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Computation of matrix gamma function

Joao R. Cardoso1, Amir Sadeghi2

1Coimbra Polytechnic – ISEC, Coimbra, PortugalInstitute of Systems and Robotics – Coimbra

jocar@isec.pt

2Islamic Azad University, Tehran, Iran

Advances in NLA, Univ. Manchester, May 2019

Joao R. Cardoso, Amir Sadeghi Computation of matrix gamma functionAdvances in NLA, Univ. Manchester, May 2019 1

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Motivation

Matrix gamma function is an important matrix function (connectionswith differential equations and with other special functions, such asbeta function and Bessel function);

We haven’t found any research involving its numerical computation.

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Revisiting scalar gammaDefinition

Γ(z) =

∫ ∞0

e−ttz−1dt, Re(z) > 0.

(definition using Euler’s integral)

By analytic continuation, we can extend this definition to all complexnumbers except the non-positive integers 0,−1,−2, . . ..

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-4 -3 -2 -1 0 1 2 3 4 5

x

-100

-80

-60

-40

-20

0

20

40

60

80

(x)

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Joao R. Cardoso, Amir Sadeghi Computation of matrix gamma functionAdvances in NLA, Univ. Manchester, May 2019 5

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Scalar gammaProperties

Connection with factorial

Γ(n + 1) = n!

(n positive integer)

Translation formulaΓ(z + 1) = z Γ(z)

Reflection formula

Γ(z) =π

Γ(1− z) sin(π z), z /∈ Z

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Computation of scalar gamma

Lanczos method: a formula involving a partial fraction expansion dueto Cornelius Lanczos (1964); most popular (available in NumericalRecipes);

Stirling’s formula: Asymptotic expansion due to Abraham de Moivre(1733); also studied by James Stirling.

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Matrix gammaDefinition

Γ(A) =∫∞

0 e−ttA−I dt,

where A has eigenvalues with positive real parts and

tA−I := exp((A− I ) log t)

(Jodar & Cortes, 1998)

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Computation of Matrix gamma

Major difficulties

Methods for the matrix gamma based just on the substitution of thevariable z by a matrix A in Lanczos and Stirling approximations give verypoor results.

Scalar Lanczos method just holds for z with positive real part; if z hasnegative real part, the reflection gamma formula should be used.Scalar asymptotic Stirling formula gives good results if the real part of z issufficiently large.

How to proceed for matrices having small and large eigenvalues oreigenvalues with positive and negative real parts?

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Lanczos formula for matrices

log[Γ(A)] = 0.5 log(2π)I + (A− 0.5I ) log(A + (α− 0.5) I )− (A + (α− 0.5) I ) +

log

[c0(α)I +

m∑k=1

ck(α) (A + (k − 1)I )−1 + εα,m(A)

]

, where Re(λ) > 0.

(Logarithmic version to avoid overflow)

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Lanczos method

If A has spectrum entirely contained in the open right-half plane,compute Γ(A) by Lanczos formula (e.g, with m = 10 terms andα = 9, as recommended by Lanczos);

If σ(A) doesn’t contain negative integers and lies entirely on the openleft-half plane, use reflection property combined with Lanczos formula

(Not recommended when A has simultaneously eigvs with positive andnegative real parts)

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Stirling approximation for matrices

Γ(A) ≈

(s∏

k=1

(A + (k − 1)I )

)−1

eSm,s(A),

where

Sm,s(A) := (A + (s − 0.5)I ) log(A + (s − 1)I )− (A + (s − 1)I ) +

0.5 log(2π) +m∑

k=1

B2k

2k(2k − 1) (A + (s − 1)I )2k−1,

where Re(λ) > 0.

(B2k : Bernoulli numbers)

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Matrix gammaStirling approximation

Algorithm. This algorithm evaluates Γ(A) using the Stirling formula, with m = 12, andA ∈ Cn×n is a nonsingular matrix with spectrum satisfying one and only one of the followingconditions: (i) σ(A) is contained in the closed right-half plane; or (ii) σ(A) does not containnegative integers and lies on the open left-half plane.

1 if Re(trace(A)) ≥ 0

2 z = trace(A)/n;

3 if Im(z) ≥ 8.3 or 1− Re(z +

√8.32 − Im(z)2)

)≤ 0

4 s = 0

5 else

6 s = d1− Re(z) +√

8.32 − Im(z)2e7 end

8 Compute Γ(A) by Stirling formula;

9 else

10 S = sin(πA);

11 Compute G = Γ(I − A) by Stirling formula;;

12 Γ(A) ≈ π(SG)−1;

13 end

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Matrix reciprocal gamma

∆(A) = (Γ(A))−1 =∞∑k=0

akAk ,

a1 = 1, a2 = γ (here γ stands for the Euler-Mascheroni constant) and thecoefficients ak (k ≥ 2) are given recursively by

ak =a2ak−1 −

∑k−1j=2 (−1)jζ(j)ak−j

k − 1,

with ζ(.) being the Riemann zeta function.

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-6 -4 -2 0 2 4 6

x

-4

-3

-2

-1

0

1

2

3

4

5

(x)

Reciprocal gamma

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Matrix reciprocal gammaError bound

Lemma

If A ∈ Cn×n with ‖A‖ ≤ 1, then∥∥∥∥∥∆(A)−m∑

k=1

akAk

∥∥∥∥∥ .4

π2

∞∑k=m+1

√k!

(m + 1)!(k −m − 1)!.

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Matrix reciprocal gamma

Algorithm. This algorithm approximates Γ(A) , where A is a non-singularmatrix with no negative integers eigenvalues, by the reciprocal gammafunction series combined with the Gauss multiplication formula.

1 µ = 3;2 if ρ(A) ≤ µ3 ∆ =

∑50k=1 akA

k ;4 Γ(A) ≈ (∆)−1;5 else6 Compute r =

⌈ρ(A)−1µ−1

⌉;

7 ∆ =∑50

k=0 ak(Ar

)k;

8 for p = 1 : r − 1

9 Compute ∆ = ∆∑50

k=0 ak

(A+pI

r

)k;

10 end11 ∆ = (2π)

r−12 r0.5 I−A ∆;

12 Γ(A) ≈ (∆)−1;13 endJoao R. Cardoso, Amir Sadeghi Computation of matrix gamma function

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Schur-Parlett method

Algorithm. This algorithm approximates Γ(A) by Schur-Parlett methodcombined with Lanczos, Stirling and reciprocal algorithms.

1 Compute a Schur decomposition A = UTU∗, where the blocks Tii inthe diagonal of T are well separated (δ = 0.1); (Some codes availablein Higham’s MFTOOLBOX)

2 Approximate Gii = Γ(Tii ) by one of the previous three algs;

3 Solve the Sylvester equations arising in Parlett’s recurrence, in orderto compute all the blocks Gij , with i < j ;

4 Γ(A) ≈ UGU∗, where G = [Gij ].

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Experiments

2 4 6 8 10 12 14 16 18 20

10-15

10-14

10-13

10-12

10-11

10-10

10-9

cond (A)u

par-lanczos

par-reciprocal

par-stirling

Figure: Relative errors for 20 test matrices together with the relative conditionnumber of Γ(A) times the unit roundoff of MATLAB.

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Experiments

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

10-15

10-14

10-13

10-12

10-11

10-10

10-9

cond (A)u

par-lanczos

par-reciprocal

par-stirling

Figure: Relative errors for gallery(’moler’,12,a), by varying a from 0.1 to 1.

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Experiments

15 16 17 18 19 20 21 22 23 24

n

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

cond (A)u

par-lanczos

par-reciprocal

par-stirling

Figure: Relative errors for wilkinson(n) with size n increasing as n = 15, . . . , 24.

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Conclusions

Likewise the scalar case, the Lanczos methods shows a goodperformance in terms of a combination between accuracy andcomputational cost;

However, the technique based on the reciprocal gamma functioncombined with the Gauss multiplication formula, gives very goodresults in terms of accuracy, with the advantage of being rich inmatrix-matrix multiplications;

To make the three approximations of matrix gamma effective, theywere combined with the Schur-Parlett method.

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Further details in

Joao R. Cardoso, Amir Sadeghi, Computation of matrix gamma function,BIT Numerical Mathematics, (2019)(article in press, available online athttps://doi.org/10.1007/s10543-018-00744-1).

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