Research Article Riemannian Gradient Algorithm...

Research ArticleRiemannian Gradient Algorithm for the NumericalSolution of Linear Matrix Equations

Xiaomin Duan12 Huafei Sun1 and Xinyu Zhao34

1 School of Mathematics Beijing Institute of Technology Beijing 100081 China2 School of Science Dalian Jiaotong University Dalian 116028 China3 School of Mechanical Engineering Beijing Institute of Technology Beijing 100081 China4 School of Materials Science and Engineering Dalian Jiaotong University Dalian 116028 China

Correspondence should be addressed to Huafei Sun huafeisunbiteducn

Received 6 August 2013 Revised 10 December 2013 Accepted 10 December 2013 Published 6 January 2014

Academic Editor Zhi-Hong Guan

Copyright copy 2014 Xiaomin Duan et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

ARiemannian gradient algorithmbased on geometric structures of amanifold consisting of all positive definitematrices is proposedto calculate the numerical solution of the linear matrix equation 119876 = 119883 + sum

119898

119894=1119860119879

119894119883119860119894 In this algorithm the geodesic distance on

the curved Riemannian manifold is taken as an objective function and the geodesic curve is treated as the convergence path Alsothe optimal variable step sizes corresponding to the minimum value of the objective function are provided in order to improvethe convergence speed Furthermore the convergence speed of the Riemannian gradient algorithm is compared with that of thetraditional conjugate gradientmethod in two simulation examples It is found that the convergence speed of the provided algorithmis faster than that of the conjugate gradient method

1 Introduction

The linear matrix equation

119876 = 119883 +

119898

sum

119894=1

119860119879

119894119883119860119894 (1)

where 1198601 1198602 119860

119898are arbitrary 119899 times 119899 real matrices

119898 is a nonnegative integer and 119860119879

119894denotes the transpose

of the matrix 119860119894 arises from many fields such as the

control theory the dynamic programming and the stochasticfiltering [1ndash4] In the past decades there has been increasinginterest in the solution problems of this equation In thecase of 119898 = 1 some numerical methods such as Bartels-Stewart method Hessenberg-Schur method and Schur andQR decompositions method were proposed in [5 6] Basedon the Kronecker product and the fixed point theorem inpartially ordered sets some sufficient conditions for theexistence of a unique symmetric positive definite solutionare given in [7 8] Ran and Reurings ([7 Theorem 33] and

[9 Theorem 31]) pointed out that if 119876 minus sum119898

119894=1119860119879

119894119876119860119894is a

positive definite matrix then there exists a unique solutionand it is symmetric positive definite Recently under thecondition that (1) is consistent Su and Chen presented anefficient numerical iterative method based on the conjugategradient method (CGM) [10]

In addition based on geometric structures on a Rie-mannian manifold Duan et al proposed a natural gradientdescent algorithm to solve algebraic Lyapunov equations [1112] Following them we investigate the solution problemof (1) in the view of Riemannian manifolds Note that thissolution of (1) is a symmetric positive definite matrix andthe set of all the symmetric positive definite matrices can beconsidered as a manifold Thus it is more convenient toinvestigate the solution problem with the help of these geo-metric structures on this manifold To address such a needa new framework is presented in this paper to calculate thenumerical solution which is based on the geometric struc-tures on the Riemannian manifold of positive definitesymmetric matrices We will first describe briefly some

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 507175 7 pageshttpdxdoiorg1011552014507175

2 Journal of Applied Mathematics

fundamental knowledge on manifold Then a Riemanniangradient algorithm is proposed where the geodesic distanceon the curved Riemannian manifold is taken as an objectivefunction and the geodesic curve is treated as the convergencepath Also the optimal variable step sizes corresponding tothe minimum value of the objective function are providedin order to improve the convergence speed Finally thebehavior of the provided algorithm and the traditional CGMis compared and demonstrated in two simulation examples

2 A Survey of Some Geometrical Concepts

In the present section we briefly survey the geometry ofsmoothmanifolds by recalling concepts as tangent spaces theRiemannian gradient geodesic curves and geodesic distanceMore details can be found in [13 14] Specially these conceptsof manifold consisting of all symmetric positive definitematrices have also been introduced in [15ndash20]

21 Riemannian Manifolds Let us denote a Riemannianmanifold byM Its tangent space at point119884 isin M is denoted by119879119884M Given any pair of points 119881 119882 isin 119879

119884M an inner product

⟨119881 119882⟩119884

isin R is definedThe specification of the inner productfor a Riemannian manifold turns it into a metric space Infact the length of a curve 120601

119884119881 [0 1] rarr M such that

120601119884119881

(0) = 119884 isin M and 120601119884119881

(0) = 119881 isin 119879119884M is given by

ℓ (120601119884119881

) = int

1

0

radic⟨ 120601119884119881

(119905) 120601119884119881

(119905)⟩120601119884119881(119905)

d119905 (2)

Given arbitrary 119884 isin M and 119881 isin 119879119884M the curve 120574

119884119881

[0 1] rarr M of shortest length is called geodesic curve Suchminimal length ℓ(120574

119884119881) is called geodesic distance between

two points namely 120574119884119881

(0) = 119884 and 120574119884119881

(1) The Riemanniandistance between two points is denoted by

d (120574119884119881

(0) 120574119884119881

(1)) = ℓ (120574119884119881

) (3)

It is obvious that if any pair of points on manifold M can beconnected by a geodesic curve then it is possible to measurethe distance between any given pair of points on it Given aregular function 119891 M rarr R its Riemannian gradient nabla

119884119891

in the direction of the vector 119881 isin 119879119884M measures the rate of

change of the function119891 in the direction119881 Namely given anysmooth curve 120601

119884119881 [0 1] rarr M such that 120601

119884119881(0) = 119884 isin M

and 120601119884119881

(0) = 119881 the Riemannian gradient nabla119884119891 is the unique

vector in 119879119884M such that

⟨119881 nabla119884119891⟩119884

=dd119905

119891 (120601119884119881

(119905))1003816100381610038161003816119905=0

(4)

Instead of the Euclidean space by using the Riemanniangradient the optimization method may be readily extendedto smooth manifolds [21 Chapter 7] For this purpose let usconsider the differential equation on manifoldM

119910 = minusnabla119910(119905)

119891 119910 (0) = 119910 isin M (5)

If the function 119891 M rarr R is bounded and the smoothmanifoldM is compact then the solution of such differential

equation tends to a local minimum of the function 119891 in Mdepending on the initial point 119910 In fact it is achieved bydefinition of the Riemannian gradient

dd119905

119891 (119910 (119905)) = ⟨ 119910 (119905) nabla119910(119905)

119891⟩119910(119905)

= minus⟨nabla119910(119905)

119891 nabla119910(119905)

119891⟩119910(119905)

le 0

(6)

22 Manifold of Symmetric Positive Definite Matrices Let usdenote the general linear group by GL(119899) which consists ofall nonsingular real matrices And 119883 gt 0 means that 119883 is asymmetric positive definite matrix For a different notation119883 minus 119884 gt 0 we will use 119883 gt 119884 And the manifold consisting ofall symmetric positive definite matrices is defined by S+(119899) =

119883 isin R119899times119899 | 119883119879

= 119883 119883 gt 0 The tangent space at a point119883 isin S+(119899) is given by 119879

119883S+(119899) = 119881 isin R119899times119899 | 119881

119879

= 119881On manifold S+(119899) the Riemannian metric at the point 119883 isdefined by

⟨1198811 1198812⟩119883

= tr [119883minus1

1198811119883minus1

1198812] (7)

where 1198811 1198812

isin 119879119883S+(119899) and tr denotes the trace of the

matrix Then manifold S+(119899) with the Riemannian metric(7) becomes a Riemannian manifold Moreover it is a curvedmanifold since thismetric is invariant under the group actionof GL(119899) The geodesic curve at the point 119883 isin S+(119899) in thedirection 119881 isin 119879

119883S+(119899) can be expressed as

120574119883119881

(119905) = 11988312 exp (119905119883

minus12

119881119883minus12

) 11988312

(8)

Obviously this geodesic curve is entirely contained in man-ifold The Hopf-Rinow theorem implies that manifold S+(119899)

with the Riemannianmetric (7) is geodesically complete ([22Theorem 28]) This means for any given pair 119883 119884 isin S+(119899)we can find a geodesic curve 120574

119883119881(119905) such that 120574

119883119881(0) = 119883

and 120574119883119881

(1) = 119884 by taking the initial velocity 120574119883119881

(0) =

11988312 log(119883

minus12

119884119883minus12

)11988312

isin 119879119883S+(119899) According to (2) the

geodesic distance d(119883 119884) can be computed explicitly by

d2 (119883 119884) = tr [log2 (119883minus12

119884119883minus12

)] (9)

3 Riemannian Gradient Algorithm

Some researches show that the solution equation (1) is uniquepositive definite if 119860

119894and 119876 satisfy ([7 Theorem 33] and [9

Theorem 31])

119876 gt

119898

sum

119894=1

119860119879

119894119876119860119894 (10)

For convenience we set L(119883) = 119883 + sum119898

119894=1119860119879

119894119883119860119894 And it

is obvious that L(119883) is positive definite for any 119883 isin S+(119899)Our purpose is to seek a matrix 119883 isin S+(119899) so that L(119883) isas close as possible to the given matrix 119876 We figure out somekey points in designing such an algorithm as follows

(1) For some 119883 the difference between the given positivedefinite matrix 119876 and the matrix L(119883) should bedetermined and calculated firstly

Journal of Applied Mathematics 3

(2) Then the Riemannian gradient descent algorithm canbe used to adjust 119883 so that the difference will be assmall as possible

Note that the Riemannian manifold S+(119899) is geodesicallycomplete hence we take the geodesic distance between119876 andL(119883) as the objective function 119869(119883) namely

119869 (119883) = d2 (119876L (119883)) (11)

Then the exact solution of (1) may be defined as

119883lowast

= arg min119883isinS+(119899)

119869 (119883) (12)

In order to find aminimizer for the criterion d2(119876L(119883)) wemay make use of the differential equation (5) The minimumis apparently achieved for a value 119883

lowastsuch that

nabla119883

119869 (119883)1003816100381610038161003816119883=119883lowast

= 0 (13)

It is necessary to solve the optimization problem (12) by anumerical optimization algorithm that takes the geometry ofthe Riemannian manifold S+(119899) into account In particularstarting from an initial guess 119883

0 it is possible to provide an

iterative learning algorithm that generates a pattern 119883119905

isin

S+(119899) at any learning step 119905 isin N Following [21 23] suchsequence may be generated by moving from each point 119883

119905isin

S+(119899) to the next point 119883119905+1

along a short geodesic curvein the opposite direction of the Riemannian gradient of theobjective function 119869(119883

119905) Namely if we set

119881119905

= nabla119883119905

119869 (119883119905) (14)

then the Riemannian gradient algorithmmay be expressed as

119883119905+1

= 120574119883119905minus119881119905

(120578) (15)

where 120578 denotes any suitable step-size schedule that drives theiterative algorithm (15) to be convergent Moreover with theaim to employ the learning algorithm (14) and (15) we firstneed to compute the Riemannian gradient of the objectivefunction 119869(119883

119905) to be optimized In the definition of the

Riemannian gradient (4) the generic smooth curve 120601119884119881

maybe replaced with a geodesic curve We obtain the followingtheorem

Theorem 1 TheRiemannian gradient of the objective function119869(119883) at point 119883 is given by

nabla119883

119869 (119883) = 2119883 ( log (119876minus1

L (119883))L(119883)minus1

+

119898

sum

119895=1

119860119895log (119876

minus1

L (119883))L(119883)minus1

119860119879

119895) 119883

(16)

Proof First we compute the derivative of the real-valuedfunction

119869 (119904 (119905)) = tr [log2 (119876minus12

L (119904 (119905)) 119876minus12

)] (17)

with respect to 119905 where 119904(119905) = 11988312 exp(119905119883

minus12

119881119883minus12

)11988312

is the geodesic curve emanating from 119883 in the direction 119881 =

119904(0) Using Proposition 21 in [16] and some properties of thetrace we have

dd119905

119869 (119904 (119905))|119905=0

= 2 tr[

[

( log (119876minus1

L (119883))L(119883)minus1

+

119898

sum

119895=1

119860119895log (119876

minus1

L (119883))L(119883)minus1

119860119879

119895) 119881]

]

= ⟨119881 2119883 ( log (119876minus1

L (119883))L(119883)minus1

+

119898

sum

119895=1

119860119895log (119876

minus1

L (119883))L(119883)minus1

119860119879

119895) 119883⟩

119883

(18)

From (4) it is shown that formula (16) is valid

Remark 2 In fact note that log(119860minus1

119861119860) = 119860minus1

(log119861)119860 forall 119860 isin GL(119899) and for all 119861 isin S+(119899) then nabla

119883119869(119883) can be

rewritten as

nabla119883

119869 (119883)

= 2119883 (L(119883)minus12 log (L(119883)

12

119876minus1

L(119883)12

)L(119883)minus12

+

119898

sum

119895=1

119860119895L(119883)

minus12

times log (L(119883)12

119876minus1

L(119883)12

)L(119883)minus12

119860119879

119895) 119883

(19)

Therefore it can be shown that nabla119883

119869(119883) belongs to 119879119883S+(119899)

Corollary 3 TheRiemannian gradientnabla119883

119869(119883) has the uniquezero point 119883

lowast Furthermore for the suitable step-size 120578 the

sequence 119883119905generated by (15) converges to the solution 119883

lowastof

(1)

Proof Obviously 119883lowastis a zero point of the Riemannian gra-

dient nabla119883

119869(119883) If there exists 1198831015840

lowast= 119883lowastsuch that the iterative

process (15) stops then it means nabla1198831015840

lowast

119869(1198831015840

lowast) = 0 which is

equivalent to

log (119876minus1

L (1198831015840

lowast))L(119883

1015840

lowast)minus1

+

119898

sum

119895=1

119860119895log (119876

minus1

L (1198831015840

lowast))L(119883

1015840

lowast)minus1

119860119879

119895= 0

(20)


On (20) the matrices are multiplied by 1198831015840

lowastand the traces of

the matrices are taken then we have

tr [log (119876minus1

L (1198831015840

lowast))] = 0 (21)

Also if we replace 1198831015840

lowastwith 119883

lowastand follow the similar

procedure as the above equation then the following equalitiesare valid

tr[log (119876minus1

L (1198831015840

lowast))L(119883

1015840

lowast)minus1

(119883lowast

+

119898

sum

119894=1

119860119879

119894119883lowast119860119894)]

= tr [log (119876minus1

L (1198831015840

lowast))L(119883

1015840

lowast)minus1

119876]

= 0

(22)

Combining (21) and (22) it can be obtained that


L (1198831015840

lowast))] minus tr [log (119876

minus1

L (1198831015840

lowast)) 119876]


L (1198831015840

lowast)) (119868 minus L(119883

1015840

lowast)minus1

119876)]


L (1198831015840

lowast) 119876minus12

)

times (119868 minus 11987612

L(1198831015840

lowast)minus1

11987612

)]

= 0

(23)

Note that the matrix 11987612L(119883

1015840

lowast)minus1

11987612 is positive defi-

nite hence its orthogonal similarity diagonalization can beexpressed as 119880Λ119880

119879 where 119880 is an orthogonal matrix Λ =

diag (1205821 120582

119899) and 120582

119894gt 0 (1 le 119894 le 119899) Thus we have

tr [(Λ minus 119868) logΛ] =

119899

sum

119894=1

(120582119894minus 1) log 120582

119894

= 0 (24)

Note that the second equality of (24) is equivalent to

119899

prod

119894=1

120582119894

120582119894minus1 = 1 (25)

Thus we have 1205821

= 1205822

= sdot sdot sdot = 120582119899

= 1 That means that 1198831015840

lowastis

also the solution of (1) which is contrary to the uniquenessof the solution Therefore the Riemannian gradient nabla

119883119869(119883)

has the unique zero point 119883lowast This completes the proof of

Corollary 3

Now the Riemannian gradient algorithm with a constantstep-size (RGACS) becomes

119883119905+1

= 11988312

119905exp (minus120578119883

minus12

119905nabla119883119905

119869 (119883119905) 119883minus12

119905) 11988312

119905 (26)

where the initial value 1198830

isin S+(119899) The step-size 120578 corres-ponding to the best convergence speed can be obtained exper-imentally and the details will be described in our followingsimulations

31 Riemannian Gradient Algorithm with Variable Step SizesIn order to improve the convergence speed of the RGACSan optimal variable step-size schedule 120578

119905will be considered

and defined as follows The step-size 120578119905should be evaluated

in such a way that the objective function 119869(119883119905+1

) at step 119905 + 1

is reduced as much as possible with respect to 119869(119883119905) We may

regard 119869(119883119905+1

) as a function of the step-size 120578119905that

119869 (119883119905+1

) = d2 (119876L (120574119883119905minus119881119905

(120578119905))) (27)

Thus we can optimize the value of 120578119905to ensure that the

difference 119869(119883119905) minus 119869(119883

119905+1) could be as large as possible Since

it is difficult to solve this nonlinear problem exactly we use asuboptimal approximation in practice Under the hypothesisthat the step-size value 120578

119905is small enough we may invoke the

expansion of the function 119869(119883119905+1

) around the point 120578119905

= 0 asfollows

119869 (119883119905+1

) asymp 119869 (119883119905) + 1198621119905

120578119905+

1

21198622119905

1205782

119905 (28)

Then the optimal step-size 120578lowast

119905 corresponding to the maxi-

mum of the difference 119869(119883119905) minus 119869(119883

119905+1) can be expressed as

120578lowast

119905asymp minus

1198621119905

1198622119905

(29)

The coefficients 1198621119905 and 119862

2119905can be calculated using the

first-order and the second-order derivatives of the function119869(119883119905+1

) with respect to the parameter 120578119905at the point 120578

119905= 0

On the basis of the above discussion we may obtain theoptimal step-size

120578lowast

119905asymp

tr [(119883L (log (119876minus1L (119883

119905))L(119883

119905)minus1

))2

]

tr [(L(119883119905)minus1

L (nabla119883119905

119869 (119883119905)))2

]

(30)

Then the Riemannian gradient algorithm with variablestep-sizes (RGAVS) can be written explicitly as

119883119905+1

= 11988312

119905exp (minus120578

lowast

119905119883minus12

119905nabla119883119905

119869 (119883119905) 119883minus12

119905) 11988312

119905 (31)

where 1198830

isin S+(119899) is the initial value

32 Volume Feature of Riemannian Gradient Algorithm TheRGACS shows an interesting volume feature that the ratiodet(119883

119905+1) det(119883

119905) always keeps a certain relationship In

order to prove the volume feature let us recall two propertiesof determinant operator det namely for each 119860 119861 isin GL(119899)there hold

det (119860119861) = det (119860) det (119861)

det (exp (119860)) = exp (tr (119860)) (32)


By applying the determinant operator to both sides of (31)we obtain

det (119883119905+1

)

= det (119883119905) exp (tr [minus120578119883

minus12

119905nabla119883119905

119869 (119883119905) 119883minus12

119905])

= det (119883119905) exp(tr[

[

120578 ( log (L(119883119905)minus1

119876)L(119883119905)minus1

119883119905

+ log (L(119883119905)minus1

119876)L(119883119905)minus1

times

119898

sum

119895=1

119860119879

119895119883119905119860119895)]

]

)

= det (119883119905) exp (tr [120578 log (L(119883

119905)minus1

119876)])

= det (119883119905) det ((L(119883

119905)minus1

119876)120578

)

(33)

Therefore the following property is valid

det (119883119905+1

)

det (119883119905)

= (det (119876)

det(L (119883119905))

120578

(34)

This equation shows that the determinant of (1) isclosely linked with that of the sequence 119883

119905 Moreover

the ratio det(119883119905+1

) det(119883119905) is directly proportional to

det(119876) det(L(119883119905)) for the fixed step-size 120578 In the case of

the RGAVS the similar conclusion can also be obtained

4 Simulations

Two simulation examples are given to compare the con-vergence speed of the Riemannian gradient algorithm (theRGACS and the RGAVS) with those of the traditional CGMThe error criterion used in these simulations is120588(L(119883)minus119876) lt

10minus15 where 120588(sdot) denotes the spectral radius of the matrix

Example 4 First we consider the matrices equation

119876 = 119883 + 119860119879

11198831198601

+ 119860119879

21198831198602

+ 119860119879

31198831198603

(35)

in the case of 119899 = 5 And 119876 1198601 1198602 1198603are as follows

119876 = (

46350 00698 03104 01990 02918

00698 40376 00742 00694 00690

03104 00742 42242 01554 01826

01990 00694 01554 41472 01572

02918 00690 01826 01572 42134

)

1198601

= (

00834 00007 00293 00142 00139

00141 02975 00154 00289 00339

00244 00152 00317 00270 00117

00251 00097 00140 00135 00240

00158 01499 00275 00077 00157

)

02 03 04 06 07 08050

20

40

60

80

100

120

140

160

Step size

Itera

tions

Figure 1 Step sizes versus the iterations in the RGACS

1198602

= (

02624 00166 00454 00172 00160

00893 00608 00242 00029 00061

00054 00190 00589 00036 00160

00278 00050 00255 00689 00048

00319 00064 00103 00219 00018

)

1198603

= (

81622 06020 04505 08258 01067

27943 70630 00838 05383 09619

03112 06541 62290 09961 00046

05285 06892 09133 10782 07749

01656 07482 01524 04427 90073

)

(36)

which satisfy the condition (10) Both the RGACS andRGAVS are used and compared with the traditional CGMand 119876 is taken as the initial value 119883

0in the following

simulation In the RGACS case the step sizes versus theiterations are shown in Figure 1 It can be found that theiterations will reduce gradually as the step-size changes fromzero to a critical value then the iterations will increasegradually above this critical valueTherefore the critical valueabout 064 can be selected experimentally as an optimal step-size and used in the RGACS

Figure 2 shows the convergence comparison between theRGACS and the RGAVS and the traditional CGM It can befound that the RGAVS possesses themost steady convergenceand the fastest convergence speed and it needs 23 iterationsto obtain the numerical solution of (35) as follows

(

00856 minus00265 00151 minus01065 00148

minus00265 00970 00166 minus01581 minus00023

00151 00166 01591 minus03608 00283

minus01065 minus01581 minus03608 23761 minus01723

00148 minus00023 00283 minus01723 00654

)

(37)

The case of the CGM realizes the given error through 33iterations and the slowest one among them is the RGACSwith 31 steps By comparison it is shown that the convergence


5 10 15 20 25 30 35Iteration

Erro

r

RGACSRGAVS

CGM

105

100

10minus5

10minus10

Figure 2 Comparison of convergence speeds (log scale)

speed of the RGAVS is faster than both those of the RGACSand the traditional CGM in solving (35)

Example 5 A 20-order linear matrix equation with 119898 =

2 is also considered here following the similar procedureas Example 4 where 119876 isin S+(20) and 119860

1 1198602

isin R20times20

satisfy the condition (10) The simulation indicates that thenumerical solution needs 26 45 and 37 iterations in theRGAVS the RGACS (120578 = 074) and the traditional CGMrespectively Therefore the convergence speed of the RGAVSis also better than those of the RGACS and the CGM

5 Conclusion

Using geometric structures of manifold consisting of all sym-metric positive definite matrices the Riemannian gradientalgorithm is developed to calculate the numerical solutionfor the linear matrix equation 119876 = 119883 + sum

119898

119894=1119860119879

119894119883119860119894 In this

algorithm the geodesic distance on the curved Riemannianmanifold is taken as an objective function and the geodesiccurve is treated as the convergence path Also the variablestep sizes algorithm corresponding to the minimum valueof the objective function is provided in order to improvethe convergence speed in the constant step-size case Finallythe convergence speed of the Riemannian gradient algorithmhas been compared with the traditional conjugate gradientmethod by two simulation examples in 5-order and 20-orderlinear matrix equations respectively It is shown that theconvergence speed of the Riemannian gradient algorithmwith variable step sizes is faster than that of the conjugate gra-dient method Although the Riemannian gradient algorithmis used here for the linear matrix equation in the real numberfield it may be generalized to the complex case as well

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to express their sincere thanks tothe referees for their valuable suggestions which greatlyimproved the presentation of this paper This subject issupported by the National Natural Science Foundations ofChina (nos 61179031 10932002 and 51105033)

References

[1] M Berzig ldquoSolving a class of matrix equations via the Bhaskar-Lakshmikantham coupled fixed point theoremrdquo Applied Math-ematics Letters vol 25 no 11 pp 1638ndash1643 2012

[2] J C Engwerda ldquoOn the existence of a positive definite solutionof the matrix equation 119883 + 119860

119879

119883minus1

119860 = 119868rdquo Linear Algebra and ItsApplications vol 194 pp 91ndash108 1993

[3] W L Green and E W Kamen ldquoStabilizability of linear sys-tems over a commutative normed algebra with applications tospatially-distributed and parameter-dependent systemsrdquo SIAMJournal onControl andOptimization vol 23 no 1 pp 1ndash18 1985

[4] W Pusz and S L Woronowicz ldquoFunctional calculus forsesquilinear forms and the purification maprdquo Reports on Math-ematical Physics vol 8 no 2 pp 159ndash170 1975

[5] C-Y Chiang E K-W Chu and W-W Lin ldquoOn the 995333-Sylvester equation 119860119883 plusmn 119883

995333

119861995333

= 119862rdquo Applied Mathematics andComputation vol 218 no 17 pp 8393ndash8407 2012

[6] Z Gajic and M T J Qureshi Lyapunov Matrix Equation inSystem Stability and Control vol 195 Academic Press LondonUK 1995

[7] A C M Ran and M C B Reurings ldquoThe symmetric linearmatrix equationrdquo Electronic Journal of Linear Algebra vol 9 pp93ndash107 2002

[8] M C B Reurings Symmetric matrix equations [PhD thesis]Vrije Universiteit Amsterdam The Netherlands 2003

[9] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrixequationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004

[10] Y Su and G Chen ldquoIterative methods for solving linear matrixequation and linear matrix systemrdquo International Journal ofComputer Mathematics vol 87 no 4 pp 763ndash774 2010

[11] X Duan H Sun L Peng and X Zhao ldquoA natural gradi-ent descent algorithm for the solution of discrete algebraicLyapunov equations based on the geodesic distancerdquo AppliedMathematics and Computation vol 219 no 19 pp 9899ndash99052013

[12] X Duan H Sun and Z Zhang ldquoA natural gradient descentalgorithm for the solution of Lyapunov equations based on thegeodesic distancerdquo Journal of Computational Mathematics vol219 no 19 pp 9899ndash9905 2013

[13] M Spivak A Comprehensive Introduction to Differential Geom-etry vol 1 Publish or Perish Berkeley Calif USA 3rd edition1999

[14] S Lang Fundamentals of Differential Geometry vol 191Springer 1999


[15] F Barbaresco ldquoInteractions between symmetric cones andinformation geometrics Bruhat-tits and Siegel spaces modelsfor high resolution autoregressive Doppler imageryrdquo in Emerg-ing Trends in Visual Computing vol 5416 of Lecture Notes inComputer Science pp 124ndash163 2009

[16] M Moakher ldquoA differential geometric approach to the geo-metric mean of symmetric positive-definite matricesrdquo SIAMJournal on Matrix Analysis and Applications vol 26 no 3 pp735ndash747 2005

[17] M Moakher ldquoOn the averaging of symmetric positive-definitetensorsrdquo Journal of Elasticity vol 82 no 3 pp 273ndash296 2006

[18] A Schwartzman Random ellipsoids and false discovery ratesstatistics for diffusion tensor imaging data [PhD thesis] StanfordUniversity 2006

[19] J D Lawson and Y Lim ldquoThe geometric mean matricesmetrics and morerdquo The American Mathematical Monthly vol108 no 9 pp 797ndash812 2001

[20] R Bhatia Positive definite Matrices Princeton Series in AppliedMathematics Princeton University Press Princeton NJ USA2007

[21] C Udriste Convex Functions and Optimization Methods onRiemannian Manifolds vol 297 ofMathematics and its Applica-tions Kluwer Academic Publishers Dordrecht Germany 1994

[22] M P D Carmo Riemannian Geometry Springer 1992[23] D G Luenberger ldquoThe gradient projection method along

geodesicsrdquoManagement Science vol 18 pp 620ndash631 1972

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


fundamental knowledge on manifold Then a Riemanniangradient algorithm is proposed where the geodesic distanceon the curved Riemannian manifold is taken as an objectivefunction and the geodesic curve is treated as the convergencepath Also the optimal variable step sizes corresponding tothe minimum value of the objective function are providedin order to improve the convergence speed Finally thebehavior of the provided algorithm and the traditional CGMis compared and demonstrated in two simulation examples

2 A Survey of Some Geometrical Concepts

In the present section we briefly survey the geometry ofsmoothmanifolds by recalling concepts as tangent spaces theRiemannian gradient geodesic curves and geodesic distanceMore details can be found in [13 14] Specially these conceptsof manifold consisting of all symmetric positive definitematrices have also been introduced in [15ndash20]

21 Riemannian Manifolds Let us denote a Riemannianmanifold byM Its tangent space at point119884 isin M is denoted by119879119884M Given any pair of points 119881 119882 isin 119879

119884M an inner product

⟨119881 119882⟩119884

isin R is definedThe specification of the inner productfor a Riemannian manifold turns it into a metric space Infact the length of a curve 120601

119884119881 [0 1] rarr M such that

120601119884119881

(0) = 119884 isin M and 120601119884119881

(0) = 119881 isin 119879119884M is given by

ℓ (120601119884119881

) = int

1

0

radic⟨ 120601119884119881

(119905) 120601119884119881

(119905)⟩120601119884119881(119905)

d119905 (2)

Given arbitrary 119884 isin M and 119881 isin 119879119884M the curve 120574

119884119881

[0 1] rarr M of shortest length is called geodesic curve Suchminimal length ℓ(120574

119884119881) is called geodesic distance between

two points namely 120574119884119881

(0) = 119884 and 120574119884119881

(1) The Riemanniandistance between two points is denoted by

d (120574119884119881

(0) 120574119884119881

(1)) = ℓ (120574119884119881

) (3)

It is obvious that if any pair of points on manifold M can beconnected by a geodesic curve then it is possible to measurethe distance between any given pair of points on it Given aregular function 119891 M rarr R its Riemannian gradient nabla

119884119891

in the direction of the vector 119881 isin 119879119884M measures the rate of

change of the function119891 in the direction119881 Namely given anysmooth curve 120601

119884119881 [0 1] rarr M such that 120601

119884119881(0) = 119884 isin M

and 120601119884119881

(0) = 119881 the Riemannian gradient nabla119884119891 is the unique

vector in 119879119884M such that

⟨119881 nabla119884119891⟩119884

=dd119905

119891 (120601119884119881

(119905))1003816100381610038161003816119905=0

(4)

Instead of the Euclidean space by using the Riemanniangradient the optimization method may be readily extendedto smooth manifolds [21 Chapter 7] For this purpose let usconsider the differential equation on manifoldM

119910 = minusnabla119910(119905)

119891 119910 (0) = 119910 isin M (5)

If the function 119891 M rarr R is bounded and the smoothmanifoldM is compact then the solution of such differential

equation tends to a local minimum of the function 119891 in Mdepending on the initial point 119910 In fact it is achieved bydefinition of the Riemannian gradient

dd119905

119891 (119910 (119905)) = ⟨ 119910 (119905) nabla119910(119905)

119891⟩119910(119905)

= minus⟨nabla119910(119905)

119891 nabla119910(119905)

119891⟩119910(119905)

le 0

(6)

22 Manifold of Symmetric Positive Definite Matrices Let usdenote the general linear group by GL(119899) which consists ofall nonsingular real matrices And 119883 gt 0 means that 119883 is asymmetric positive definite matrix For a different notation119883 minus 119884 gt 0 we will use 119883 gt 119884 And the manifold consisting ofall symmetric positive definite matrices is defined by S+(119899) =

119883 isin R119899times119899 | 119883119879

= 119883 119883 gt 0 The tangent space at a point119883 isin S+(119899) is given by 119879

119883S+(119899) = 119881 isin R119899times119899 | 119881

119879

= 119881On manifold S+(119899) the Riemannian metric at the point 119883 isdefined by

⟨1198811 1198812⟩119883

= tr [119883minus1

1198811119883minus1

1198812] (7)

where 1198811 1198812

isin 119879119883S+(119899) and tr denotes the trace of the

matrix Then manifold S+(119899) with the Riemannian metric(7) becomes a Riemannian manifold Moreover it is a curvedmanifold since thismetric is invariant under the group actionof GL(119899) The geodesic curve at the point 119883 isin S+(119899) in thedirection 119881 isin 119879

119883S+(119899) can be expressed as

120574119883119881

(119905) = 11988312 exp (119905119883

minus12

119881119883minus12

) 11988312

(8)

Obviously this geodesic curve is entirely contained in man-ifold The Hopf-Rinow theorem implies that manifold S+(119899)

with the Riemannianmetric (7) is geodesically complete ([22Theorem 28]) This means for any given pair 119883 119884 isin S+(119899)we can find a geodesic curve 120574

119883119881(119905) such that 120574

119883119881(0) = 119883

and 120574119883119881

(1) = 119884 by taking the initial velocity 120574119883119881

(0) =

11988312 log(119883

minus12

119884119883minus12

)11988312

isin 119879119883S+(119899) According to (2) the

geodesic distance d(119883 119884) can be computed explicitly by

d2 (119883 119884) = tr [log2 (119883minus12

119884119883minus12

)] (9)

3 Riemannian Gradient Algorithm

Some researches show that the solution equation (1) is uniquepositive definite if 119860

119894and 119876 satisfy ([7 Theorem 33] and [9

Theorem 31])

119876 gt

119898

sum

119894=1

119860119879

119894119876119860119894 (10)

For convenience we set L(119883) = 119883 + sum119898

119894=1119860119879

119894119883119860119894 And it

is obvious that L(119883) is positive definite for any 119883 isin S+(119899)Our purpose is to seek a matrix 119883 isin S+(119899) so that L(119883) isas close as possible to the given matrix 119876 We figure out somekey points in designing such an algorithm as follows

(1) For some 119883 the difference between the given positivedefinite matrix 119876 and the matrix L(119883) should bedetermined and calculated firstly




119869 (119883) = d2 (119876L (119883)) (11)


119883lowast

= arg min119883isinS+(119899)

119869 (119883) (12)


lowastsuch that

nabla119883

119869 (119883)1003816100381610038161003816119883=119883lowast

= 0 (13)




isin


119905isin




119881119905

= nabla119883119905

119869 (119883119905) (14)


119883119905+1

= 120574119883119905minus119881119905

(120578) (15)






nabla119883

119869 (119883) = 2119883 ( log (119876minus1

L (119883))L(119883)minus1

+

119898

sum

119895=1

119860119895log (119876

minus1

L (119883))L(119883)minus1

119860119879

119895) 119883

(16)


119869 (119904 (119905)) = tr [log2 (119876minus12

L (119904 (119905)) 119876minus12

)] (17)


minus12

119881119883minus12

)11988312



dd119905

119869 (119904 (119905))|119905=0

= 2 tr[

[

( log (119876minus1

L (119883))L(119883)minus1

+

119898

sum

119895=1

119860119895log (119876

minus1

L (119883))L(119883)minus1

119860119879

119895) 119881]

]

= ⟨119881 2119883 ( log (119876minus1

L (119883))L(119883)minus1

+

119898

sum

119895=1

119860119895log (119876

minus1

L (119883))L(119883)minus1

119860119879

119895) 119883⟩

119883

(18)



119861119860) = 119860minus1


119883119869(119883) can be

rewritten as

nabla119883

119869 (119883)

= 2119883 (L(119883)minus12 log (L(119883)

12

119876minus1

L(119883)12

)L(119883)minus12

+

119898

sum

119895=1

119860119895L(119883)

minus12


119876minus1

L(119883)12

)L(119883)minus12

119860119879

119895) 119883

(19)


119869(119883) belongs to 119879119883S+(119899)





lowastof

(1)


dient nabla119883

119869(119883) If there exists 1198831015840



lowast

119869(1198831015840


equivalent to

log (119876minus1

L (1198831015840

lowast))L(119883

1015840

lowast)minus1

+

119898

sum

119895=1

119860119895log (119876

minus1

L (1198831015840

lowast))L(119883

1015840

lowast)minus1

119860119879

119895= 0

(20)






L (1198831015840

lowast))] = 0 (21)


lowastwith 119883




L (1198831015840

lowast))L(119883

1015840

lowast)minus1

(119883lowast

+

119898

sum

119894=1

119860119879

119894119883lowast119860119894)]


L (1198831015840

lowast))L(119883

1015840

lowast)minus1

119876]

= 0

(22)



L (1198831015840


minus1

L (1198831015840

lowast)) 119876]


L (1198831015840

lowast)) (119868 minus L(119883

1015840

lowast)minus1

119876)]


L (1198831015840


)

times (119868 minus 11987612

L(1198831015840

lowast)minus1

11987612

)]

= 0

(23)


1015840

lowast)minus1




diag (1205821 120582

119899) and 120582



119899

sum

119894=1

(120582119894minus 1) log 120582

119894

= 0 (24)


119899

prod

119894=1

120582119894

120582119894minus1 = 1 (25)


= 1205822

= sdot sdot sdot = 120582119899

= 1 That means that 1198831015840

lowastis


119883119869(119883)


Corollary 3


119883119905+1

= 11988312

119905exp (minus120578119883

minus12

119905nabla119883119905

119869 (119883119905) 119883minus12

119905) 11988312

119905 (26)







) at step 119905 + 1


regard 119869(119883119905+1


119869 (119883119905+1

) = d2 (119876L (120574119883119905minus119881119905

(120578119905))) (27)








= 0 asfollows

119869 (119883119905+1

) asymp 119869 (119883119905) + 1198621119905

120578119905+

1

21198622119905

1205782

119905 (28)





120578lowast

119905asymp minus

1198621119905

1198622119905

(29)





119905= 0


120578lowast

119905asymp

tr [(119883L (log (119876minus1L (119883

119905))L(119883

119905)minus1

))2

]

tr [(L(119883119905)minus1

L (nabla119883119905

119869 (119883119905)))2

]

(30)


119883119905+1

= 11988312

119905exp (minus120578

lowast

119905119883minus12

119905nabla119883119905

119869 (119883119905) 119883minus12

119905) 11988312

119905 (31)

where 1198830



119905+1) det(119883



det (119860119861) = det (119860) det (119861)

det (exp (119860)) = exp (tr (119860)) (32)



det (119883119905+1

)

= det (119883119905) exp (tr [minus120578119883

minus12

119905nabla119883119905

119869 (119883119905) 119883minus12

119905])

= det (119883119905) exp(tr[

[

120578 ( log (L(119883119905)minus1

119876)L(119883119905)minus1

119883119905

+ log (L(119883119905)minus1

119876)L(119883119905)minus1

times

119898

sum

119895=1

119860119879

119895119883119905119860119895)]

]

)

= det (119883119905) exp (tr [120578 log (L(119883

119905)minus1

119876)])

= det (119883119905) det ((L(119883

119905)minus1

119876)120578

)

(33)


det (119883119905+1

)

det (119883119905)

= (det (119876)

det(L (119883119905))

120578

(34)


119905 Moreover

the ratio det(119883119905+1




4 Simulations




119876 = 119883 + 119860119879

11198831198601

+ 119860119879

21198831198602

+ 119860119879

31198831198603

(35)


119876 = (

46350 00698 03104 01990 02918

00698 40376 00742 00694 00690

03104 00742 42242 01554 01826

01990 00694 01554 41472 01572

02918 00690 01826 01572 42134

)

1198601

= (

00834 00007 00293 00142 00139

00141 02975 00154 00289 00339

00244 00152 00317 00270 00117

00251 00097 00140 00135 00240

00158 01499 00275 00077 00157

)

02 03 04 06 07 08050

20

40

60

80

100

120

140

160

Step size

Itera

tions


1198602

= (

02624 00166 00454 00172 00160

00893 00608 00242 00029 00061

00054 00190 00589 00036 00160

00278 00050 00255 00689 00048

00319 00064 00103 00219 00018

)

1198603

= (

81622 06020 04505 08258 01067

27943 70630 00838 05383 09619

03112 06541 62290 09961 00046

05285 06892 09133 10782 07749

01656 07482 01524 04427 90073

)

(36)


0in the following



(

00856 minus00265 00151 minus01065 00148


00151 00166 01591 minus03608 00283


00148 minus00023 00283 minus01723 00654

)

(37)



5 10 15 20 25 30 35Iteration

Erro

r

RGACSRGAVS

CGM

105

100

10minus5

10minus10





1 1198602

isin R20times20


5 Conclusion


119898

119894=1119860119879

119894119883119860119894 In this




Acknowledgments


References



119879

119883minus1





995333

119861995333




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119869 (119883) = d2 (119876L (119883)) (11)


119883lowast

= arg min119883isinS+(119899)

119869 (119883) (12)


lowastsuch that

nabla119883

119869 (119883)1003816100381610038161003816119883=119883lowast

= 0 (13)




isin


119905isin




119881119905

= nabla119883119905

119869 (119883119905) (14)


119883119905+1

= 120574119883119905minus119881119905

(120578) (15)






nabla119883

119869 (119883) = 2119883 ( log (119876minus1

L (119883))L(119883)minus1

+

119898

sum

119895=1

119860119895log (119876

minus1

L (119883))L(119883)minus1

119860119879

119895) 119883

(16)


119869 (119904 (119905)) = tr [log2 (119876minus12

L (119904 (119905)) 119876minus12

)] (17)


minus12

119881119883minus12

)11988312



dd119905

119869 (119904 (119905))|119905=0

= 2 tr[

[

( log (119876minus1

L (119883))L(119883)minus1

+

119898

sum

119895=1

119860119895log (119876

minus1

L (119883))L(119883)minus1

119860119879

119895) 119881]

]

= ⟨119881 2119883 ( log (119876minus1

L (119883))L(119883)minus1

+

119898

sum

119895=1

119860119895log (119876

minus1

L (119883))L(119883)minus1

119860119879

119895) 119883⟩

119883

(18)



119861119860) = 119860minus1


119883119869(119883) can be

rewritten as

nabla119883

119869 (119883)

= 2119883 (L(119883)minus12 log (L(119883)

12

119876minus1

L(119883)12

)L(119883)minus12

+

119898

sum

119895=1

119860119895L(119883)

minus12


119876minus1

L(119883)12

)L(119883)minus12

119860119879

119895) 119883

(19)


119869(119883) belongs to 119879119883S+(119899)





lowastof

(1)


dient nabla119883

119869(119883) If there exists 1198831015840



lowast

119869(1198831015840


equivalent to

log (119876minus1

L (1198831015840

lowast))L(119883

1015840

lowast)minus1

+

119898

sum

119895=1

119860119895log (119876

minus1

L (1198831015840

lowast))L(119883

1015840

lowast)minus1

119860119879

119895= 0

(20)






L (1198831015840

lowast))] = 0 (21)


lowastwith 119883




L (1198831015840

lowast))L(119883

1015840

lowast)minus1

(119883lowast

+

119898

sum

119894=1

119860119879

119894119883lowast119860119894)]


L (1198831015840

lowast))L(119883

1015840

lowast)minus1

119876]

= 0

(22)



L (1198831015840


minus1

L (1198831015840

lowast)) 119876]


L (1198831015840

lowast)) (119868 minus L(119883

1015840

lowast)minus1

119876)]


L (1198831015840


)

times (119868 minus 11987612

L(1198831015840

lowast)minus1

11987612

)]

= 0

(23)


1015840

lowast)minus1




diag (1205821 120582

119899) and 120582



119899

sum

119894=1

(120582119894minus 1) log 120582

119894

= 0 (24)


119899

prod

119894=1

120582119894

120582119894minus1 = 1 (25)


= 1205822

= sdot sdot sdot = 120582119899

= 1 That means that 1198831015840

lowastis


119883119869(119883)


Corollary 3


119883119905+1

= 11988312

119905exp (minus120578119883

minus12

119905nabla119883119905

119869 (119883119905) 119883minus12

119905) 11988312

119905 (26)







) at step 119905 + 1


regard 119869(119883119905+1


119869 (119883119905+1

) = d2 (119876L (120574119883119905minus119881119905

(120578119905))) (27)








= 0 asfollows

119869 (119883119905+1

) asymp 119869 (119883119905) + 1198621119905

120578119905+

1

21198622119905

1205782

119905 (28)





120578lowast

119905asymp minus

1198621119905

1198622119905

(29)





119905= 0


120578lowast

119905asymp

tr [(119883L (log (119876minus1L (119883

119905))L(119883

119905)minus1

))2

]

tr [(L(119883119905)minus1

L (nabla119883119905

119869 (119883119905)))2

]

(30)


119883119905+1

= 11988312

119905exp (minus120578

lowast

119905119883minus12

119905nabla119883119905

119869 (119883119905) 119883minus12

119905) 11988312

119905 (31)

where 1198830



119905+1) det(119883



det (119860119861) = det (119860) det (119861)

det (exp (119860)) = exp (tr (119860)) (32)



det (119883119905+1

)

= det (119883119905) exp (tr [minus120578119883

minus12

119905nabla119883119905

119869 (119883119905) 119883minus12

119905])

= det (119883119905) exp(tr[

[

120578 ( log (L(119883119905)minus1

119876)L(119883119905)minus1

119883119905

+ log (L(119883119905)minus1

119876)L(119883119905)minus1

times

119898

sum

119895=1

119860119879

119895119883119905119860119895)]

]

)

= det (119883119905) exp (tr [120578 log (L(119883

119905)minus1

119876)])

= det (119883119905) det ((L(119883

119905)minus1

119876)120578

)

(33)


det (119883119905+1

)

det (119883119905)

= (det (119876)

det(L (119883119905))

120578

(34)


119905 Moreover

the ratio det(119883119905+1




4 Simulations




119876 = 119883 + 119860119879

11198831198601

+ 119860119879

21198831198602

+ 119860119879

31198831198603

(35)


119876 = (

46350 00698 03104 01990 02918

00698 40376 00742 00694 00690

03104 00742 42242 01554 01826

01990 00694 01554 41472 01572

02918 00690 01826 01572 42134

)

1198601

= (

00834 00007 00293 00142 00139

00141 02975 00154 00289 00339

00244 00152 00317 00270 00117

00251 00097 00140 00135 00240

00158 01499 00275 00077 00157

)

02 03 04 06 07 08050

20

40

60

80

100

120

140

160

Step size

Itera

tions


1198602

= (

02624 00166 00454 00172 00160

00893 00608 00242 00029 00061

00054 00190 00589 00036 00160

00278 00050 00255 00689 00048

00319 00064 00103 00219 00018

)

1198603

= (

81622 06020 04505 08258 01067

27943 70630 00838 05383 09619

03112 06541 62290 09961 00046

05285 06892 09133 10782 07749

01656 07482 01524 04427 90073

)

(36)


0in the following



(

00856 minus00265 00151 minus01065 00148


00151 00166 01591 minus03608 00283


00148 minus00023 00283 minus01723 00654

)

(37)



5 10 15 20 25 30 35Iteration

Erro

r

RGACSRGAVS

CGM

105

100

10minus5

10minus10





1 1198602

isin R20times20


5 Conclusion


119898

119894=1119860119879

119894119883119860119894 In this




Acknowledgments


References



119879

119883minus1





995333

119861995333




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














L (1198831015840

lowast))] = 0 (21)


lowastwith 119883




L (1198831015840

lowast))L(119883

1015840

lowast)minus1

(119883lowast

+

119898

sum

119894=1

119860119879

119894119883lowast119860119894)]


L (1198831015840

lowast))L(119883

1015840

lowast)minus1

119876]

= 0

(22)



L (1198831015840


minus1

L (1198831015840

lowast)) 119876]


L (1198831015840

lowast)) (119868 minus L(119883

1015840

lowast)minus1

119876)]


L (1198831015840


)

times (119868 minus 11987612

L(1198831015840

lowast)minus1

11987612

)]

= 0

(23)


1015840

lowast)minus1




diag (1205821 120582

119899) and 120582



119899

sum

119894=1

(120582119894minus 1) log 120582

119894

= 0 (24)


119899

prod

119894=1

120582119894

120582119894minus1 = 1 (25)


= 1205822

= sdot sdot sdot = 120582119899

= 1 That means that 1198831015840

lowastis


119883119869(119883)


Corollary 3


119883119905+1

= 11988312

119905exp (minus120578119883

minus12

119905nabla119883119905

119869 (119883119905) 119883minus12

119905) 11988312

119905 (26)







) at step 119905 + 1


regard 119869(119883119905+1


119869 (119883119905+1

) = d2 (119876L (120574119883119905minus119881119905

(120578119905))) (27)








= 0 asfollows

119869 (119883119905+1

) asymp 119869 (119883119905) + 1198621119905

120578119905+

1

21198622119905

1205782

119905 (28)





120578lowast

119905asymp minus

1198621119905

1198622119905

(29)





119905= 0


120578lowast

119905asymp

tr [(119883L (log (119876minus1L (119883

119905))L(119883

119905)minus1

))2

]

tr [(L(119883119905)minus1

L (nabla119883119905

119869 (119883119905)))2

]

(30)


119883119905+1

= 11988312

119905exp (minus120578

lowast

119905119883minus12

119905nabla119883119905

119869 (119883119905) 119883minus12

119905) 11988312

119905 (31)

where 1198830



119905+1) det(119883



det (119860119861) = det (119860) det (119861)

det (exp (119860)) = exp (tr (119860)) (32)



det (119883119905+1

)

= det (119883119905) exp (tr [minus120578119883

minus12

119905nabla119883119905

119869 (119883119905) 119883minus12

119905])

= det (119883119905) exp(tr[

[

120578 ( log (L(119883119905)minus1

119876)L(119883119905)minus1

119883119905

+ log (L(119883119905)minus1

119876)L(119883119905)minus1

times

119898

sum

119895=1

119860119879

119895119883119905119860119895)]

]

)

= det (119883119905) exp (tr [120578 log (L(119883

119905)minus1

119876)])

= det (119883119905) det ((L(119883

119905)minus1

119876)120578

)

(33)


det (119883119905+1

)

det (119883119905)

= (det (119876)

det(L (119883119905))

120578

(34)


119905 Moreover

the ratio det(119883119905+1




4 Simulations




119876 = 119883 + 119860119879

11198831198601

+ 119860119879

21198831198602

+ 119860119879

31198831198603

(35)


119876 = (

46350 00698 03104 01990 02918

00698 40376 00742 00694 00690

03104 00742 42242 01554 01826

01990 00694 01554 41472 01572

02918 00690 01826 01572 42134

)

1198601

= (

00834 00007 00293 00142 00139

00141 02975 00154 00289 00339

00244 00152 00317 00270 00117

00251 00097 00140 00135 00240

00158 01499 00275 00077 00157

)

02 03 04 06 07 08050

20

40

60

80

100

120

140

160

Step size

Itera

tions


1198602

= (

02624 00166 00454 00172 00160

00893 00608 00242 00029 00061

00054 00190 00589 00036 00160

00278 00050 00255 00689 00048

00319 00064 00103 00219 00018

)

1198603

= (

81622 06020 04505 08258 01067

27943 70630 00838 05383 09619

03112 06541 62290 09961 00046

05285 06892 09133 10782 07749

01656 07482 01524 04427 90073

)

(36)


0in the following



(

00856 minus00265 00151 minus01065 00148


00151 00166 01591 minus03608 00283


00148 minus00023 00283 minus01723 00654

)

(37)



5 10 15 20 25 30 35Iteration

Erro

r

RGACSRGAVS

CGM

105

100

10minus5

10minus10





1 1198602

isin R20times20


5 Conclusion


119898

119894=1119860119879

119894119883119860119894 In this




Acknowledgments


References



119879

119883minus1





995333

119861995333




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











det (119883119905+1

)

= det (119883119905) exp (tr [minus120578119883

minus12

119905nabla119883119905

119869 (119883119905) 119883minus12

119905])

= det (119883119905) exp(tr[

[

120578 ( log (L(119883119905)minus1

119876)L(119883119905)minus1

119883119905

+ log (L(119883119905)minus1

119876)L(119883119905)minus1

times

119898

sum

119895=1

119860119879

119895119883119905119860119895)]

]

)

= det (119883119905) exp (tr [120578 log (L(119883

119905)minus1

119876)])

= det (119883119905) det ((L(119883

119905)minus1

119876)120578

)

(33)


det (119883119905+1

)

det (119883119905)

= (det (119876)

det(L (119883119905))

120578

(34)


119905 Moreover

the ratio det(119883119905+1




4 Simulations




119876 = 119883 + 119860119879

11198831198601

+ 119860119879

21198831198602

+ 119860119879

31198831198603

(35)


119876 = (

46350 00698 03104 01990 02918

00698 40376 00742 00694 00690

03104 00742 42242 01554 01826

01990 00694 01554 41472 01572

02918 00690 01826 01572 42134

)

1198601

= (

00834 00007 00293 00142 00139

00141 02975 00154 00289 00339

00244 00152 00317 00270 00117

00251 00097 00140 00135 00240

00158 01499 00275 00077 00157

)

02 03 04 06 07 08050

20

40

60

80

100

120

140

160

Step size

Itera

tions


1198602

= (

02624 00166 00454 00172 00160

00893 00608 00242 00029 00061

00054 00190 00589 00036 00160

00278 00050 00255 00689 00048

00319 00064 00103 00219 00018

)

1198603

= (

81622 06020 04505 08258 01067

27943 70630 00838 05383 09619

03112 06541 62290 09961 00046

05285 06892 09133 10782 07749

01656 07482 01524 04427 90073

)

(36)


0in the following



(

00856 minus00265 00151 minus01065 00148


00151 00166 01591 minus03608 00283


00148 minus00023 00283 minus01723 00654

)

(37)



5 10 15 20 25 30 35Iteration

Erro

r

RGACSRGAVS

CGM

105

100

10minus5

10minus10





1 1198602

isin R20times20


5 Conclusion


119898

119894=1119860119879

119894119883119860119894 In this




Acknowledgments


References



119879

119883minus1





995333

119861995333




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5 10 15 20 25 30 35Iteration

Erro

r

RGACSRGAVS

CGM

105

100

10minus5

10minus10





1 1198602

isin R20times20


5 Conclusion


119898

119894=1119860119879

119894119883119860119894 In this




Acknowledgments


References



119879

119883minus1





995333

119861995333




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Riemannian Gradient Algorithm...

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