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Continued Fraction Bag Of Tricks
Bálint Joó
School of Physics, University of Edinburgh
UKQCD Collaboration
In collaboration with
Robert Edwards
Tony Kennedy
Kostas Orginos
Urs Wenger
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Jlab
LHPC Collaboration
Edinburgh
UKQCD Collaboration
MIT
LHPC Collaboration
DESY Zeuthen UKQCD Collaboration
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The Goal
Chiral symmetry on the lattice
Massless Dirac Operator,
relation:
, must satisfy Ginsparg-Wilson
Explicit solution by Neuberger
is an auxiliary Dirac Operator (the kernel)
)
is the Matrix Sign Function (
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Traditional Implementation
Approximate Sign Function with rational function
and
.
polynomials in H of degree
with
Write approximation as a sum over poles:
: Use Multi-shift solver to apply poles
: Use nested CG Solver
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Inversion of
Application of
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Motivation for 5D approach:
Nested solver is inefficient
– Each outer solve throws away Krylov Subspace built up by inner
solve.
Can solve an enlarged 5D system: equivalent to solving by nested
solve
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Basic Idea
Consider the matrix equation:
on
Usual (4D): Invert
Higher dimensional (5D): Solve enlarged system
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From second row:
From first row:
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Enlarged system for sum over poles:
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More than just an inversion trick: Domain Wall Fermions
It has been shown (Boriçi, Edwards & Heller, Brower Nef and Orginos)
that the 5D domain wall fermion matrix can be reduced to a 4D effective operator which approximates the Overlap
LDU decomposition (see Appendix)
...
... ...
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With
..
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is rational approximation to overlap action:
is the 5D transfer matrix Hamiltonian
traditional DWF - Higham representation of
from Zolotarev approximation
“Optimal DWF” (Chiu)
is contribution from 5D Bulk (Pauli-Villars) modes
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Choice of Kernel
Shamir-like DWF: From 5D transfer Matrix:
is the Wilson Fermion Operator,
where
,
is the Domain Wall Height
Wilson/Boriçi (Overlap like) DWF:
Recent Unification of kernels: Möbius Kernel (Brower, Nef, Orginos)
is arbitrary
Shamir),
Borici,
(
chooses
scale factor
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to shift
For Möbius, with Higham (tanh) approx, use scale factor
spectrum of
to where
is a good approximation.
to where approximation be-
But can push large eigenmodes of
comes worse again
−α|λ| −|λ|
max
max
α
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|λ| α|λ|
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|λ|
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−α|λ| −|λ|
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DWF Determinant
Cancel Bulk Contribution using
:
5D DWF action, generates 4D effective action, eg for 2 flavours:
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DWF Pros and Cons
Advantages
– Admits red-black preconditioning (in 4D and 5D)
can be chosen for optimal rational approximation
–
– Well known, both from 5D and 4D perspective
Disadvantages
from Zolotarev
– Optimal
numerically expensive
– Apart from placement of Domain Wall (permutation freedom),
operators cannot be tuned
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An Alternative: Continued Fractions
Different functional form for approximating the sign function. Write:
specify approximation
Coefficients
Equivalence transformation between continued fractions
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Inversion Trick approach ( Wenger et al. hep-lat/0110070, hep-lat/0403003):
Consider:
...
with
) with even ,
For type 0 rational approx (
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Then solving
..
.
..
.
is equivalent to solving:
ie
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-s.
Large (continuous) freedom of choosing
Naive choice:
, ensures diagonal
elements are
(except for corner)
Matrix is block tridiagonal – useful: easy decompositions
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More than just an inversion trick: LDU Decomposition of Operator
Set
for convenience and consider operator (2n = 2):
are hermitian and invertible.
where
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as:
Can Decompose
with
with
and
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Decomposition works for any positive , and
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Now identify:
and
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Comments
are successive continued fractions growing from
, , ...,
“the tail inwards”
Only
is coupled to the mass term
is the continued fraction we need:
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Determinant of Continued Fraction Operator
we have that
Since
and since
we have that
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Determinants of
play same role as the bulk (Pauli–
Villars) determinant in DWF (ie: they need to be cancelled)
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Cancelling The Unwanted Determinants
Consider the operator
...
... ...
..
which by construction has
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We see immediately that
..
... ... ...
Is an effective 4D overlap operator, with
we have
and since
:
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Further, we can consider the action:
and change variables:
we have
Since
no extra Jacobean arises from changing variables.
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We have formally reduced a 5D continued fraction theory to an effective 4D overlap theory
If the
are chosen from the Higham Representation for
and
is a Domain Wall kernel (
or Möbius) this theory is exactly
equivalent to DWF.
The difference is a change in representation (Cayley Transform
form of Rational Function, v.s. Continued Fraction form of Rational
Function)
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Practicalities: PV Matrix
is problematic: (
Evaluating
case)
The
element contains a continued fraction.
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. Don’t want to evaluate this.
In general it would be
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and :
However, defining dimension reduced forms of
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...
...
...
...
...
Can form
...
...
...
...
is straightforward to evaluate (no Cont. Frac. terms)
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still have -s along diagonal so
and
so
And (finally):
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Choice of Kernel
Recall definition of
:
...
, operator is straighforward: need
(when
)
If
of
applications
has a demominator (
or Möbius) we can play additional trick:
If
):
Consider Möbius case (includes
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. (One application of Q)
, so
2.
1. Solve
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Now solve
Two step process:
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Relative Costs of Applying Operators
DWF (Shamir/Möbius)
Cont. Frac. (
)
Cont. Frac. (
)
Relative cost
per CG Iteration
compared to DWF
1
1
), with even order
Cont. Frac.: type 0 approximation (
Continued Fraction can be competitive if matrix is sufficiently better conditioned than DWF matrix for same
.
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Chiral Symmetry Breaking & Measuring
For approximate Overlap Operator, breaking of GW relation is:
to get:
Substitute in
or
All information about breaking GW relation lives in
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(Kikukawa, Brower, Neff, Orginos):
and
:
Usual DWF
with
Going through all the algebra
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.
is a matrix element of
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Can measure
fraction version of
.
cancels in
are renormalised by
and
for just effective 4D action, by using 4D partial
.
Relation between formulations:
-s
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for Shamir and
for Boriçi, with the
tuned to give the same pion mass in both cases
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. Eg:
Can have negative
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Further Random Musings on
always positive
approx never crosses
for
For low order Zolotarev Approx, big wiggles may cancel to give small
. Can I really trust
to tell me the amount of chiral symmetry
breaking?
not sufficient measure of chiral symmetry breaking (may need
to measure higher order operators in
. Eg
is always positive)
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Numerical Studies
Performed test to study efficiency of various formulations.
, 15
–
Dynamical DWF dataset
Used RBRC
Configurations
– Kostas also worked on
sets
Performed inversions with various formulations (details later)
Computed
Counted CG iterations
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Matched pion mass between operators with
kernels
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and Möbius DWF
ALL operators are even-odd preconditioned
Did not project eigenvectors of
.
Computers used: Myrinet and Gig-E PC Clusters at JLab, UK National Grid Service Myrinet PC Clusters in Leeds & Oxford, UK HPCx
IBM facility, Edinburgh BlueGene/L, Edinburgh QCDOC
Operators are coded in Chroma, using BAGEL Dslash from Peter
Boyle where appropriate (HPCx, QCDOC, BlueGene)
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Comparing Operators
DWF Kernel, Higham Approximation
Cπ(t)
0.0001
Moebius DWF (b5=1, c5=0)
Continued Fraction (HT)
Wilson Kernel, Higham Approxmation
Moebius DWF (b5=1, c5=1)
Continued Fraction (HW)
0.0001
Borici DWF
Normal Shamir DWF
1e-05
1e-05
Relative difference (%)
1e-06
0.0025
0.015
0.002
0.01
0.0015
0.001
0.005
0.0005
0
0
10
20
30
0
0
10
t
. Single Precision
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t
Single Configuration (#806). Max Difference
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between Operators with DWF and
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kernel
Matching
Tuning quark mass to get same Pion Mass, Higham Approximation
0.01
Moebius (Hw mode) Ls=12, mf=0.0115
Cont. Frac (Hw mode) Ls=12, mf=0.0115
Moebius (Shamir mode), Ls=12, mf=0.02
Cπ(t)
0.001
0.0001
1e-05
1e-06
Effective Mass of π
0
2
4
6
8
0.5
10
12
14
16
t
18
20
22
24
26
28
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Folded Correlation Function about Mid Point
0.4
0.3
0.2
0.1
0
0
2
4
8
6
10
12
14
t
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using
Computing
3
V=16 32, Ls=12,M=1.8, Shamir msea=0.02, 15 Configs, matched Borici mval=0.0115
0.0004
Moebius DWF (Midpoint/Pseudo) (H_w, Higham Approx)
Continued Fraction (∆L/Pseudo), (H_w, Higham Approx)
Cont. Frac. (∆L/Pseudo):
bare mres = 0.000110(-12,13)
mres ratio
0.0003
Moebius (Midpoint/Pseudo):
bare mres = 0.000114(-13,+14)
2
χ ./ d.o.f = 1.27
2
χ ./ d.o.f = 1.33
0.0002
0.0001
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measurements per configuration
Results with Hw kernel, Ls=12
| mres | per conf
0.01
0.001
0.0001
1e-05
1e-06
1e-07
| λmin | / | λmax |
1
0.01
0.0001
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(Configuration - 605)/50
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HT kernel
Hw kernel
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Cont Frac (Higham Approx)
Cont Frac (Zolotarev Approx)
Moebius DWF (Zolotarev Approx)
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Cost Normalised by Unscaled Shamir DWF
Ls=14
off graph
Scaled Shamir (α=1.7, HT, Higham)
Scaled Shamir (α=2, HT, Higham)
Scaled Shamir (α=2.5, HT, Higham)
Scaled Shamir (α=3, HT, Higham)
Continued Fraction (α=1, Hw, Higham)
6
Continued Fraction (Hw, Zolotarev)
Continued Fraction (α=1, HT, Higham)
Continued Fraction (HT, Zolotarev, 3 Cfg)
4
Ls=6
2
1e-06
0.0001
| mres | (mShamir/m)
0.01
1
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(b5=1, c5=1, Zolotarev)
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Borici (α=1, Hw, Higham)
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Conclusions 1: Theory and Background
Nice unified framework for 5D chiral actions. Both DWF and
Continued Fraction
– reduce to Approximate 4D Overlap Effective Action
or full Möbius (including Shamir) kernels
Separate issues of representation, approximation and kernel
Should choose one or other on basis of numerical cost
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– can accomodate
– can accomodate Higham and Zolotarev approximations for
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Conclusions 2: Numerical tests (so far)
Continued Fraction with Zolotarev coefficients.
–
Kernel becomes competitive with DWF at reasonable
Cost per application amortised by small
(Preliminary)
–
Kernel seems least costly
.
– But need to check for other approximation ranges
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Möbius appears best for kernels with denominator and Higham apkernel and Zolotarev approx.
proximation but is costly with
– scaling Shamir kernel helps reduce
for Higham approx.
Continued Fraction appears to be costly for Higham approx.
equivalence transform coeffs
– But can still try and tune
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Future/Current Work
Complete results for Cont. Frac. with Shamir kernel (in production
on Edinburgh BlueGene/L and QCDOC)
Still to examine 3rd formulation, based on Partial Fraction approximation (Neuberger and Narayanan’s “Alternative to Domain Wall
Fermions”)
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Appendix 1: Cayley form of Rational Functions
.
Write
Define “Euclidean Cayley Transform”:
Oddness of
:
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implies:
and
in factored form:
and
Writing
write:
To find form of
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Finally
And so
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Appendix 2: Domain Wall Fermions
: R. Brower, H. Neff, K. Orginos hep-lat/0409118
Möbius kernel for
Transfer Matrix Hamiltonian:
Shamir)
Borici,
chooses operator (
eg:
is arbitrary scale factor
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is the Domain Wall height
is the fermion mass, and
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Möbius 5D Operator (R. Brower, H. Neff, K. Orginos hep-lat/0409118)
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¨
Reducing Mobius
DWF to 4D effective theory
First written down by Boriçi
Approach presented here (a la Brower et al):
– Define change of variables
from new variables
.
using
– “LDU” decompose
– Define
– Diagonal piece will yield effective 4D operator
.
– Ls=4 examples for simplicity
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Definitions
New variables:
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Write
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LDU Decomposition
with
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with
i.e:
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Putting it together:
and
and switch to
Step 1: Apply
:
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-s appear
and
Step 2: Multiply in the
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)
overlap with Cayley representation:
( extra bit
to get:
and the
Step 3: Multiply in the
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with
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?
What about this
arises from Jacobian from change of variables
, so called high
corresponds to modes with
frequency, or bulk modes.
In calculations, “cancelling high-frequency bulk modes with PauliVillars fields” corresponds to removing this extra piece.
, eg 2 flavour
Usually done by working with
HMC Hamiltonian:
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Summary of Domain Wall Fermions
DW fermions are 5D realisations of Overlap fermions with Cayley
representation for the sign function and extra high frequency (PV)
modes that need to be cancelled
Approximation specified by
approximation (traditional DWF)
Physics (
behaviour) dictated by choice of
Möbius, by
.
, or in the case of
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the extra scale factor
When using Zolotarev
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Appendix 3: Details of Approximations
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Approximation: Higham (tanh), Neuberger’s Polar Approx.
Approximation is
with
for all ,
Cayley form:
– Conventional DWF.
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Approximation: Zolotarev’s approximation
:
Approximation is over range
with
come from Jacobian elliptic functions:
where the coefficients
,
is the complete elliptic integral, and
end of the approximation range.
is the lower
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can be conveniently split into partial or continued fraction form.
Cayley form is entirely trivial (according to Tony) (T.W. Chiu hep-lat/0209153)
can be scaled to cover spectrum of operator:
eg:
Approximation is rationally BAD outside approx. interval
Approximation is (minimax) optimal over the approx. interval.
Maximum error decreases exponentially with .
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Zolotarev or tanh ?
Comparison of Tanh and Zolotarev Approximations
1
0.6
2
1 - ε (x)
0.8
0.4
0.2
0
-1.4 -1.2
-1
-0.8 -0.6 -0.4 -0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0
x
0.2
0.4
0.6
0.8
1
1.2
1.4
ε(x)
1
0
tanh approx - Ls=12
zolotarev approx - Ls=12
tanh approx - Ls=16
zolotarev approx - Ls =16
tanh approx - Ls=20
tanh approx - Ls=24
-1
-1.4 -1.2
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Zoom in on Graph showing approximation errors
unscaled tanh - Ls=16
zolotarev - Ls=16, eps=0.01
unscaled tanh - Ls=20
zolotarev-Ls=20, eps=0.01
zolotarev - Ls = 24, eps=0.01
Zolotarev - Ls=12, eps=0.4
2e-05
2
1 - ε (x)
1e-05
0
-1e-05
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Maximum error as we increase the range or approximation
maximal error δ rational approximations over +/- [ε,1]
1
δ = | 1 - R(ε) |
0.01
0.0001
1e-06
1e-08
1e-10
1e-06
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Ls=12 - zolotarev
Ls=16 - zolotarev
Ls=20 - zolotarev
Ls=24 - zolotarev
Ls=12 - unscaled tanh
Ls=16 - unscaled tanh
Ls=20 - unscaled tanh
Ls=24 - unscaled tanh
0.0001
ε
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Maximum error as we change Ls
maximal error δ of type 0 Zolotarev approximation over +/- [ε,1] for various Ls
0
10
-1
ε=0.005
ε=0.001
ε=0.0001
ε=0.00001
10
-2
10
-3
10
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δ = | 1 - R(ε) |
10
-5
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error
are important to get right?
is unimportant, I can reapproximate Zolotarev with
, and the approximation will be better than
.
if
BUT which
so that for
Given an , there is some
is smaller than Zolotarev.
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Zolotarev is uniformly good for a given interval.
is good for larger .
(since it is not “uniform”)
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Maximum error is always BAD for
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