theta functions", Math. Comp. 73:247 (2004), 1417--1442. https://arxiv.org/abs/nlin/0206009
+
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@@ -97,6 +105,8 @@ References
 
 .. [EM2004] \O. Espinosa and V. Moll, "A generalized polygamma function", Integral Transforms and Special Functions (2004), 101-115.
 
+.. [EK2023] \N. D. Elkies and J. Kieffer, "A uniform quasi-linear time algorithm for evaluating theta functions in any dimension", in preparation.
+
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@@ -115,6 +125,8 @@ References
 
 .. [Gos1974] \R. W. Gosper, "Acceleration of series", MIT AI Memo no.304, (March-1974). https://dspace.mit.edu/handle/1721.1/6088
 
+.. [Got1959] \E. Gottschling, "Explizite Bestimmung der Randflächen es Fundamentalbereiches der Modulgruppe zweiten Grades'', Math. Annalen 138 (1959), 103--124. https://doi.org/10.1007/BF01342938
+
 .. [GowWag2008] \Jason Gower and Sam Wagstaff : "Square form factoring" Math. Comp. 77, 2008, pp 551-588, https://doi.org/10.1090/S0025-5718-07-02010-8
 
 .. [GraMol2010] \Torbjorn Granlund and Niels Moller : Improved Division by Invariant Integers https://gmplib.org/~tege/division-paper.pdf
@@ -147,6 +159,10 @@ References
 
 .. [Iliopoulos1989] \Iliopoulos, C. S., Worst-Case Complexity Bounds on Algorithms for Computing the Canonical Structure of Finite Abelian Groups and the Hermite and Smith Normal Forms of an Integer Matrix : SIAM J. Computation 18:4 (1989) 658
 
+.. [Igu1972] \J.-I. Igusa. *Theta functions*, Springer, 1972. https://doi.org/10.1007/978-3-642-65315-5
+
+.. [Igu1979] \J.-I. Igusa, "On the ring of modular forms of degree two over Z", Amer. J. Math. 101:1 (1979), 149--183. https://doi.org/10.2307/2373943
+
 .. [JB2018] \F. Johansson and I. Blagouchine. "Computing Stieltjes constants using complex integration", preprint (2018), https://arxiv.org/abs/1804.01679
 
 .. [JM2018] \F. Johansson and M. Mezzarobba, "Fast and rigorous arbitrary-precision computation of Gauss-Legendre quadrature nodes and weights", preprint (2018), https://arxiv.org/abs/1802.03948
@@ -193,10 +209,14 @@ References
 
 .. [Kri2013] \A. Krishnamoorthy and D. Menon, "Matrix Inversion Using Cholesky Decomposition" Proc. of the International Conference on Signal Processing Algorithms, Architectures, Arrangements, and Applications (SPA-2013), pp. 70-72, 2013.
 
+.. [LT2016] \H. Labrande and E. Thomé, "Computing theta functions in quasi-linear time in genus 2 and above", ANTS XII, Kaiserslautern, LMS J. Comp. Math 19 (2016), 163--177. https://doi.org/10.1112/S1461157016000309
+
 .. [Leh1970] \R. S. Lehman, "On the Distribution of Zeros of the Riemann Zeta-Function", Proc. of the London Mathematical Society 20(3) (1970), 303-320, https://doi.org/10.1112/plms/s3-20.2.303
 
 .. [LukPatWil1996] \R. F. Lukes and C. D. Patterson and H. C. Williams "Some results on pseudosquares" Math. Comp. 1996, no. 65, 361--372
 
+.. [MN2019] \P. Molin and C. Neurohr, "Computing period matrices and the Abel--Jacobi map of superelliptic curves", Math. Comp. 88:316 (2019), 847--888.
+
 .. [MP2006] \M. Monagan and R. Pearce. "Rational simplification modulo a polynomial ideal". Proceedings of the 2006 international symposium on Symbolic and algebraic computation - ISSAC '06. https://doi.org/10.1145/1145768.1145809
 
 .. [MPFR2012] The MPFR team, "MPFR Algorithms" (2012), https://www.mpfr.org/algo.html
@@ -211,6 +231,10 @@ References
 
 .. [Mul2000] \Thom Mulders : On Short Multiplications and Divisions, AAECC vol. 11 (2000) 69--88
 
+.. [Mum1983] \D. Mumford, *Tata Lectures on Theta I*, Birkhäuser, 1983. https://doi.org/10.1007/978-1-4899-2843-6
+
+.. [Mum1984] \D. Mumford, *Tata Lectures on Theta II*, Birkhäuser, 1984. https://doi.org/10.1007/978-0-8176-4578-6
+
 .. [NIST2012] National Institute of Standards and Technology, *Digital Library of Mathematical Functions* (2012), https://dlmf.nist.gov/
 
 .. [NakTurWil1997] \Nakos, George and Turner, Peter and Williams, Robert : Fraction-free algorithms for linear and polynomial equations, ACM SIGSAM Bull. 31 (1997) 3 11--19
@@ -267,6 +291,8 @@ References
 
 .. [StoMul1998] \Storjohann, Arne and Mulders, Thom : Fast algorithms for linear algebra modulo :math:`N` : Algorithms---{ESA} '98 (Venice), Lecture Notes in Comput. Sci. 1461 139--150
 
+.. [Str2014] \M. Streng, "Computing Igusa class polynomials", Math. Comp. 83:285 (2014), 275--309. https://doi.org/10.1090/S0025-5718-2013-02712-3
+
 .. [Str1997] \A. Strzebonski. "Computing in the field of complex algebraic numbers". Journal of Symbolic Computation (1997) 24, 647-656. https://doi.org/10.1006/jsco.1997.0158
 
 .. [Str2012] \A. Strzebonski. "Real root isolation for exp-log-arctan functions". Journal of Symbolic Computation 47 (2012) 282–314. https://doi.org/10.1016/j.jsc.2011.11.004
@@ -303,4 +329,4 @@ References
 
 .. [vdH2006] \J. van der Hoeven, "Computations with effective real numbers". Theoretical Computer Science, Volume 351, Issue 1, 14 February 2006, Pages 52-60. https://doi.org/10.1016/j.tcs.2005.09.060
 
-All referenced works: [AbbottBronsteinMulders1999]_, [Apostol1997]_, [Ari2011]_, [Ari2012]_, [Arn2010]_, [Arn2012]_, [ArnoldMonagan2011]_, [BBC1997]_, [BBC2000]_, [BBK2014]_, [BD1992]_, [BF2020]_, [BFSS2006]_, [BJ2013]_, [BM1980]_, [BZ1992]_, [BZ2011]_, [BaiWag1980]_, [BerTas2010]_, [Blo2009]_, [Bodrato2010]_, [Boe2020]_, [Bog2012]_, [Bor1987]_, [Bor2000]_, [Bre1978]_, [Bre1979]_, [Bre2010]_, [BrentKung1978]_, [BuhlerCrandallSompolski1992]_, [CGHJK1996]_, [CP2005]_, [Car1995]_, [Car2004]_, [Chen2003]_, [Cho1999]_, [Coh1996]_, [Coh2000]_, [Col1971]_, [CraPom2005]_, [DYF1999]_, [DelegliseNicolasZimmermann2009]_, [DomKanTro1987]_, [Dup2006]_, [Dus1999]_, [EHJ2016]_, [EM2004]_, [Fie2007]_, [FieHof2014]_, [Fil1992]_, [GCL1992]_, [GG2003]_, [GS2003]_, [GVL1996]_, [Gas2018]_, [Gos1974]_, [GowWag2008]_, [GraMol2010]_, [HM2017]_, [HS1967]_, [HZ2004]_, [HanZim2004]_, [Har2010]_, [Har2012]_, [Har2015]_, [Har2018]_, [Hart2010]_, [Hen1956]_, [Hoe2001]_, [Hoe2009]_, [Hor1972]_, [Iliopoulos1989]_, [JB2018]_, [JM2018]_, [JR1999]_, [Joh2012]_, [Joh2013]_, [Joh2014a]_, [Joh2014b]_, [Joh2014c]_, [Joh2015]_, [Joh2015b]_, [Joh2016]_, [Joh2017]_, [Joh2017a]_, [Joh2017b]_, [Joh2018a]_, [Joh2018b]_, [JvdP2002]_, [Kahan1991]_, [KanBac1979]_, [Kar1998]_, [Knu1997]_, [Kob2010]_, [Kri2013]_, [Leh1970]_, [LukPatWil1996]_, [MP2006]_, [MPFR2012]_, [MasRob1996]_, [Mic2007]_, [Miy2010]_, [Mos1971]_, [Mul2000]_, [NIST2012]_, [NakTurWil1997]_, [Olv1997]_, [PP2010]_, [PS1973]_, [PS1991]_, [Paterson1973]_, [PernetStein2010]_, [Pet1999]_, [Pla2011]_, [Pla2017]_, [RF1994]_, [Rad1973]_, [Rademacher1937]_, [Ric1992]_, [Ric1995]_, [Ric1997]_, [Ric2007]_, [Ric2009]_, [RosSch1962]_, [Rum2010]_, [Smi2001]_, [SorWeb2016]_, [Ste2002]_, [Ste2010]_, [Stehle2010]_, [Stein2007]_, [Sut2007]_, [StoMul1998]_, [Str1997]_, [Str2012]_, [Tak2000]_, [ThullYap1990]_, [Tre2008]_, [Tru2011]_, [Tru2014]_, [Tur1953]_, [Villard2007]_, [WaktinsZeitlin1993]_, [Wei2000]_, [Whiteman1956]_, [Zip1985]_, [Zun2023]_, [vHP2012]_, [vdH1995]_, [vdH2006]_
+All referenced works: [AbbottBronsteinMulders1999]_, [Apostol1997]_, [Ari2011]_, [Ari2012]_, [Arn2010]_, [Arn2012]_, [ArnoldMonagan2011]_, [BBC1997]_, [BBC2000]_, [BBK2014]_, [BD1992]_, [BF2020]_, [BFSS2006]_, [BJ2013]_, [BM1980]_, [BZ1992]_, [BZ2011]_, [BaiWag1980]_, [BerTas2010]_, [Blo2009]_, [Bodrato2010]_, [Boe2020]_, [Bog2012]_, [Bol1887]_, [Bor1987]_, [Bor2000]_, [Bre1978]_, [Bre1979]_, [Bre2010]_, [BrentKung1978]_, [BuhlerCrandallSompolski1992]_, [CFG2017]_, [CFG2019]_, [CGHJK1996]_, [CP2005]_, [Car1995]_, [Car2004]_, [Chen2003]_, [Cho1999]_, [Coh1996]_, [Coh2000]_, [Col1971]_, [CraPom2005]_, [DHBHS2004]_, [DYF1999]_, [DelegliseNicolasZimmermann2009]_, [DomKanTro1987]_, [Dup2006]_, [Dus1999]_, [EHJ2016]_, [EM2004]_, [EK2023]_, [Fie2007]_, [FieHof2014]_, [Fil1992]_, [GCL1992]_, [GG2003]_, [GS2003]_, [GVL1996]_, [Gas2018]_, [Gos1974]_, [GowWag2008]_, [GraMol2010]_, [HM2017]_, [HS1967]_, [HZ2004]_, [HanZim2004]_, [Har2010]_, [Har2012]_, [Har2015]_, [Har2018]_, [Hart2010]_, [Hen1956]_, [Hoe2001]_, [Hoe2009]_, [Hor1972]_, [Iliopoulos1989]_, [Igu1972]_, [Igu1979]_, [JB2018]_, [JM2018]_, [JR1999]_, [Joh2012]_, [Joh2013]_, [Joh2014a]_, [Joh2014b]_, [Joh2014c]_, [Joh2015]_, [Joh2015b]_, [Joh2016]_, [Joh2017]_, [Joh2017a]_, [Joh2017b]_, [Joh2018a]_, [Joh2018b]_, [JvdP2002]_, [Kahan1991]_, [KanBac1979]_, [Kar1998]_, [Knu1997]_, [Kob2010]_, [Kri2013]_, [LT2016]_, [Leh1970]_, [LukPatWil1996]_, [MN2019]_, [MP2006]_, [MPFR2012]_, [MasRob1996]_, [Mic2007]_, [Miy2010]_, [Mos1971]_, [Mul2000]_, [Mum1983]_, [Mum1984]_, [NIST2012]_, [NakTurWil1997]_, [Olv1997]_, [PP2010]_, [PS1973]_, [PS1991]_, [Paterson1973]_, [PernetStein2010]_, [Pet1999]_, [Pla2011]_, [Pla2017]_, [RF1994]_, [Rad1973]_, [Rademacher1937]_, [Ric1992]_, [Ric1995]_, [Ric1997]_, [Ric2007]_, [Ric2009]_, [RosSch1962]_, [Rum2010]_, [Smi2001]_, [SorWeb2016]_, [Ste2002]_, [Ste2010]_, [Stehle2010]_, [Stein2007]_, [Sut2007]_, [StoMul1998]_, [Str2014]_, [Str1997]_, [Str2012]_, [Tak2000]_, [ThullYap1990]_, [Tre2008]_, [Tru2011]_, [Tru2014]_, [Tur1953]_, [Villard2007]_, [WaktinsZeitlin1993]_, [Wei2000]_, [Whiteman1956]_, [Zip1985]_, [Zun2023]_, [vHP2012]_, [vdH1995]_, [vdH2006]_