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1
Numerical Laplace Transform Inversion and Selected Applications
Patrick O. Kano, Ph.D.
March 5, 2010
2
Outline The talk is organized as follows:
1. Basic definitions and analytic inversion
2. Issues in numerical Laplace transform inversion
3. Introduce three of the most commonly known numerical inversion procedures
1. Talbot’s Method
2. Weeks’ Method
3. Post’s Formula
4. Illustrate through applications 1. Pulse propagation in dispersive materials
2. Calculation of the matrix exponential
5. Future directions and conclusions
Overview
Contributions
3
Laplace Transform Definition
Laplace transform solution methods are a standard of mathematics, physics, and engineering undergraduate education.
Difficult Time Dependant
Problem
Solve SimplerLaplace Space
Problem
Time Dependant
Solution
)(0)(1
tfsFtfE LL
?
A Laplace transform is a tool to make a difficult problem into a
simpler one.
4
A sufficient existence condition is that f(t) be piecewise continuous for nonnegative values of t
of exponential order
Intuitively, the Laplace transform can be viewed as the continuous analog to a power series.
tMetf
Mt 0 allfor )(such that
, and enonnegativexist There
Laplace Transform Definitions (s)sdttfestfLsF st Re where allfor )()()()(
0
2
)(
Transform Notetf
dtetfxa st
n
ex
tfa
tn
nn
sn
00
)(
5
Laplace Transform Inversion How does one return from the Laplace space representation to the time domain?
We can alleviate some of the suspense at the very beginning by cheerfully confessing that there is no single answer to this question.
Instead, there are many particular methods geared to appropriate situations.
This is the usual situation in mathematics and science and, hardly necessary to add, a very fortunate situation for the brotherhood.
Richard Bellman
6
Analytic Inversion The analytic inversion of the Laplace transform is a well-known application
of the theory of complex variables. For isolated singularities, the Bromwich contour is the standard approach.
i
i
stdsesFi
tf
)(
2
1)(
Realσ
Abscissa of convergence
Isolated singularities
Imaginary
Laplace transform inversion is not a unique operation. In practice, one can assume that the analytic inverse is well-defined.
Lerch’s theorem
0 ,0 allfor then,
)(
transformLaplace same thehave and )(function twoif
2
0
1
21
21
dttftf
fLfLsF
tftf
7
Numerical Inversion Issues The numerical inversion of the Laplace transform is an inherently ill-
posed problem.
To combat these numerical issues one may use 2 tactics
1. Fixed-point high precision variables.
2. Use of multiple algorithms, each with efficacy for certain classes of functions.
dsesFi
tf st)(2
1)(
Inherent sensitivity due to the multiplication by a exponential function of time.
Algorithmic and finite precision errors can lead to exponential divergence of numerical solutions.
8
Mathematica ARPREC
An Arbitrary Precision Computation Package Lawrence Berkley National Laboratory D. Bailey, Y. Hida, X. Li, B. Thompson
Fixed-Point High Precision Variables
High precision variables are required for most inversion methods.
This requirement is important consequences: Numerical LT methods are typically slower than other time-
propagation methods. Implementation requires either
An environment where high precision variables are innate. Additional high precision variable software packages.
Double Precision: 10-308-10308
GMP GNU Multiple Precision Arithmetic
Library MP
Matlab Based Toolbox Ben Barrows Matlab file exchange
10308~e709σ Large for even modest times.
9
Numerical Inversion
Post’s Formula Alternative to integration; arises from Laplace’s method Post (1930), Gaver (1966), Valko-Abate (2004)
The Weeks method Laguerre polynomial expansion Ward (1954), Weeks (1966), Weideman (1999)
Fourier series expansion Fourier related method Koizumi (1935), Dubner-Abate (1968), DeHoog-Knight-Stokes (1982), D’Amore (1999)
Talbot’s method Deformed contour method Talbot (1979), Weideman & Trefethen (2007)
Euler1707-1783
Laplace1749-1827
Heaviside1850-1925
Time
Post1930
Weeks1966
FourierSeries1968
Talbot1979
10
Talbot’s Method (1979) Talbot’s method is based on a deformation of the
Bromwich contour. The idea is to replace the contour with one which
opens towards the negative real axis.
Talbot’s method requires that
jj sKs
ssF
iessingularit allfor Im
as 0)(
-
real are ,,
cotis
11
Talbot’s Method The method is easily implemented in Mathematica.
Precision Run Time
10 0.047
20 0.141
40 0.391
80 1.625
3
/12 )()(
t
etfesF
ts
1 , ,0 21
dd
dssFe
itf
id
ds
is
st
)(2
1)(
cot1cot
-,1,0
cot
The Talbot method answers are accurate up-to the computation precision for time t=1.
Timeval = 1;
Rval = 1/2;
Flap[s_]=Exp[-2*Sqrt[s]];
Tfunexact[t_] =Exp[-1/t]/Sqrt[Pi*t*t*t];
Valexact = N[Tfunexact[Timeval],1000]
STalbot[r_,a_]=r*a*Cot[a]+I*r*a;
dsda[r_,a_]=I*r*(1+I*(a+Cot[a]*(a*Cot[a]-1)));
TimeDfun[r_,t_]:=1/(2*Pi*I)*NIntegrate[Exp[STalbot[r,a]*t]*Flap[STalbot[r,a]]*dsda[r,a],{a,-Pi,Pi},WorkingPrecision20];
{Timeval,Approxval}=Timing[TimeDfun[Rval,Timeval]]
RelError = Abs[Approxval-Valexact]/Valexact
12
Talbot’s Method
Attempts have been made to automate the selection: “Algorithm 682: Talbot’s method of the Laplace inversion problems”, Murli &
Rizzardi, 1990. [FORTRAN]
This is an active area of research. Optimizing Talbot’s contours for the inversion of the Laplace transform, A.
Weideman, 2006 Parabolic and Hyperbolic contours for computing the Bromwich integral, A.
Weideman & L.N. Trefethen, 2007
The primarily difficulty lies in the selection of appropriate values for the contours parameters.
2cos)(2
)(2
ttfs
ssF
1 , ,0 21
Mathematica’s adaptive integration fails for the same parameter values.
13
Post’s Formula (1930)
There are two features of Post’s formula which are particularly attractive It contains no parameters, save the order of the derivative and the
precision of the computations. The inversion is performed using
Only real values for s Without priority knowledge of poles
Post’s formula manifests the same inherent ill-posedness from which all numerical inversion procedures suffer. Errors are amplified multiplicative factor grows quickly with the order of
the derivative q The method converges slowly One needs an expression or approximation for the higher order derivatives
of F(s)
tqs
q
qqq
qsF
ds
d
t
q
qtf
/
1
)(!
1lim)(
Emil Post’s inversion procedure provides an alternative to Bromwich contour integration
14
Derivation Post’s formula can be
derived using Laplace’s method
0
2
20
0
2
2
0
0
00
k
2~)I(k,
behaviour asymptotic thehas
I(k)
integral the,k as
then,0 if and0 if
he
dτ
gd
dhe
dτ
gd
dτ
dg
kg
kg
2
1
2
1
ln12
22
2
0
0
/ln
0
0
288
1
12
11)(
!
1~)(
288
1
12
112k!
formula sStirling'With
)(k2~)(1
)(2
~)(1
1
d
gd ,ln/ , ,Assign
)(1
at evaluate and Rearrange
1)(
respect to with derivative theTake
)(
0
kktkF
t
k
ktf
kkkek
tfetkF
t
k
tfek
tt
kF
ttgfht
dfetkF
tks
dfesF
sk
dfesF
kkk
kk
kkk
k
tkkk
tkkk
skkk
th
s
Approximate
15
Derivatives Finite differences an obvious method by which to approximate the
derivatives of a reasonably behaved function.
The Gaver functionals can be computed by a recursive algorithm:
)(lim)(
)2ln(2)2ln()(
1
)2ln(2)2ln(1)(
01
tftf
tjqF
j
q
q
q
tqtf
nxFxnFnxF
tqF
q
q
tqtf
q
j
j
q
qqq
Gaver-Post Formula1966
np p,1 1G
n1 2ln2ln
111
np
0
n
pn
p
n
Gp
nG
p
n
tnF
tnG
qqGtf )(q
16
Derivatives Post’s formula does not require a finite difference
approximation. For a particular function form, e.g. composition of two
functions, a tailored method may be more robust.
qqq
m
m
q
q
pmqm
mpq
mpq
pq
pq
q,p
q
pp
p
q
q
xgB
dx
gdB
qB
B
Bdx
gd
m
qB
dx
gd,,
dx
gd,
dx
dgBg(x)
dx
fdf(g(x))
dx
d
)(
1for 0
1
1
1
,
1,
0,
0,0
1,
1
1,
1
1
2
2
0
Bell Polynomials of the Second Kindwww.mathworks.com/matlabcentral/fileexchange/14483
Faa di Bruno’s formula
17
Acceleration Sequence acceleration methods be used to greatly increase accuracy
The proper application of an acceleration convergence method requires some additional knowledge about the series.
Post’s formula is logarithmically convergent
Slow Sequence {fq(t)}
Linear or NonlinearTransform
Fast Sequence {hq(t)}
eConvergencLinear 10
eConvergenc cLogarithmi1lim
)()(
)()( 1
ca
tftf
tftfa
q
q
q
11
lim)()(
)()(lim
)()()(~)(
1
211
q
q
tftf
tftf
q
tc
q
tctftf
q
q
q
Gaver (1966)
18
Acceleration Wynn-rho algorithm is well suited to logarithmically convergent
sequences. Studies have shown that it is useful for the Post formula:
NSum in Mathematica implements these acceleration methods.
Post Inversion Formula and Sequence AccelerationUA VIGRE Project 2009J. Cain & B. Berman
11
01
1
21
32
22
212
20
21
12-Q
12
111
10
11
02-Q
02
010
00
01
02-Q
0
0
0
0
by Q,even for ,lim)(
function for theion approximatan yields algorithm The
QQQ
Q
Q
f
f
f
f
ftf
kQq
Qk
k qkq
kqk
qk
1,,0
3,,0
1 1111
"AlternatingSigns" method fo r s ummands with a lternating s igns
"EulerMaclaurin" E ulerM ac laurin s ummation method
"WynnEpsilon" Wy nn eps ilon ex trapo la tion method
19
Application of Post’s Formula
Rapid computation of the distribution of an initial optical pulse in a fixed dielectric medium with a nontrivial material dispersion relation.
NSF Grant ITR-0325097An Integrated Simulation Environment for High-Resolution Computational Methods in Electromagnetics with Biomedical ApplicationsMoysey Brio, et. al.
sr
Material
Biological materials often have a dielectric constantwhich is a complex function of wavelength.
Input Pulse of Light Out Pulse of a Different Shape
Create databases of pre-computed tables which can be used by devices which must operate in real-time.
20
Cole-Type Dispersion Relation
Many real world materials can be described by a Cole-type dispersion model.
,0
,0
,0
),0[
]1,0(
)1,0[
1)(
01
n
n
n
n
nba
n
nr
b
a
sss
nn
02.0
958.7)(
105.3
30.0
05.53)(
100.4
10.0
96.7)(
0.100
10.0
96.7)(
0.32
4
4
74
3
3
43
2
2
2
1
1
1
a
ms
a
ns
a
ns
a
ps
Brain
White Matter
A standard method used in computational optics is to incorporate the dispersion relation by means of an associated difference equation.
For fractional coefficients, it is not clear how to translate into an associated equation.
Fractional aCoefficients
1
02.0
0.4
b
21
Maxwell’s equations are the starting point for this analysis.
In the Laplace space, the convolution and derivatives become multiplications.
dxEttxEtxD
DE
txt
HtxE
txt
DtxH
txH
txD
t
,,,
nt displaceme theand strength field electricbetween assumption General
,,
,,
0,
0,
0
00
0
Maxwell’s Equations
Temporal Convolution
)()()(0
sGsFdtgfLt
),()(),(
0
sxEsdtxELt
22
Maxwell’s equations now have a simpler form.
Eliminating the magnetic field H from the problem,
One obtains the wave equation in Laplace space
Maxwell’s Equations
0,,,
0,,)(1,
0,
0,
,)(1,
0
00
0
txHsxHssxE
txEsxEsssxH
sxH
sxE
sxEssxD
)0,()0,(),()(1),( 00000022
txt
EtxEssxEsssxE
)0,()0,( 0
2
txt
EtxH
HHH
23
One can more succinctly state this last equation as
Applying a Fourier transform yields the desired solution in the joint space
002
22
22
1 )(1)(
)0,()0,(),(
),(1
),()(),(
css
txt
EtxEssxV
sxVc
sxEsc
ssxE
r
r
Maxwell’s Equations
222 )(
),(),(
kcss
skVskE
r
24
Database Coefficients in the Joint Space
The solution in a dielectric medium can be characterized by one coefficient α and its time derivative.
Compute high order derivatives of α(k,s) and β(k,s) beta derivatives are trivially obtain from the alpha derivatives.
skssk
kcsssk
tkEt
sktkEskskE
r
,, )(
1,
)0,(,)0,(,),(
222
For a given dispersion relation εr(s), the coefficients are pre-computed and stored in a matrix of k vs time.
skqDsksDskD qqq ,,, 1
The crux of the problem is the arbitrary precision calculation of the q-th derivative of α .
25
Derivative Approaches1.1. Standard Gaver-Wynn-RhoStandard Gaver-Wynn-Rho
Finite Differences + Wynn-Rho Acceleration A brute force application entails a computation for each k and s.
2.2. Gaver-PostGaver-Post Finite Differences + Wynn-Rho Acceleration The arbitrary precision computation of the dispersion relation εr(s) is time consuming. Dispersion relation is independent of k More efficient to store εr(s) and call for each k evaluation of α.
3.3. Bell-PostBell-Post Analytic Derivatives + Wynn-Rho Acceleration Store εr(s) and its derivatives. Use Faa di Bruno’s formula for the qth derivative of the computation of two functions.
tjqsk
j
n
q
q
tqtk
q
j
j
q
)2ln(,
2)2ln(),(
01
26
Bell-Post Method The problem of determining the time dependence of α(k,t) and thus
the electric field is reduced to evaluating the susceptibility function and its arbitrary order derivatives.
sDnnssnDsDssgD
sgDsgDsDgBkcss
pskD
kcsssg
ssf
dx
gd,,
dx
gd,
dx
dgBg(x)
dx
fdf(g(x))
dx
d
kcsssk
rn
rn
rnn
q
p
pqpqp
r
pq
r
pq
pq
q,p
q
pp
p
q
q
r
212
0
12,1222
222
1
1
2
2
0
222
)1(2)(
Rule Leibniz
)(,),(),(1!
),(
)(
1)(
)(
1,
27
Cole-Type Dispersion Relation
For white brain matter the derivatives of εr(s) can be found by using the Faa di Bruno formula.
,0
,0
,0
),0[
]1,0(
)1,0[
1)(
01
n
n
n
n
nba
n
nr
b
a
sss
nn
kbk
j
kk
pap
j
ap
kqkq
q
k
kq
q
b
a
nn
q
q
nq
q
qr
q
gjbgf
sjasg
sgggBsgfds
sgfd
ssf
ssg
sgfds
d
s
q
ds
sdn
1)1()(
1)(
)(,,,)())((
1)(
)(
)(!1)(
1
0
11
0
1
121,
0
1
10
28
Mathematic Implementation Flow DiagramInputs
1. Choose [qmin,qmax]2. Inversion time t3. Take an explicit expression for εr(s) and its derivatives4. A set of wavenumber k
Evaluate εr(s) and its derivatives at s=q/t
Compute s2εr(s) and its derivatives via Leibniz’s rule
Compute the Bell polynomials from the recursion relation
For k, compute s2εr(s) + c2k2
Compute the qth and (q-1)th derivatives of α(k,s)
Compute the qth derivative of β(k,s)
Approximate the inversion coefficients via Post’s formula
Apply Wynn-rho acceleration
Repeat for each q
29
Brain White Matter: Run Time
The Bell-Post and Gaver-Post methods are faster than a standard Gaver. The acceleration dominates over the sequence computation times. The time follows a polynomial growth with q-max.
Time=(16/3)t0
100 Digits Precision
pAqT Case Log A p
Bell-Post
-0.592 1.670
Gaver-Post
-0.189 1.438
Brute
Gaver
0.148 1.655
30
Brain White Matter: Accuracy
Time=(16/3)t0
100 Digits Precision
The Bell-Post and Gaver-Post methods have comparable accuracy At higher precision and Post formula derivative orders.
31
The Weeks Method (1966) The Weeks’ method is one of the most well known algorithms for the
numerical inversion of a scalar Laplace space function. It popularity is due, in part, to the fact that it returns an explicit
expression for the time domain function. The Weeks method assumes that
a smooth time domain function of bounded exponential growth can be expressed as the limit of an expansion in scalar Laguerre
polynomials.
nxn
nx
n
N
nn
btn
tN
NN
xedx
d
n
exL
btLeaetf
tftf
!
2)(
)(lim
1
0
The coefficients {an}
1. contain the information particular to the Laplace space function
2. may be complex scalar, vectors, or matrices
3. time independent
32
The Weeks Method Two free scaling parameters σ and b, must be
selected according to the constraints that b>0 ensures that the Laguerre polynomials are well
behaved for large t σ>σ0-abscissa of convergence
Laguerre Polynomials
33
The Weeks Method The computation of the coefficients begins with a
Bromwich integration in the complex plane.
Assume the expansion
Equate the two expressions
dyiyFee
tf
iys
iytt
2)(
y- , , 0
0
2)(n
nbt
nt btLeaetf
dyiyFebtLea iyt
nn
btn
2
12
0
34
Key Weeks Method Facts It is known that the weighted Laguerre coefficients have the
Fourier representation.
Performing the appropriate substitution, assuming it is possible to interchange the sum and integral equating integrands leaves
dybiy
biyebtLe n
niyt
nbt
12
12
01
nn
n
n iyFbiy
biya
35
Moebius Transformation One may apply a transformation from complex
variable s to a new complex variable w
biy
biywiyσs
bs
bsw
, or with
Isolated singularities of F(s) in the s-half-plane are mapped to the exterior of the unit circle in the w-plane.
36
W-plane Representation With the change of variables, one obtains
a power series in w whose radius of convergence is greater than 1.
The function is analytic on the unit circle.
Numerically, the evaluation of the integral can be computed very accurately using the midpoint rule
1
1
1
2
0 w
wbF
w
bwa
n
nn
d
e
ebF
e
bea
i
i
iin
n
1
1
1
2
2
1
M
m
e
ebF
e
be
M
ea m
M
Mmi
i
iin
Min
n m
m
m
m
1
1
1
2
2 2/1
2/1
2/1
2
37
Matrix Exponential Application
An application of the Weeks method is to the calculation of the matrix exponential.
00 )(0 , xetxx)(tx xAdt
xd At
What does it mean “the
exponential of a matrix”?
!3!2
32 AAAIeA
Why don’t we just calculate this?
“Nineteen Dubious Ways to Compute the Exponential of a Matrix”,SIAM Review 20, C. B. Moler & C. F. Van Loan, 1978.
“Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later”, SIAM Review 45, C. B. Moler & C. F. Van Loan, 2003.
Inverse Laplace Transform (#12)
Bromwich
At AsIi
e dse2
1 st1
Resolvent Matrix of A
Apply the Weeks method
38
Matrix Exponential Application
Matlab: Pade’ approximation with scaling and squaring (#3)
Matlab demos expmdemo1: Pade’ + Scaling + Squaring in an m-file expmdemo2: Taylor Series expmdemo3: Similarity Transformation
j
q
jpq
jp
jpq
pq
pqB
nn
AA
Bjqjqp
qjqpBD
Bjpjqp
pjqpBN
BD
BNe
ee
0
0
22
)!(!!
!!)(
)!(!!
!!)(
)(
)( expm
39
Beam Propagation Equation Nonparaxial scalar beam propagation equation
Discretisation in space yields a set of ODEs
The Laplace transform in z yields
uzyxnkx
iuiz
u
),,(2202
2
uzyxnkDiuiz
u),,(22
0
2
2220
1 0)(ˆ
InkD
A
uAIIisIsu
u = a component of the electric field
40
Beam Propagation Equation The Laplace space function is of a matrix exponential
The issue is how to pick the Laguerre polynomial parameters σ and b. Weeks’ original suggestions
Error-Estimate Motivated Approach Weideman Method
minimization of the error estimate as a function of σ and b
Min-Max Method Maximum the radius of convergence as a function of σ and b
Bromwich
Mt MsIi
e
AIIiMuMsIsu
dse2
1
0)(ˆ
st1
1
tcoefficienexpansion #21
0max 0
N
t
N b
t,σσ
Expensive Matrix
Inversions
41
Error Estimates A straight forward error estimate yields three contributions
Discretisation error: Finite integral sampling Truncations error: Finite number of Laguerre polynomials Round-off error: Finite computation precision
1
0
22N
nFn
NnFn
ttotal aaeE
Truncation Round-Off
Weideman Method
Min-Max Method
Midpoint rule on circular contour
1
)(
)()(
2
1
1
1
2
rr
rKT
r
rKdw
w
rKa
NE
nrw
nFn
radiuscontour circular plane-wr
matrixresolvent theof norm
rK
42
Beam Propagation Equation Example Multi-mode interference coupler
σ b Maximum
Solution
Absolute Error
Weeks 1 32 12.28
Min-Max 20 10 0.00119
Weideman 11.84 16.79 0.000425
By proper selection of the parameters, it is possible to perform the calculations in double precision.
43
Pathological Matrices An application is the exponential of special matrices.
6x6 Pei MatrixMaximum Element Relative Error
32 Coefficients
2 1 1 1 1 11 2 1 1 1 11 1 2 1 1 11 1 1 2 1 11 1 1 1 2 11 1 1 1 1 2
Weeks’
MinMax(2 Search Methods)
Weideman Eigenvalues 1N+1 (7)
gallery(‘pei’,6)
www.math.arizona.edu/~brio/WEEKS_METHOD_PAGE/pkanoWeeksMethod.html
44
Future Directions
NLAP-CL: Robust Parallel Numerical Laplace Transform Inversion via a C-CUDA Library and Application to Optical Pulse Propagation•Mosey Brio – University of Arizona•Patrick Kano – Applied Energetics, Inc.•Paul Dostert – Coker College
Extend NSF supported Post’s Formula Work to 2D and 3D
Mathematica is too slow.
(CUDA)NVIDIA’s Compute Unified Device Architecture
December 2009 – NSF proposal submitted
Tables of coefficients for
multiple dispersive materials
NLAP requires multiple simple arithmetic computations in high precision.
MATLAB C-MEX Files MATLABNLAP
Front End
45
Summary & Conclusions The purpose the of the presentation was to provide some insight and
illustrate applications of numerical Laplace transform inversion.
Standard methods Talbot’s Method Post’s Formula The Weeks method
Illustrated two examples Calculation of the matrix exponential Optical pulse propagation in dispersive media
Numerical Laplace transform inversion is a topic multitude of nuances to provide avenues for further research popularity increase as computing power improves potential for practical application in diverse fields
Great utility and intellectual merit to the development of a general numerical Laplace inversion toolbox.
46
Sources Numerical Inversion of the Laplace Transform, Bellman, Kalaba, Lockett, 1966.
Peter Valko’s NLAP website: www.pe.tamu.edu/valko/public%5Fhtml/NIL/
“Numerical inversion of Laplace transforms using Laguerre functions”, W. Weeks, Journal of the ACM, vol. 13, no. 3, pp.419-429, July 1966.
“The accurate numerical inversion of Laplace transforms”, J. Inst. Math. Appl., vol. 23, 1979.
“Application of Weeks method for the numerical inversion of the Laplace transform the matrix exponential”, P. Kano, M. Brio, J. Moloney, Comm. Math. Sci., 2005.
“Application of Post’s formula to optical pulse propagation in dispersive media”, P. Kano, M. Brio, Computers and Mathematics with Applications, 2009.
47
BACKUPS
BACKUP
48
Laguerre Polynomials An unstable approach to obtain the time domain
function is to generate the Laguerre polynomials is to use the recurrence relation
A backward Clenshaw algorithm is a stable method.
0
1
0
11
2)(
1)(
1)(
)()(12)(1
nnn
tb
nnn
btLaetf
xxL
xL
xnLxLxnxLn
49
Analytic Inversion
2order 0
1
1)(
2
2
s
es
sg
ssF
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50
Application of Post’s Formula Post’s formula was implemented to
invert the Fourier-Laplace space coefficients which arise from the solution of the optical dispersive wave equation.
We considered three implementations Standard Gaver-Wynn-Rho Gaver-Post Bell-Post
51
Brain White MatterRelative Error, Bell-Post, k=kmax Relative Error, Gaver-Post, k=kmax
The Bell-Post method performs well at modest values for the
precision and order of the Post formula approximation.
At higher precision and Post formula approximation order, the Gaver-Post has an accuracy/unit run time comparable or
better than the Bell-Post method.
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