Simulation Algorithms for Lattice QCD A D KennedySchool of Physics, The University of Edinburgh

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ContentsMonte Carlo methodsFunctional Integrals and QFTCentral Limit TheoremImportance SamplingMarkov ChainsConvergence of Markov ChainsAutocorrelationsHybrid Monte CarloMDMCPartial Momentum RefreshmentSymplectic IntegratorsDynamical fermionsGrassmann algebrasReversibilityPHMCRHMCChiral FermionsOn-shell chiral symmetryNeubergers OperatorInto Five DimensionsKernelSchur ComplementConstraintApproximationtanhRepresentationContinued Fraction Partial FractionCayley TransformChiral Symmetry BreakingNumerical StudiesConclusionsApproximation TheoryPolynomial approximationWeierstra theorem polynomialss theorem polynomials approximationElliptic functionsLiouvilles theoremWeierstra elliptic functionsExpansion of elliptic functionsAddition formulJacobian elliptic functionsModular transformationss formulaArithmetico-Geometric meanConclusionsBibliographyHasenbusch Acceleration

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Functional Integrals and QFTThe Expectation value of an operator is defined non-perturbatively by the Functional IntegralThe action is S ()Continuum limit: lattice spacing a 0Thermodynamic limit: physical volume V Defined in Euclidean space-timeLattice regularisation

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Monte Carlo methods: IMonte Carlo integration is based on the identification of probabilities with measuresThere are much better methods of carrying out low dimensional quadratureAll other methods become hopelessly expensive for large dimensionsIn lattice QFT there is one integration per degree of freedomWe are approximating an infinite dimensional functional integral

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Monte Carlo methods: II

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Central Limit Theorem: I

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Central Limit Theorem: IINote that this is an asymptotic expansion

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Central Limit Theorem: IIIConnected generating functionDistribution of the average of N samples

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Central Limit Theorem: IV

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Central Limit Theorem: V

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Importance Sampling: I

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Importance Sampling: II

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Importance Sampling: III

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Importance Sampling: IV1 Construct cheap approximation to |sin(x)|2 Calculate relative area within each rectangle5 Choose another random number uniformly4 Select corresponding rectangle3 Choose a random number uniformly6 Select corresponding x value within rectangle7 Compute |sin(x)|

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Importance Sampling: VBut we can do better! With 100 rectangles we have V = 16.02328561 With 100 rectangles we have V = 0.011642808

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Markov Chains: I Deterministic evolution of probability distribution P: Q Q State space (Ergodic) stochastic transitions P:

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Convergence of Markov Chains: IThe sequence Q, PQ, PQ, PQ, is Cauchy

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Convergence of Markov Chains: II

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Convergence of Markov Chains: III

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Markov Chains: IIUse Markov chains to sample from QSuppose we can construct an ergodic Markov process P which has distribution Q as its fixed point Start with an arbitrary state (field configuration)Iterate the Markov process until it has converged (thermalized)Thereafter, successive configurations will be distributed according to QBut in general they will be correlatedTo construct P we only need relative probabilities of statesDo not know the normalisation of QCannot use Markov chains to compute integrals directlyWe can compute ratios of integrals

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Markov Chains: IIIIntegrate w.r.t. y to obtain fixed point conditionSufficient but not necessary for fixed pointSufficient but not necessary for detailed balance

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Markov Chains: IVComposition of Markov stepsLet P1 and P2 be two Markov steps which have the desired fixed point distributionThey need not be ergodicThen the composition of the two steps P2P1 will also have the desired fixed pointAnd it may be ergodicThis trivially generalises to any (fixed) number of stepsFor the case where P1 is not ergodic but (P1 ) n is the terminology weakly and strongly ergodic are sometimes used

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Markov Chains: VThis result justifies sweeping through a lattice performing single site updatesEach individual single site update has the desired fixed point because it satisfies detailed balanceThe entire sweep therefore has the desired fixed point, and is ergodicBut the entire sweep does not satisfy detailed balanceOf course it would satisfy detailed balance if the sites were updated in a random orderBut this is not necessaryAnd it is undesirable because it puts too much randomness into the system

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Markov Chains: VICoupling from the PastPropp and Wilson (1996)Use fixed set of random numbersFlypaper principle: If states coalesce they stay together forever Eventually, all states coalesce to some state with probability one Any state from t = - will coalesce to is a sample from the fixed point distribution

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Autocorrelations: IExponential autocorrelationsThe unique fixed point of an ergodic Markov process corresponds to the unique eigenvector with eigenvalue 1All its other eigenvalues must lie within the unit circle

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Autocorrelations: IIIntegrated autocorrelationsConsider the autocorrelation of some operator

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Autocorrelations: III

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Hybrid Monte Carlo: IIn order to carry out Monte Carlo computations including the effects of dynamical fermions we would like to find an algorithm whichUpdate the fields globallyBecause single link updates are not cheap if the action is not localTake large steps through configuration spaceBecause small-step methods carry out a random walk which leads to critical slowing down with a dynamical critical exponent z=2Does not introduce any systematic errors

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Hybrid Monte Carlo: IIA useful class of algorithms with these properties is the (Generalised) Hybrid Monte Carlo (HMC) methodIntroduce a fictitious momentum p corresponding to each dynamical degree of freedom qFind a Markov chain with fixed point exp[-H(q,p) ] where H is the fictitious Hamiltonian p2 + S(q)The action S of the underlying QFT plays the rle of the potential in the fictitious classical mechanical systemThis gives the evolution of the system in a fifth dimension, fictitious or computer timeThis generates the desired distribution exp[-S(q) ] if we ignore the momenta p (i.e., the marginal distribution)

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Hybrid Monte Carlo: IIIThe HMC Markov chain alternates two Markov stepsMolecular Dynamics Monte Carlo (MDMC)(Partial) Momentum RefreshmentBoth have the desired fixed pointTogether they are ergodic

Hybrid Monte Carlo: III

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MDMC: IIf we could integrate Hamiltons equations exactly we could follow a trajectory of constant fictitious energyThis corresponds to a set of equiprobable fictitious phase space configurationsLiouvilles theorem tells us that this also preserves the functional integral measure dp dq as requiredAny approximate integration scheme which is reversible and area preserving may be used to suggest configurations to a Metropolis accept/reject testWith acceptance probability min[1,exp(-H)]

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MDMC: IIWe build the MDMC step out of three partsA Metropolis accept/reject step with y being a uniformly distributed random number in [0,1]

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Partial Momentum RefreshmentThe Gaussian distribution of p is invariant under F The extra momentum flip F ensures that for small the momenta are reversed after a rejection rather than after an acceptanceFor = / 2 all momentum flips are irrelevant

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Hybrid Monte Carlo: IVSpecial casesThe usual Hybrid Monte Carlo (HMC) algorithm is the special case where = / 2 = 0 corresponds to an exact version of the Molecular Dynamics (MD) or Microcanonical algorithm (which is in general non-ergodic)the L2MC algorithm of Horowitz corresponds to choosing arbitrary but MDMC trajectories of a single leapfrog integration step ( = ). This method is also called Kramers algorithm.

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Hybrid Monte Carlo: VFurther special casesThe Langevin Monte Carlo algorithm corresponds to choosing = / 2 and MDMC trajectories of a single leapfrog integration step ( = ).The Hybrid and Langevin algorithms are approximations where the Metropolis step is omittedThe Local Hybrid Monte Carlo (LHMC) or Overrelaxation algorithm corresponds to updating a subset of the degrees of freedom (typically those living on one site or link) at a timeHybrid Monte Carlo: V

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Symplectic Integrators: IBaker-Campbell-Hausdorff (BCH) formula

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Symplectic Integrators: IIIn order to construct reversible integrators we use symmetric symplectic integrators

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Symplectic Integrators: IIIThe basic idea of such a symplectic integrator is to write the time evolution operator as

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Symplectic Integrators: IV

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Symplectic Integrators: VFrom the BCH formula we find that the PQP symmetric symplectic integrator is given byIn addition to conserving energy to O ( ) such symmetric symplectic integrators are manifestly area preserving and reversible

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Symplectic Integrators: VIFor each symplectic integrator there exists a Hamiltonian H which is exactly conservedFor the PQP integrator we have

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Symplectic Integrators: VIISubstituting in the exact forms for the operations P and Q we obtain the vector field

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Symplectic Integrators: VIIINote that H cannot be written as the sum of a p-dependent kinetic term and a q-dependent potential term

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Dynamical fermions: IFermion fields are Grassmann valuedRequired to satisfy the spin-statistics theoremEven classical Fermion fields obey anticommutation relationsGrassmann algebras behave like negative dimensional manifolds

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Grassmann algebras: IGrassmann algebrasLinear space spanned by generators {1,2,3,} with coefficients a, b, in some field (usually the real or complex numbers)Algebra structure defined by nilpotency condition for elements of the linear space = 0There are many elements of the algebra which are not in the linear space (e.g., 12)Nilpotency implies anticommutativity + = 00 = = = ( + ) = + + + = + = 0Anticommutativity implies nilpotency, 2 = 0 Unless the coefficient field has characteristic 2, i.e., 2 = 0

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Grassmann algebras: IIA natural antiderivation is defined on a Grassmann algebraLinearity: d(a + b) = a d + b d Anti-Leibniz rule: d() = (d) + P()(d)Grassmann algebras have a natural grading corresponding to the number of generators in a given productdeg(1) = 0, deg(i ) = 1, deg(ij ) = 2, ...All elements of the algebra can be decomposed into a sum of terms of definite grading

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Grassmann algebras: IIIThere is no reason for this function to be positive even for real coefficientsDefinite integration on a Grassmann algebra is defined to be the same as derivation

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Grassmann algebras: IVWhere Pf(a) is the PfaffianPf(a) = det(a)Despite the notation, this is a purely algebraic identityIt does not require the matrix a > 0, unlike its bosonic analogueGaussian integrals over Grassmann manifolds

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Dynamical fermions: IIPseudofermionsDirect simulation of Grassmann fields is not feasibleThe problem is not that of manipulating anticommuting values in a computerThe overall sign of the exponent is unimportant

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Dynamical fermions: IIIAny operator can be expressed solely in terms of the bosonic fieldsE.g., the fermion propagator is

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Dynamical fermions: IVThe fermion kernel must be positive definite (all its eigenvalues must have positive real parts) otherwise the bosonic integral will not convergeThe new bosonic fields are called pseudofermionsThe determinant is extensive in the lattice volume, thus again we get poor importance sampling

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Dynamical fermions: V

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Dynamical fermions: VIIt is not necessary to carry out the inversions required for the equations of motion exactlyThere is a trade-off between the cost of computing the force and the acceptance rate of the Metropolis MDMC stepThe inversions required to compute the pseudofermion action for the accept/reject step does need to be computed exactly, howeverWe usually take exactly to by synonymous with to machine precision

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Reversibility: IAre HMC trajectories reversible and area preserving in practice?The only fundamental source of irreversibility is the rounding error caused by using finite precision floating point arithmeticFor fermionic systems we can also introduce irreversibility by choosing the starting vector for the iterative linear equation solver time-asymmetricallyWe do this if we to use a Chronological Inverter, which takes (some extrapolation of) the previous solution as the starting vectorFloating point arithmetic is not associativeIt is more natural to store compact variables as scaled integers (fixed point)Saves memoryDoes not solve the precision problem

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Reversibility: IIData for SU(3) gauge theory and QCD with heavy quarks show that rounding errors are amplified exponentiallyThe underlying continuous time equations of motion are chaotic exponents characterise the divergence of nearby trajectoriesThe instability in the integrator occurs when H 1Zero acceptance rate anyhow

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Reversibility: IIIIn QCD the exponents appear to scale with as the system approaches the continuum limit = constantThis can be interpreted as saying that the exponent characterises the chaotic nature of the continuum classical equations of motion, and is not a lattice artefactTherefore we should not have to worry about reversibility breaking down as we approach the continuum limitCaveat: data is only for small lattices, and is not conclusive

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Reversibility: IVData for QCD with lighter dynamical quarksInstability occurs close to region in where acceptance rate is near oneMay be explained as a few modes becoming unstable because of large fermionic force Integrator goes unstable if too poor an approximation to the fermionic force is used

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PHMCPolynomial Hybrid Monte Carlo algorithm: instead of using polynomials in the multiboson algorithm, Frezzotti & Jansen and deForcrand suggested using them directly in HMCPolynomial approximation to 1/x are typically of order 40 to 100 at presentNumerical stability problems with high order polynomial evaluationPolynomials must be factoredCorrect ordering of the roots is importantFrezzotti & Jansen claim there are advantages from using reweighting

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RHMCRational Hybrid Monte Carlo algorithm: the idea is similar to PHMC, but uses rational approximations instead of polynomial onesMuch lower orders required for a given accuracyEvaluation simplified by using partial fraction expansion and multiple mass linear equation solvers1/x is already a rational function, so RHMC reduces to HMC in this caseCan be made exact using noisy accept/reject step

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Chiral Fermions

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On-shell chiral symmetry: IIt is possible to have chiral symmetry on the lattice without doublers if we only insist that the symmetry holds on shell

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On-shell chiral symmetry: II

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Neubergers Operator: IWe can find a solution of the Ginsparg-Wilson relation as follows

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Into Five DimensionsH Neuberger hep-lat/9806025A Borii hep-lat/9909057, hep-lat/9912040, hep-lat/0402035A Borii, A D Kennedy, B Pendleton, U Wenger hep-lat/0110070R Edwards & U Heller hep-lat/0005002 (T-W Chiu) hep-lat/0209153, hep-lat/0211032, hep-lat/0303008R C Brower, H Neff, K Orginos hep-lat/0409118 Hernandez, Jansen, Lscher hep-lat/9808010

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Neubergers Operator: IIIs DN local?It is not ultralocal (Hernandez, Jansen, Lscher)It is local iff DW has a gapDW has a gap if the gauge fields are smooth enoughIt seems reasonable that good approximations to DN will be local if DN is local and vice versaOtherwise DWF with n5 may not be local

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Neubergers Operator: IIIFour dimensional space of algorithmsRepresentation (CF, PF, CT=DWF)Constraint (5D, 4D)

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Kernel

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Schur ComplementIt may be block diagonalised by an LDU factorisation (Gaussian elimination)

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Constraint: ISo, what can we do with the Neuberger operator represented as a Schur complement?Consider the five-dimensional system of linear equations

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Constraint: IIand we are left with just det Dn,n = det DN

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Approximation: tanhPandey, Kenney, & Laub; Higham; NeubergerFor even n (analogous formul for odd n)j

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Approximation: j

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Approximation: ErrorsThe fermion sgn problemApproximation over 10-2 < |x| < 1Rational functions of degree (7,8)

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Representation: Continued Fraction I

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Representation: Continued Fraction II

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Representation: Partial Fraction IConsider a five-dimensional matrix of the form (Neuberger & Narayanan)

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Representation: Partial Fraction IICompute its LDU decomposition

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Representation: Partial Fraction III

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Representation: Cayley Transform ICompute its LDU decompositionNeither L nor U depend on C

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Representation: Cayley Transform IIIn Minkowski space a Cayley transform maps between Hermitian (Hamiltonian) and unitary (transfer) matrices

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Representation: Cayley Transform IIIP-P-P+P+

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Representation: Cayley Transform IV

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Representation: Cayley Transform VWith some simple rescaling

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Representation: Cayley Transform VIIt therefore appears to have exact off-shell chiral symmetryBut this violates the Nielsen-Ninomiya theoremq.v., Pelissetto for non-local versionRenormalisation induces unwanted ghost doublers, so we cannot use DDW for dynamical (internal) propagatorsWe must use DN in the quantum action insteadWe can us DDW for valence (external) propagators, and thus use off-shell (continuum) chiral symmetry to manipulate matrix elements

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Chiral Symmetry Breakingmres is just one moment of L G is the quark propagator

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Numerical StudiesMatched mass for Wilson and Mbius kernelsAll operators are even-odd preconditionedDid not project eigenvectors of HW

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Comparison of RepresentationConfiguration #806, single precision

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Matching m between HS and HW

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Computing mres using L

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mres per Configuration

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Cost versus mres

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Conclusions: IRelatively goodZolotarev Continued FractionRescaled Shamir DWF via Mbius (tanh)Relatively poor (so far)Standard Shamir DWFZolotarev DWF (Chiu)Can its condition number be improved?Still to doProjection of small eigenvaluesHMC5 dimensional versus 4 dimensional dynamicsHasenbusch acceleration5 dimensional multishift?Possible advantage of 4 dimensional nested Krylov solversTunnelling between different topological sectorsAlgorithmic or physical problem (at =0)Reflection/refraction

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Approximation TheoryWe review the theory of optimal polynomial and rational approximations, and the theory of elliptic functions leading to s formula for the sign function over the range R={z : |z|1}. We show how Gau arithmetico-geometric mean allows us to compute the coefficients on the fly as a function of . This allows us to calculate sgn(H) quickly and accurately for a Hermitian matrix H whose spectrum lies in R.

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Polynomial approximationWhat is the best polynomial approximation p(x) to a continuous function f(x) for x in [0,1] ?

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Weierstra theorem

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polynomials

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s theoremThe error |p(x) - f(x)| reaches its maximum at exactly d+2 points on the unit interval

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s theorem: NecessitySuppose p-f has less than d+2 extrema of equal magnitudeThen at most d+1 maxima exceed some magnitudeAnd whose magnitude is smaller than the gapThe polynomial p+q is then a better approximation than p to f This defines a gapLagrange was born in Turin!

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s theorem: SufficiencyThus p p = 0 as it is a polynomial of degree dTherefore p - p must have d+1 zeros on the unit interval

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polynomialsThe notation is an old transliteration of !Convergence is often exponential in d

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rational functionss theorem is easily extended to rational approximationsRational functions with nearly equal degree numerator and denominator are usually bestConvergence is still often exponential Rational functions usually give a much better approximation

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rationals: ExampleUsing a partial fraction expansion of such rational functions allows us to use a multishift linear equation solver, thus reducing the cost significantly.This is accurate to within almost 0.1% over the range [0.003,1]This appears to be numerically stable.

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Polynomials versus rationalsOptimal L approximation cannot be too much better (or it would lead to a better L2 approximation)This has L2 error of O(1/n)

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Elliptic functionsElliptic function are doubly periodic analytic functionsJacobi showed that if f is not a constant it can have at most two primitive periods

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Liouvilles theorem

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Liouvilles theorem: Corollary I

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Liouvilles theorem: Corollary IIIt follows that there are no non-constant holomorphic elliptic functionsIf f is an analytic elliptic function with no poles then f(z)-a could have no zeros either

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Liouvilles theorem: Corollary IIIWhere n and n and the number of times f(z) winds around the origin as z is taken along a straight line from 0 to or respectively

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Weierstra elliptic functions: IA simple way to construct a doubly periodic analytic function is as the convergent sumNote that Q is an odd function

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Weierstra elliptic functions: IIThe only singularity is a double pole at the origin and its periodic imagesThe name of the function is , and the function is called the Weierstra function, but what is its name called?(with apologies to the White Knight)

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Weierstra elliptic functions: III

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Weierstra elliptic functions: IVSimple pole at the origin with unit residue

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Weierstra elliptic functions: VThat was so much fun well do it again.No poles, and simple zero at the origin

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Expansion of elliptic functions: ILet f be an elliptic function with periods and whose poles and zeros are at j and j respectively

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Expansion of elliptic functions: IIBy Liouvilles theorem f/g is an elliptic function with no poles, and is thus a constant C

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Addition formul: ISimilarly, any elliptic function f can be expanded in partial fraction form in terms of Weierstra functions.

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Addition formul: II

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Jacobian elliptic functions: I

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Jacobian elliptic functions: II

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Jacobian elliptic functions: IIIWe are hiding the fact that the Weierstra functions depend not only on z but also on periods and

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Jacobian elliptic functions: IV

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Modular transformations: ISince the period lattices are the same, elliptic functions with these periods are rationally expressible in terms of each otherThis is called a first degree transformation

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Modular transformations: IIElliptic functions with these periods must also be rationally expressible in terms of ones with the original periods, although the inverse may not be trueThis is called a transformation of degree n

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Modular transformations: IIIThis is easily done by taking sn(z;k) and scaling z by some factor 1/M and choosing a new parameter The periods for z are thus 4LM and 2iLM, so we must have LM=K and LM=K/n

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Modular transformations: IV

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Modular transformations: VLikewise, the parameter is found by evaluating the identity at z = K + iK/n

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s formula: IReal period 4K, K=K(k)Imaginary period 2iK, K=K(k) , k2+k2=1Fundamental region

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s formula: II

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Arithmetico-Geometric mean: ILong before the work of Jacobi and Weierstra

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Gau transformationThis is just a special case of the degree n modular transformation we used before

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Arithmetico-Geometric mean: II

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Arithmetico-Geometric mean: III

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Arithmetico-Geometric mean: IVWe have a rapidly converging recursive function to compute sn(u;k) and K(k) for real u and 0