Estimating genetic parameters using an animal model with imaginary effects

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Estimating genetic parameters using an animal model

with imaginary effects

R Thompson, K Meyer

To cite this version:

Original

article

Estimating

genetic

parameters

using

an

animal model with

imaginary

effects

R

_Thompson

₁

K

_Meyer

₂

1

AFRC

Institute

_of

Animal

_Physiology

and _Genetics,

_Edinburgh

Research

_Station,

Roslin, Midlothian, EH25 _9PS;

2

Institute _of Animal _{Genetics, University of Edinburgh,}

_King’s

_Buildings,

Edinburgh EH9 _3JN, UK

(Received

8 _{July 1988; accepted} 24 November

₁₉₈₉₎

Summary - The estimation of additive _genetic variance

_by

maximum likelihood is

discussed. The extension of reduced animal _models, when _parents are not inbred to

allow the use of

_existing

_algorithms for maximum likelihood estimation of additive _genetic

variance, is described. This involves the introduction of _imaginary effects with

_negative

variance, and leads to _{computation using complex} arithmetic. Methods are _developed to

allow the _computation to be carried out

_using

_only real arithmetic. This method has

computational advantages

when

_only

a small proportion of animals have

offspring.

genetic parameter / animal _{model / estimation /} maximum likelihood

R.ésumé - Estimation de _{paramètres génétiques} selon un modèle animal incluant

des effets _{imaginaires -} On discute l’estimation de la variance _génétique additive par le

maximum de vraisemblance. On décrit l’extension des _algorithmes

_disponibles

d’estimation

par le maximum de vraisemblance de la variance génétique additive, à la situation d’un

modèle animal réduit au cas où les _parents ne _{sont pas consanguins.} Ceci _{passe par}

l’introduction

_d’effets

_imaginaires, de variance

_négative,

et conduit à des calculs en

arithmétique complexe. Des méthodes sont _{développées pour n’utiliser que l’arithmétique}

réelle. Cette méthode _présente des _{avantages numériques quand} une

_faible

_proportion

seulement des animaux ont des descendants.

paramètre génétique / modèle animal _/ estimation _/ maximum de vraisemblance

INTRODUCTION

Additive

_genetic

variance and

_heritability

have most

_commonly

been estimated in

animal

_breeding

data information from collateral

_relatives,

such as half-sibs or

full-sibs,

or non-collateral

relatives,

such as

parent-offspring.

Covariances

generated

by

these

_{relationships provide}

the most information for

_estimating

additive ge-netic variance.

_However,

there is interest in

_combining

these alternative estimates

*

(Nicholas

and

_Hill,

_1974),

and also

_using

the additional information from other

genetic

relationships.

Linear models can be formed that contain

_genetic

and environmental effects for

each

_animal,

_commonly

called individual animal models

_(eg,

_Quaas

and

_Pollak,

1980).

In

_theory

at

_least,

standard or restricted maximum likelihood estimation

(ML

or

_REML)

_procedures

can be

_applied

to individual animal models for variance component estimation

_(Harville,

_1977).

One

_problem

is that this model _generates

as _many

_equations

as there are animals in the

_pedigree

and so there have been

attempts to reduce the

_{computational}

effort.

_Thompson

₍₁₉₇₇₎

considered the case

of two

_generations

and showed how to

_develop

REML

_{estimating equations just}

for animals in the first

_generation.

More

_generally,

this estimation can

_usually

be

interpreted using

predicted

breeding

values.

_Quaas

and Pollak

₍₁₉₈₀₎

showed that

a reduced animal model

_(RAM)

has

_advantages

in

_{calculating predicted breeding}

values.

_{Notably, only}

_equations

for animals with

_offspring

are needed. Henderson

(1986)

and Sorensen and

_Kennedy

₍₁₉₈₆₎

showed that REML and minimum norm

quadratic

(MINQUE)

estimation can be

_expressed

with

_{advantage using}

a RAM.

However,

these

_(REML)

methods are iterative

_and,

as

presented,

_require

the inversion of a matrix with as many rows and columns as the number of _parents in each round of iteration. For some variance component

problems,

this

computa-tional burden can be reduced

_{by using orthogonal}

transformations to

_{give simpler}

equations,

either in terms of a

_diagonal

matrix

_(Patterson

and

_Thompson,

1971;

Dempster

et

_al,

_1984),

or in terms

_{of a tridiagonal}

matrix

_(Smith

and

_Graser,

_1986).

These results are not

_{directly applicable}

to estimation methods based on a RAM.

This paper shows how a RAM can be modified

by

the introduction of extra random effects with

_negative

_variances,

so that the

_resulting

matrices involved in estimation can be

_{tridiagonalized using}

existing

algorithms

based on Householder transforma-tions.

_Complex

numbers are needed to take account of

_negative

_variances,

but a new

algorithm

is

_given

that takes account of the

_special

structure of the matrices

involved,

and reduces the need for arithmetic based on

complex

numbers.

THE MODEL

If additive

_genetic

covariances are the

_only

source of covariances between

_records,

then a linear model for the individual animal model

_(IAM)

can be written as:

with

_E(y)

=

Xb,

E(a) = E(e)

= 0 and

V(a)

=

Ao, 2,

V(e)

=

Ia5

and

cov(a,e’)

=

0,

where _y,

b,

a and e denote the vectors of

_{observations,}

fixed

effects,

animal effects and residual errors. X and Z are the

corresponding

incidence

matrices,

and for

_simplicity,

we assume X had r columns and rank r. With

_single

records _per

_animal,

Z =

I

nt

,

where m denotes the number of observations. A is the

numerator

_relationship

matrix

_(NRM)

between animals.

As described

_{by Quaas}

and Pollak

_(1980),

with a

_RAM,

the vector of animals is

divided into _parents,

_ie,

animals which have

_offspring

_(subscript

P in the

_following),

and _non-parents,

_ie,

animals without _progeny

_(denoted

_{by subscript}

_N).

The additive

_genetic

value for _non-parents is then

_partitioned

into contributions from

with the residual e in

eqn(l)

into a new residual _error,

_gives

the RAM as a

reparameterisation

of eqn(1):

Let p denote the number of parents and _q the number of _non-parents, then

(p

+ q =

_m).

_Z

_N

_is then _{a matrix of order}

qxp, with elements z2! =

0.5,

if j

is

a _parent of

_i,

and zero otherwise. This

_{representation,}

_{strictly speaking,}

assumes

all _parents of _non-parents are in the data and one record per individual.

However,

eqn(2)

can be

_easily

_modified,

if that is not the case. The residual error for a

non-parent has variance

at

v

=

a5

+

_0.5

_QA

_,

if both _parents are

_known,

and variance

at

v

=

QE

+0.75

QA

,

if

_only

one _parent is

known, provided

they

are not inbred. In the

following,

it is assumed that there is

_{equal parental}

information for all _non-parents

and that parents are non inbred

_ie,

that

a 2 w

is constant.

Estimation

_equations,

based on the

RAM, usually

have to be reconstructed after each variance iteration because the ratio of error to additive variance

_changes.

Therefore,

for

_{computational}

_reasons, it is desirable to have a model where the

vector of residuals has

_homogeneous

variance. This can be achieved

_by

a further

reparameterisation,

adding

effects to either _parents or _non-parents. As there are

usually

fewer _parents than _non-parents, fewer

_equations

will be

_generated

if

_eqn(2)

is

_expanded

to:

The variance of e, is such that

_var(e

_j

₎

+

_-

_var(e

_P

_e

_j

₎

=

var(ep)

=

Ipas

and

var(ep —

e

l

)

=

lo,2&dquo;

so that ep — ej and eN have the same variance.

Hence,

var(e

1

)

_

-

C

2

UA

Ip,

with

2

c

= 0.75 and 0.50 for one and both _parents

_known,

respectively.

Because of the

_assumption

that there is

_{equal parental}

information for all _non-parents,

c

2

is the same for all e, values.

Normally,

variances are assumed to

be

_positive

so there is a

_{slight difficulty}

in

_interpreting

el with a

_negative

variance.

However,

if aD are effects with variance

_{a A 2 Ip,}

then

_defining

ej =

_ica

_D

_,

where

i =

i

,

then the variance of _ej =

,

A Ip.

2

2

0

C

-

_Hence,

_{ei can} be

_thought

of as

imaginary

effects as

_they

are a

_multiple

of i.

Hence,

eqn(3)

can be written in terms of real effects as:

with e =

_{[(ep - e’) eN]&dquo;}

This

_gives

mixed model

_equations

_(MME)

as:

where A denotes the variance ratio

_!yt,/!A,

and

_Ap

is the numerator

_relationship

Eliminating

b from

_eqn(4)

_gives

equations

of the form

where

Hence,

The _component matrices

_H

_u

_,

H

12

and

H

22

are all real.

Equation

(7)

gives predictors

of the effects ap and aD. Note that ap is real and a

D is

imaginary,

so that iaR = _aD with _aR

real,

and the

predictor

for _el =

-ca

R

is a real

quantity,

as would be

_anticipated.

REML

_equations

to estimate

_QA

and

Q

yy

are

(in

terms of real

_quantities):

where r is the rank of

_{(X’ p Xp}

+

_{X%XN ) ,}

and n is the number of elements in both

ap and a

D

.

These

_equations

involve

_a

₂

_A

and

_Q

_yy

_{through A =}

_!yy/QA

on both sides

of eqns(8)

and

_(10),

and have to be solved

_iteratively.

To

_simplify

_eqns(8)

and

₍₁₀₎

and avoid the inversion associated with

_Ap

_l

_,

Smith and Graser

₍₁₉₈₆₎

and

_Meyer

₍₁₉₈₇₎

suggest

writing

Ap

as LL’ and

_using

aL =

L-

l

ap

in a model

equivalent

to

eqn(3).

(A

I

is not to be confused with A or

_Ap,

numerator

_relationship

_matrices).

This iterative scheme

_requires

the inversion of

_(A

_I

_{+ IA)}

in each iteration. Smith and Graser

₍₁₉₈₆₎

have shown that these

_computations

can be reduced

_{by writing}

A

l

as

_(PA!P’)

where P is an

_orthogonal

matrix and

_A*

_is

a

_tridiagonal

matrix.

Using algebra

similar to that of Smith and Graser

_(1986),

it can be shown that

quantities

arising

in

_eqns(11)

and

₍₁₂₎

can be written in terms of

_(A*

_{+ >’1)-}

₁

and For

_example:

A!

can be found

using

a _sequence of Householder

_{transformations, using complex}

arithmetic. In the

_Appendix,

it is shown how the

_special

structure of

A

l

with real

quadrants

(L’BL

and

_D)

and

_{imaginary quadrants}

_(iL’C)

and

_(iL’C)’

can be used

to

_give

an

algorithm

for

_A*

_and

q

*

and to evaluate

_eqns(12)

to

_(16),

_{using only}

real arithmetic. It should be noted that this

_{computational}

_strategy is

_using

an

existing

iterative scheme and is

_manipulating

the

_equations

to reduce the number

of

_{computations.}

NUMERICAL EXAMPLE

To illustrate the formation of the model and mixed model

_equations,

we use an

example

of 5 observations of the same sex with individuals 2 and

_3,

the

offspring

of

individual 1,

and individuals 4 and

_5,

the

_offspring

of individual 2. There is 1 fixed

with the observation vector coded so that observations on _parents

(1

and

₂₎

occur first. The variance of ei is

_a

₅

+

_3/4<r!,

as animals have

_only

one known parent. The variance of

_(y¡)

=

QA

-

3/4<7!

₊

a5

₊

3/4

Q

A

=

a5

₊

!A,

_var(

_y

_{s) _}

1/4

QA

+

QE

+ 3/4

QA

=

QE

+

QA

.

and

_eqn(5)

becomes:

Eliminating

61 and

b

2

gives:

This shows the structure of

_eqn(6)

with two real

_quadrants

and two

_imaginary

quadrants

in the matrix on the left hand side. For this

_example,

estimates of the effects can be found

_by

_partitioning

the coefficient matrix into 2 x 2 real and

imaginary

parts and

_using

results on inverses of

_partitioned

matrices.

DISCUSSION

This is a novel

_approach

with the

_{advantage of working}

with matrices of size

_2p,

rather than

_(n+p),

and the

_computations

involved,

are of the order of

_(2p)

₃

_,

rather

than

_(n

+

_p)

₃

_.

This

_technique

_is,

_therefore,

of more use in

_populations

with a low

proportion

of animals used as _{parents. The}

technique

can be

easily

extended to

estimates 2 multivariate residual and additive variance _components matrices when

measurements are taken on all animals

_using

the

_{procedures developed by Meyer}

(1985).

Two

_assumptions

are made.

_First,

that all _non-parents have

_{equal parental}

information. If this

assumption

is not

_satisfied,

_non-parents with unknown _parents

can have

_dummy

_parents inserted into the model. In most animal

_breedings

sets

where

_inbreeding

is

_{consciously avoided,}

the second

assumption

(parents

are not

inbred)

is

_unlikely

to be

_important.

There is more

_likely

to be concern about residual variance

_homogeneity,

than about

_{inbreeding generated genetic}

variance

homogeneity.

obvious

_competitor,

but it is difficult to _say

_precisely

when each method is to

be

_preferred.

The time

required,

depends

on the number of

_animals,

structure of

population

the sequence used in

calculating

the

likelihood,

the number of variates

measured and the

_speed

of convergence of the iterative

procedure.

Their comments

suggest that our method could be

_advantageous

if the number of _parents is less

than 350.

The method has been

_presented

for a model with additive

_genetic

_covariances,

but in many cases, other components, such as litter

_variances,

need estimation. In such cases, the

procedure

can be used for a

_given

ratio of litter variance to residual variance and

_repeated

for different values of the

_ratio,

in a similar manner to that

suggested

by

Smith and Graser

_(1986).

REFERENCES

Dempster AP,

Selwyn

MR,

Patel

_IM,

Roth AJ

₍₁₉₈₄₎

Statistical and

_{computational}

aspects of mixed model

_{analysis. Appl}

Stat

_33,

203-214

Graser

_H-U,

Smith

_SP,

Tier B

₍₁₉₈₇₎

A derivative-free

_approach

for

_estimating

variance _components in animal models

_by

restricted maximum likelihood. J Anim

Sci

64,

1362-1370

Harville DA

₍₁₉₇₇₎

Maximum likelihood

_approaches

to variance _component esti-mation and to related

_problems.

J Am Stat Assoc

_72,

320-338

Henderson CR

₍₁₉₈₆₎

Estimation of variances in Animal Model and Reduced Animal Model for

_single

traits and

_single

records. J

_Dairy

Sci

_60,

1394-1402

Meyer

K

₍₁₉₈₅₎

Maximum likelihood estimation of variances _components for a multivariate mixed model with

_equal

_design

matrices. Biometrics

_41,

153-166

Meyer

K

₍₁₉₈₇₎

A note on the use of an

_equivalent

model to account for

_{relationships}

between animals in

_estimating

variance components. J Anirrc Breed Genet

104,

163-168

Nicholas

_FW,

Hill WG

₍₁₉₇₄₎

Estimation of

_{heritability by}

both

_regression

of

offspring

on _parent and intra-class correlation of sibs in one

_experiment.

Biometrics

30,

447-68

Patterson

_HD,

_Thompson

R

₍₁₉₇₁₎

_Recovery

of inter-block information when block

sizes are

_unequal.

Biorn.etrika

58,

545-554

Quaas

RL,

Pollack EJ

₍₁₉₈₀₎

Mixed model

_methodology

for farm and ranch beef

cattle

_testing

_programs. J Anirn, Sci

_51,

1277-1287

Smith

_SP,

Graser H-U

₍₁₉₈₆₎

_Estimating

variance _components in a class of mixed models

_by

restricted maximum likelihood. J

_Dairy

Sci

_69,

1156-1165

Sorensen

_{DA, Kennedy}

BW

₍₁₉₈₆₎

_Analysis

of selection

_experiments

_using

mixed

model

_methodology.

J Anim Sci

_63,

245-258

Thompson

R

₍₁₉₇₇₎

The estimation of

_heritability

with unbalanced data. II. Data available on more than two

_{generations. Biometrics, 33,}

497-504

APPENDIX

Reduction of

_complex

matrices to

_tridiagonal

form

A square matrix is said to be

_tridiagonal

_(in

the first t

_rows)

if the

_only

non-zero

elements are in the _r,

_{s elements,}

where s = _{r y-}

_1,

_{r, or r + 1}

(r

_<

t).

A real

symmetric

matrix

_A

_i

_,

of size n x _n, can be reduced to

_tridiagonal

form

An-1

by

a sequence of

(n &mdash; 2)

Householder transformations

(for

example,

Wilkinson and

Reinsch,

1971).

In this sequence of Householder

transformations,

at the

t

th

_stage, the

transfor-mation

P

t

is chosen to make the

tt!’

row of

At+,

contain all zero

_elements,

_except

possibly

those in the

_{(t &mdash; 1),}

t, and

_(t

+

₁₎

columns. This

operation

will be called

pivoting

on the

t

th

row. The non-zero elements in the

_j

_th

₍₁

_{< j}

<

_n)

row of

_A

_n

_-

_i

are in the

(j - 1)t’

and

(j

+

&dquo;

t

1)

columns,

indicating

the

_previous

_{(j -1)}

and next

( j

+

_{1 )}

pivots.

For the

_{algorithm presented}

in this paper, a

complex

matrix

A

l

of the form:

where

_{B, C, D}

are real matrices

_[of

size

_(n,

x

_n

_l

_{), (n,}

x

₎

_n

₂

and

(n

2

x

_n

₂

_)]

is to

be

_{tridiagonalized.}

In order to illustrate the

_method,

a numerical

example

will be

_given.

The matrix Al

satisfying

eqn(A1)

with:

will be

_used,

with n, = n2 = 4

(for

_convenience,

when matrices _are

symmetrical,

only

the lower

_triangular

_part is

_given).

The sequence of Householder

transfor-mations derived for real matrices could be used

but,

as a _square root of a sum

of squares, at, is used and as this could be

negative,

this would involved

com-plex arithmetic,

which is

_{computationally considerably}

more

_demanding

than real

arithmetic. A modification of the

_{tridiagonalization}

is now

_presented

which avoids

There are two _stages involved. In the

_first,

a sequence of Householder transfor-mation is used that

_gives

_At+,

=

t

Pt,

A

t

P

where

At+,

and

P

t

are found to have

the same form as A in

eqn(A1)

with

_quadrants

of real and

_imaginary

numbers. This involves

_changing

the

_ordering

of the

_pivoting.

The rows of

A

l

are

_split

into two sets 1 to nl and

(1 + n

i

)

to

_(n,

_{+ n}

₂

_),

_{corresponding}

to the division into

_quadrants

in

_eqn(Al).

The

_pivoting

is started on row 1 and continues in the first set until

at <

0,

then we

pivot

on the first row of the second set and continue in this set

until

_again

at < 0. The process is then

repeated

until

_{nl + n2 - 1 = n -}

1 rows

have been

_pivoted

on. This

_{procedure produces}

a matrix

_At+,

with as _many non-zero elements as the usual

_procedure,

but

_A

_t+i

is not

_{necessarily tridiagonal.}

For

example,

for the numerical

_example,

the

_ordering

of

_pivoting

is found to be rows _1,

2, 5, 3, 6, 7,

giving

A

7

.

There are _atmost, 3 non-zero elements in each row of

_A

₇

_.

In the second

_{stage, A}

_n

_-

_i

will be

permuted

to

_give

a

tridiagonal

matrix.

The first _stage is now illustrated

_using

recursive _arguments. Some

_housekeeping

notation is needed to

_{identify pivoted}

rows. In the real case, the index t was related

to the number of transformations carried out

_{(t &mdash; 1),}

the rows which have been

transformed

_{(1... t - 1)}

and the next row

_(t)

to be

_pivoted.

The

_complex

case is

more

_complicated;

for

_example,

formation of

A

4

for the numerical

example

involves

pivoting

on row

_1,

2 and 5 in _turn, and the next row to be

_pivoted

is row 3. It is

convenient to define

_R

_t

_,

_S

_t

and

T

t

to indicate that

_previous

_operations

have used the first

_(R

_t

_-

₁₎

rows of the first set of rows and the first

(S

t

-

1)

rows of the

second set of rows as

_pivots

and that

_T

_t

is the next

_{pivot. K}

_t

is used to indicate if the next

_pivot

_(T

_t

₎

is in the first set of rows

(K

t

=

1)

or in the second _set of

rows

_(K

_t

=

2).

Within the _{two sets} of

rows, the rows are transformed in sequence

so that

T

t

=

_R

_t

_,

if

_K

_t

=

_1,

and

_T

_t

=

n

1 +

_S

_t

_,

if

_K

_t

= 2.

_Hence,

in the numerical

example,

R4

= 2 + 1; S

4

= 5 - 4 + 1; T

4

= 3; K

4

=

1; R

5

= 4. As a total of

(t - 1)

rows of

_A

_l

have been

_pivoted,

this

_equals

the sum of rows

_pivoted

in the two sets; i _e,

_(R

_t

_-

₁₎

+

_(S

_t

_-

₁₎

= _{t -} 1.

_Initially,

_t =

_{1, R}

l

=

_{1, S}

l

= 1 and the first row in

the first set can be used as the first

_pivot,

and so,

T

i

=

_{1, K}

_l

_r 1.

Two

_slightly

different

_strategies

are needed for

_K

_t

= 1 and

K

t

= 2. The

derivation of

_At+1

is

_given

first

_{when K}

_t

= 1. To

simplify

notation in this

t

th

stage, let r =

Rt,

_{s =}

S

t

, k

=

_,

_K

_t+1

and suppose

At

is of the form

(Al).

As

K

t

=

_1,

then the next row to be

_pivoted

is _r, so

_T

_t

= _r; the number of rows in the

two sets

_already

used as

_pivots

in

_At+,

will be r, and s -

_1,

so that

_R

_t+1

= _{r +} 1

A matrix

_A

_t+i

is

_required,

so that

_At+,

=

P

t

A

t

Pt.

Using

the same _argument

as in usual Householder transformations:

with terms in

_(A2)

to

_(A4)

_expressed

as real numbers where

possible.

A sum of

squares ot is

needed,

_given

by:

with

_b

_rm

_,

crm and

d

rm

being

elements of the real matrices

B,

C, D,

that

together

form

_At

_{(using (A1)).}

The elements of

_f,

and

_,

₂

_f

added to ut, and Ut2, are

_given

by,

There are two

_{possibilities}

in

_(A6):

if Qt > 0, then usual Householder transfor-mation is

used,

and the next

_pivot

will be the

_(r

+

_1)t!’

row in

_{the first}

set

_(k

=

1),

and the

_(r,

r +

₁₎

element of

_At+,

will be non-zero.

_However,

if at <

0,

then the

(n,

+

_s)

_th

row is used as a

_pivot

_(k

=

2)

in order that

At+1

and

P

t

have the _same

form as

(Al).

Then

_At+1

can be calculated

_using:

1- - ..

--For the numerical

_example

with T1 =

_1,

then from

(A5)

and

(A7),

_<ri =

0.375;

f

l

=

_-0.6124;

f

2

=

_0;

k =

_K

₂

= _1.

So from

_(A4):

Hence, using

(A10),

A

2

has the form of

_(Al)

with:

The first column of

_A

₂

has

_only

1 non-zero

_{off diagonal}

element;

the

_(2,1)

element.

A

2

has R

2

= 2 and

S

2

=

1;

_as

only

the first row has been

pivoted.

As _{172 <}

0,

then

K

2

=

_1,

and the _{next row to} be

pivoted

is 2 and

T

2

= 2.

Equations

(A5)

and

(A7)

show that a2 =

-75.8750;

f

l

=

_0.0000;

f

2

=

8.7106; K

3

= k = 2.

Formulae are now

_presented

to show how to

_pivot

on a row in the second set of

At

i e, K

t

= 2. These formulae _are similar _to

(A2)

to

(A10),

with some

changes

in

Then,

Using

(All)

to

_(A19),

matrix

_A

₄

can be constructed from:

A

7 is indicated above.

Hence,

a matrix has been constructed

_with,

at _most,

_{2(n - 1)}

_{off diagonal}

elements. In this

_example,

there are

_12,

as the

_(6,4)

and

_(4,6)

elements are zero. The second _stage is to transform

_At-1

to

_tridiagonal

form

_{At_}

₁

_.

This can be

simply

achieved

_by

_recording

the rows and columns in the order that the

_pivoting

was carried _out;

_{ie, T}

_t

=

(t

= _{1, ... ,}

n),

where

_T

_n

is the row not included in the set

(Ti

7

T

2

, ...

)T

n

-

1

)

7

and is either n1 or

(n,

+

_n

₂

)-In the numerical

_example:

T

1

=

_1,

T2 = _2,

T

3

=

5,

T4 =

_3,

T5 =

6, T

6

=

4,

T

Formation of

_equations

₍₁₂₎

to

₍₁₅₎

If

_(A*

₊

_AI)

and

_q

_*

are real

_matrices,

say G and q, then Smith and Graser

(1986)

suggest

writing

G as U’DU

(where

U is a lower

triangular

matrix with

diagonal

elements

_1,

and D is a

_diagonal

_matrix)

and

_give

an

algorithm

to find the non-zero

elements of D and

U;

ie,

D

j

and

_U!,!.

and the other elements of D and G are zero.

The elements of a+ =

G-

l

q

can be found

by

eliminating

the elements of a+ in turn, and

_{back-substituting}

to find

_at.

That is in the first

_place

form:

Expression

(13)

can be found from either

aj

+

2 / D

j

or

atat

+

.

In our case,

(A*

+

AI), q

*

and a+ have real and

_imaginary

terms. The indices

K

j

indicate if the elements of

_(A*

₊

_AI),

_q

_*

and a+ are real or

_{imaginary. Suppose}

Then Smith and Graser’s

_algorithm

can be used

_again,

and in terms of real

arithmetic if 3 extra sets of coefficients are defined. There are 4 cases to consider:

The real coefficients associated with U and D are now

given by:

The real coefhcients of

_a

₊

_,

a, can be found from

For _instance, for the numerical

_example: