review of paper - jun, 27th

IntroductionStandard Forward Recursions

Modified Forward MethodsNumerical Tests

Summary

A unified approach to the Clenshawsummation and the recursive computation of

very high degree and order normalisedassociated Legendre functions

(Holmes and Featherstone, 2002)

Eduardo da Silva Lema

Division of Earth and Planetary SciencesDepartment of Geophysics

Geodesy Labo.

June 27, 2012E.S. Lema Geoid Seminar



Summary

Outline

1 Introduction

2 Standard Forward Recursions

3 Modified Forward Methods

4 Numerical Tests

E.S. Lema Geoid Seminar



Summary

Legendre Polynomials

Spherical Harmonics expansion of gravity potentialMaximum degree is increasing

Wenzel (1998), degree 1800EGM2008, degree 2190

High degree SH is important for other disciplines as wellMeteorologyQuantum PhysicsElectronic Engineering

Computation of Legendre Polynomials is the mainchallenge involved




Summary

Spherical Harmonics Expansion

Truncated SH reduce to sums of ALFs or their derivativesIn fact: S(d) = cΣmΩ

(d)m

where: Ω(d)m = Σα=1,2

X (d)mαcosmα, forα = 1

X (d)mαsinmα, forα = 2

and: X (d)mα = ΣnEnmαP(d)

nm (θ)

P(d)nm (θ) are the fully normalized ALF




Summary

SH expansion of Earth’s Gravity

In general. . .

V (r , θ, λ) = GMr +

GMr Σn

(ar

)nΣm

(

Cnm1cosmλ+ Cnm2sinmλ)

P(d)nm (θ)

Or, alternatively. . .

V (r , θ, λ) = GMr +

GMr Σm

[

cosmλΣn(a

r

)n Cnm1P(d)nm (θ)sinmλΣn

(ar

)n Cnm2P(d)nm (θ)

]

So that:Enmα =

(ar

)n Cnmα, α = 1,2Xmα = Σn

(ar

)n CnmαPnm(θ)




Summary

Numerical considerations

Standard recursions compute ALF directlySimpleWork well for low degrees

Issues for ultra-high degrees (e.g. 2700)∣

∣Pnm(θ)∣

∣ can range over 5000 orders of mag.IEEE standard ranges doesn’t span these values (about620 o. mag.)Underflow, i.e., values so small that are set to zeroOverflow, i.e., values so large that are set to NaN




Summary

Clenshaw’s method

Standard alternative for ultra-high degreesStableComputes both S and S(1)

But. . .

Derivation is based on matrix algebra

Recursions are quite simple to implement

Doesn’t compute individual ALF




Summary

Forward Column

Non-sectoral-values

Pnm(θ) = anmtPn−1,m(θ)− bnmPn−2,m(θ), ∀n > mwhere t = cosθ

Sectoral values, Pmm(θ):

Serve as seed for the above recursion

Are computed by the following recursion:

Pmm(θ) = u√

2m+12m Pm−1,m−1(θ), ∀m > 1

where u = sinθP0,0(θ) = 1P1,1(θ) =

√3u




Summary

Forward Column

Figure: Schematic of forward column recursion




Summary

Forward Row

Non-sectoral-values

Pnm(θ) =1√

j

(

gnmtu Pn,m+1(θ)− hnmPn,m+2(θ)

)

, ∀n > m

where t = cosθ, u = sinθ, j = 2 for m = 0, j = 1 for m > 0

Sectoral values, Pmm(θ)

Serve as seed for the above recursion

Are computed by the following recursion:

Pmm(θ) = u√

2m+12m Pm−1,m−1(θ), ∀m > 1

P0,0(θ) = 1P1,1(θ) =

√3u




Summary

Forward Row

Figure: Schematic of forward row recursion




Summary

First derivatives

Forward column

P(1)nm(θ) = 1

u

(

ntPnm(θ)− fnmPn−1,m(θ))

, ∀n ≥ m

Forward row

P(1)nm(θ) = m t

u Pnm(θ)− enmPn,m+1(θ), ∀n ≥ m




Summary

Numerical problems with standard forward methods

In double precision standards it underflows wheneverM>190020o ≤ θ ≤ 160o

um → 0 as m increasesStorage capacity of dp is exceeded

Corresponding ALF are masked and don’t influence finalSH




Summary

Modified Forward Row

As in Libbrecht (1985)

Pnm(θ)um = 1√

j

(

gnmt Pn,m+1(θ)

um+1 − hnmu2 Pn,m+2(θ)

um+2

)

, ∀n > m

And the corresponding seeds. . .

Pmm(θ)um =

√

2m+12m

Pm−1,m−1(θ)

um−1 , ∀m > 1

and P1,1(θ)

u1 =√

3

First derivatives

P(1)nm (θ)um = m t

uPnm(θ)

um − enmu Pn,m+1(θ)

um+1 , ∀n ≥ m




Summary

Modified Forward Row

The entries X (d)mα

um and Ω(d)mα

um are used to compute SH

To avoid underflow, Horner’s scheme is used to factorizeSH

Horner’s scheme

S(d) = cΩ(d)0 + cΣM≥m≥2

(

Ω(d)m

um u +Ω

(d)m−1

um−1

)

u




Summary

First Modified Forward Column

Pnm(θ)um = anmt Pn−1,m(θ)

um − bnmPn−2,m(θ)

um , ∀n > m

P(1)nm (θ)um = 1

u

(

nt Pn−1,m(θ)um − fnm

Pn−2,m(θ)um

)

, ∀n ≥ m




Summary

Second Modified Forward Column

Pnm(θ)

Pmm(θ)= anmt Pn−1,m(θ)

Pmm(θ)− bnm

Pn−2,m(θ)

Pmm(θ), ∀n > m

P(1)nm (θ)

Pmm(θ)= 1

u

(

nt Pn−1,m(θ)

Pmm(θ)− fnm

Pn−2,m(θ)

Pmm(θ)

)

, ∀n ≥ m




Summary

Overflow in modified forward methods

Underflow is avoided byModifying algorithms as Libbrecht (1985)Computing SH using Horner’s scheme

In light of double precision standardsOverflow is still unavoidable for ultra-high degreesA global scale factor equaling 10−280 must be applied toALF

SH is obtained by multiplying S(d) by 10280




Summary

Overview

3 proposed methods are compared with standardClenshaw’s methods

MFC-1, MFC-2 and MFR vs RC, CLEN-1, CLEN-2

In the Tests they took M=2700 and evaluatedRelative numerical precisionNumerical efficiencyAccuracy




Summary

Relative numerical precision

It is defined as: RP =∣

∣

∣

s(d)(double)−s(d)(extended)s(d)(extended)

∣

∣

∣

s(d) = Σm,nP(d)nm (θ)

Control values ares(d)(extended)Computed using CLEN-2Extended double precision

In the end, it is always < 10−9




Summary


Figure: Relative Precision of CLEN-1 and CLEN-2 for d=0




Summary


Figure: Relative Precision of MFC and CLEN-2 for d=0




Summary


Figure: Relative Precision of RC and CLEN-2 for d=0




Summary


Figure: Relative Precision of MFR and CLEN-2 for d=0




Summary


Figure: Relative Precision of CLEN-1 and CLEN-2 for d=1




Summary


Figure: Relative Precision of MFC and CLEN-2 for d=1




Summary


Figure: Relative Precision of RC and CLEN-2 for d=1




Summary


Figure: Relative Precision of MFR and CLEN-2 for d=1




Summary

Numerical efficiency

Evaluated by computing the elapsed CPU times ofs = Σm,nPnm(θ)

s and s(1) = Σm,nP(1)nm (θ)

M=2700, θ




Summary

Numerical efficiency

Figure: CPU time for M=2700




Summary

Accuracy

Evaluated based on the identitiesΣM = Σ0≤m≤n≤M

(

Pnm(θ))2

= (M + 1)2, ∀θ

ΣM∗ = Σ0≤m≤n≤M

(

P(1)nm(θ)

)2= M(M+1)2(M+1)

4 , ∀0 < θ <

180o

whereΣ2700 = 7,295,401Σ2700∗ = 13,305,717,113,850




Summary

Accuracy

Thus, defined as followsNA =

∣

∣

∣

Σ2700(comp)−7,295,4017,295,401

∣

∣

∣

NA∗ =∣

∣

∣

Σ2700∗(comp)−13,305,717,113,85013,305,717,113,850

∣

∣

∣

Computed value is never equal to real one due touncertainties

On the other hand they never exceeded 10−11




Summary

Accuracy

Figure: Accuracy of MFR and MFC for d=0




Summary

Accuracy

Figure: Accuracy of MFR and MFC for d=1




Summary

Summary

We need SH and its maximum degree is increasing.Standard methods to compute ALF don’t work forultra-high degrees.

Underflow and Overflow.

Standard Clenshaw’s methods have a more complicatedformulation and don’t give ALF.

OutlookProposed MFR makes use of CLEN’s Horner’s schemewhile keeping it as simple as Libbrecht (1985).MFR outperformed all in Num. eff and all but CLEN-2 in RP.Its uncertainty is < 10−12.


review of paper - jun, 27th

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