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Numerical Methods Part: Cholesky and Decomposition http://numericalmethods.eng.usf.edu

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Chapter 04.09: Cholesky and Decomposition

Major: All Engineering Majors

Authors: Duc Nguyen

http://numericalmethods.eng.usf.edu Numerical Methods for STEM undergraduates

http://numericalmethods.eng.usf.edu 5 11/14/2011

Lecture # 1 TLDL


(1)

1

Introduction

6

http://numericalmethods.eng.usf.edu

][]][[ bxA =

= known coefficient matrix, with dimension

where

][A nn×

][b = known right-hand-side (RHS) 1×n vector

][x = unknown 1×n vector.

http://numericalmethods.eng.usf.edu 7

Symmetrical Positive Definite (SPD) SLE

A matrix can be considered as SPD if either of

(a) If each and every determinant of sub-matrix is positive, or..

the following conditions is satisfied:

for any given vector (b) If

As a quick example, let us make a test a test to see if the given matrix

nnA ×][

),...,2,1( niAii =,0>AyyT 0][ 1

≠×ny

−−−

−=

110121

012][A is SPD?


Symmetrical Positive Definite (SPD) SLE

Based on criteria (a):

The given matrix 33× is symmetrical, because

Furthermore,

[ ] 022det 11 >==×A

[ ]

032112

det 22

>=

−−

=×A

jiij aa =


[ ]

01110121

012det 33

>=

−−−

−=×A

Hence is SPD. ][A


Based on criteria (b): For any given vector

0

3

2

1 ≠

=

yyy

y , one computes

[ ]

( ) { }( ) { }32232221221

3223

2221

21

3

2

1

321

2

2222

110121

012

yyyyyyy

yyyyyyy

yyy

yyy

Ayyscalar T

−+++−=

−++−=

−−−

−=

=


( ) ( ) 023221221 >−++−= yyyyyscalar

hence matrix is SPD ][A

12

Step 1: Matrix Factorization phase

][][][ UUA T=

=

33

2322

131211

332313

2212

11

333231

232221

131211

0000

00

uuuuuu

uuuuu

u

aaaaaaaaa

Multiplying two matrices on the right-hand-side (RHS) of Equation (3), one gets the following 6 equations

1111 au =11

1212 u

au =11

1313 u

au =

( )21

2122222 uau −=

22

13122323 u

uuau −= ( )21

223

2133333 uuau −−=

(2)

(3)

(4)

(5)


( ) 21

1

1

2

−= ∑

−

=

i

kkiiiii uau

ii

i

kkjkiij

ij u

uuau

∑−

=−

=

1

1

Step 1.1: Compute the numerator of Equation (7), such as

∑−

=−=

1

1

i

kkjkiij uuaSum

Step 1.2 If is an off-diagonal term (say ) iju ji <then (See Equation (7)). Else, if is a

iiij u

Sumu = ijudiagonal term (that is, ), then ji = Sumuii =

(6)

(7)

(See Equation (6))


As a quick example, one computes:

55

47453735272517155757 u

uuuuuuuuau −−−−=

Thus, for computing , one only needs )7,5( == jiuto use the (already factorized) data in columns )5(# =iand of , respectively. )7(# =j ][U

(8)


k = 1

k = 2

k = 3

k = 4

iiu iju

ju4ju3ju2

ju1

iu4iu3iu2

iiu

i = 5

Col. # i=5 Col. # j=7

Figure 1: Cholesky Factorization for the term iju


Step 2: Forward Solution phase Substituting Equation (2) into Equation (1), one gets:

][]][[][ bxUU T =Let us define:

][][][ yxU ≡Then, Equation (9) becomes:

][][][ byU T =

=

3

2

1

3

2

1

332313

2212

11

000

bbb

yyy

uuuuu

u

(9)

(10)

(11)

(12)


1111 byu =

11

11 u

by =

From the 2nd row of Equation (12), one gets

2222112 byuyu =+

22

11222 u

yuby −=Similarly

33

22311333 u

yuyuby −−=

(13)

(14)

(15)


In general, from the row of Equation (12), one has thj

jj

j

iiijj

j u

yuby

∑−

=−

=

1

1 (16)


Step 3: Backward Solution phase

As a quick example, one has (See Equation (10)):

=

3

2

1

3

2

1

33

2322

131211

000

yyy

xxx

uuuuuu

(17)


From the last (or ) row of Equation (17), rdthn 3=

one has 3333 yxu =

hence

33

33 u

yx =

Similarly: 22

32322 u

xuyx −=

and

11

31321211 u

xuxuyx −−=

(18)

(19)

(20)


In general, one has:

jj

n

jiijij

j u

xuyx

∑+=

−= 1 (21)


TLDLA ]][][[][ =For example,

=

10010

1

000000

101001

32

3121

33

22

11

3231

21

333231

232221

131211

lll

dd

d

lll

aaaaaaaaa

Multiplying the three matrices on the RHS of Equation (23), one obtains the following formulas for

the “diagonal” , and “lower-triangular” matrices: ][D ][L

(22)

(23)


∑−

=−=

1

1

2j

kkkjkjjjj dlad

×

−= ∑

−

= jj

j

kjkkkikijij d

ldlal 11

1

(24)

(25)


Step1: Factorization phase TLDLA ]][][[][ =

Step 2: Forward solution and diagonal scaling phase

Substituting Equation (22) into Equation (1), one gets:

][][]][][[ bxLDL T =Let us define:

][][][ yxL T =

(22, repeated)

(26)


Also, define: ][]][[ zyD =

=

3

2

1

3

2

1

33

22

11

000000

zzz

yyy

dd

d

nifordzyii

ii ......,,3,2,1, ==

Then Equation (26) becomes:

][]][[ bzL =

(29)

(30)


=

3

2

1

3

2

1

3231

21

101001

bbb

zzz

lll

niforzLbzi

kkikii ......,,3,2,1

1

1=−= ∑

−

=

(31)

(32)


Step 3: Backward solution phase

=

3

2

1

3

2

1

32

3121

10010

1

yyy

xxx

lll

1......,,1,;1

−=−= ∑+=

nniforxlyxn

ikkkiii


Numerical Example 1 (Cholesky algorithms)

Solve the following SLE system for the unknown vector ? [ ]x ][]][[ bxA =

where [ ]

−−−

−=

110121

012A

=

001

][b


Solution:

The factorized, [ ]Uupper triangular matrix can be computed by either referring to Equations (6-7), or looking at Figure 1, as following:

][1

0414.10

7071.0414.1

1

414.12

11

1313

11

1212

1111

Uofrow

uau

uau

au

=

=

=

−=

−=

=

==

=


( )

( ){ }( )

( )( )

][2

8165.0225.1

07071.01225.1

1

225.17071.02

2

1312

22

11

123

23

2

21

212

21

11

1

22222

Uofrow

uuU

uuau

u

uau

i

kkjki

i

kki

−=

−−−=

×−−=

−=

=

−−=

−=

−=

∑

∑

=−

=

=−

=


( )

{ }( ) ( )

][3

5774.08165.001 22

21

223

21333

21

21

1

23333

Uofrowuua

uaui

kki

=

−−−=

−−=

−= ∑=−

=

Thus, the factorized matrix

[ ]

−

−=

5774.0008165.0225.1007071.0414.1

U


The forward solution phase, shown in Equation (11), becomes:

[ ] [ ] [ ]byU T =

=

−−

001

5774.08165.000225.17071.000414.1

3

2

1

yyy


( )( )( )

( )( ) ( )( )( )

5774.05774.0

4082.08165.07071.000

4082.0225.1

7071.07071.00

7071.0

414.11

33

223113

21

13

3

22

112

11

12

2

11

11

==

=−=−==−=

−=

==

=−=−=

−=

=

=

=

∑

∑

=−

=

=−

=

uyuyu

u

yuby

uyu

u

yuby

uby

jj

j

iiij

jj

j

iiij


The backward solution phase, shown in Equation (10), becomes:

[ ][ ] [ ]yxU =

=

−

−

5774.04082.07071.0

5774.0008165.0225.1007071.0414.1

3

2

1

xxx


( )( )

1225.1

18165.04082.0

15774.05774.0

22

3232

3

312

33

3

3

=

−−=

−=

−=

=

=

=

=

∑=

=+=

uxuy

u

xuyx

uy

uy

x

jj

N

jiijij

jj

j


( )( ) ( )( )

1414.1

1017071.07071.011

3132121

3

211

=

−−−=

−−=

−=

∑=

=+=

uxuxuy

u

xuyx

jj

N

jiijij

Hence

=

111

][x


Numerical Example 2 ( Algorithms) TLDL

Using the same data given in Numerical Example 1, find the unknown vector ][x by algorithms? TLDLSolution:

The factorized matrices and can be computed

from Equation (24), and Equation (25), respectively.

][D ][L


[ ] [ ]LandDofmatricesofColumn

dal

da

d

ldlal

alwaysl

a

dlad

jj

j

kjkkkik

j

kkkjk

1

020

5.021

)!(12

11

3131

11

21

01

121

21

11

11

01

1

21111

=

=

=

−=

−=

=

−=

===

−=

∑

∑

=−

=

=−

=


( ) ( )

( )( )( )

[ ] [ ]LandDmatricesofColumn

d

ldlal

alwaysl

dl

dlad

j

k

j

kkkjk

2

6667.05.1

5.0201

)!(15.1

25.02

2

22

11

121113132

32

22

211

221

11

1

22222

−=

−−−=

−=

==

−−=

−=

−=

∑

∑

=−

=

=−

=


( ) ( ) ( ) ( )[ ] [ ]LandDmatricesofColumndldl

dladj

kkkjk

3

3333.05.16667.0201

122

2223211

231

21

1

23333

=−−−=

−−=

−= ∑=−

=


Hence

[ ]

=

3333.00005.10002

D

and

[ ]

−−=

16667.00015.0001

L


The forward solution shown in Equation (31), becomes:

[ ][ ] [ ]bzL =

=

−−

001

1667.00015.0001

3

2

1

zzz

or,

∑−

=−=

1

1

i

kkikii zlbz (32, repeated)


Hence

( )( )

( )( ) ( )( )3333.0

5.06667.0100

5.015.00

1

23213133

12122

11

=−−−=

−−==

−−=−===

zLzLbz

zLbzbz


The diagonal scaling phase, shown in Equation (29), becomes

[ ][ ] [ ]zyD =

=

3333.05.0

1

3333.00005.10002

3

2

1

yyy


or

ii

ii d

zy =

Hence

13333.03333.0

3333.05.15.0

5.021

33

33

22

22

11

11

===

===

===

dzy

dzy

dzy


The backward solution phase can be found by referring to Equation (27)

[ ] [ ] [ ]yxL T =

=

−

−

1333.05.0

100667.01005.01

3

2

1

xxx

∑+=

−=N

ikkkiii xlyx

1(28, repeated)


Hence

( )

( )( ) ( )( )1

1015.05.0

116667.03333.0

1

1

33122111

2

33222

33

=−−−=

−−==

×−−=−=

==

xxlxlyx

x

xlyx

yx


Hence

[ ]

=

=

111

3

2

1

xxx

x


Re-ordering Algorithms For Minimizing Fill-in Terms [1,2].

During the factorization phase (of Cholesky, or Algorithms ), many “zero” terms in the original/given

matrix will become “non-zero” terms in the factored matrix

TLDL

][A][U . These new non-zero terms are often

called as “fill-in” terms (indicated by the symbol ) F It is, therefore, highly desirable to minimize these fill-in terms , so that both computational time/effort

and computer memory requirements can be substantially reduced.


For example, the following matrix and vector are given:

][A ][b

[ ]

=

11001020440030006604010088500345110720007112

A

=

14477094

129121

][b

(33)

(34)


The Cholesky factorization matrix , based on the original matrix (see Equation 33) and Equations (6-7), or Figure 1, can be symbolically computed as:

[ ]

××

×××

×××××××

=

000000000

00000

0000

FFF

FFF

U

][U][A

(35)


IPERM (new equation #) = {old equation #} such as, for this particular example:

=

123456

654321

IPERM

(36)

(37)


Using the above results (see Equation 37), one will be able to construct the following re-arranged matrices:

[ ]

=

11270002711054300588001040660003004402010011

*A

and

=

12112994704714

][ *b

(38)

(39)


Remarks: In the original matrix (shown in Equation 33), the nonzero term (old row 1, old column 2) = 7 will move to new location of the new matrix (new row 6, new column 5) = 7, etc.

The non zero term (old row 3, old column 3) = 88 will move to (new row 4, new column 4) = 88, etc.

The value of (old row 4) = 70 will be moved to (or located at) (new row 3) = 70, etc

b*b

A*A

A*A *A


Now, one would like to solve the following modified system of linear equations (SLE) for ],[ *x

][]][[ *** bxA =

rather than to solve the original SLE (see Equation1). The original unknown vector can be easily recovered from and ,shown in Equation (37).

}{x][ *x [ ]IPERM

(40)


[ ]

×××

××××××

×××

=

000000000

00000000000

000

*

FU

The factorized matrix can be “symbolically” computed from as (by referring to either Figure 1, or Equations 6-7):

][ *U][ *A

(41)


4. On-Line Chess-Like Game For Reordering/Factorized Phase [4].

Figure 2 A Chess-Like Game For Learning to Solve SLE.


(A)Teaching undergraduate/HS students the process how to use the reordering output IPERM(-), see Equations (36-37) for converting the original/given matrix , see Equation (33), into the new/modified matrix , see Equation (38). This step is reflected in Figure 2, when the “Game Player” decides to swap node (or equation) (say ) with another node (or equation ) , and click the “CONFIRM” icon!

][A][ *A

""i 2=i"" j


Since node is currently connected to nodes hence swapping node with the above nodes will “NOT” change the number/pattern of “Fill-in” terms. However, if node is swapped with node then the fill-in terms pattern may change (for better or worse)!

"2" =i;8,7,6,4=j 2=i

"" j2=i

,5,3,1 ororj =


(B) Helping undergraduate/HS students to understand the “symbolic” factorization” phase, by symbolically utilizing the Cholesky factorized Equations (6-7). This step is illustrated in Figure 2, for which the “game player” will see (and also hear the computer animated sound, and human voice), the non-zero terms (including fill-in terms) of the original matrix to move to the new locations in the new/modified matrix .

][A][ *A


(C) Helping undergraduate/HS students to understand the “numerical factorization” phase, by numerically utilizing the same Cholesky factorized Equations (6-7).

(D) Teaching undergraduate engineering/science students and even high-school (HS) students to “understand existing reordering concepts”, or even to “discover new reordering algorithms”


5. Further Explanation On The Developed Game

1. In the above Chess-Like Game, which is available on-line [4], powerful features of FLASH computer environments [3], such as animated sound, human voice, motions, graphical colors etc… have all been incorporated and programmed into the developed game-software for more appealing to game players/learners.


2. In the developed “Chess-Like Game”, fictitious monetary (or any kind of ‘scoring system”) is rewarded (and broadcasted by computer animated human voice) to game players, based on how he/she swaps the node (or equation) numbers, and consequently based on how many fill-in terms occurred. In general, less fill-in terms introduced will result in more rewards! ""F


3. Based on the original/given matrix , and existing re-ordering algorithms (such as the Reverse Cuthill- Mckee, or RCM algorithms [1-2]) the number of fill-in terms can be computed (using RCM algorithms). This internally generated information will be used to judge how good the players/learners are, and/or broadcast “congratulations message” to a particular player who discovers new “chess-like move” (or, swapping node) strategies which are even better than RCM algorithms!

][A

)"("F


4. Initially, the player(s) will select the matrix size ( , or larger is recommended), and the percentage (50%, or larger is suggested) of zero-terms (or sparsity of the matrix). Then, “START Game” icon will be clicked by the player.

88×


5. The player will then CLICK one of the selected node (or equation) numbers appearing on the computer screen. The player will see those nodes which are connected to node (based on the given/generated matrix ). The player then has to decide to swap node with one of the possible node

""i"" j

""i][A

""i "" j


After confirming the player’s decision, the outcomes/ results will be announced by the computer animated human voice, and the monetary-award will (or will NOT) be given to the players/learners, accordingly. In this software, a maximum of $1,000,000 can be earned by the player, and the “exact dollar amount” will be INVERSELY proportional to the number of fill-in terms occurred (as a consequence of the player’s decision on how to swap node with another node ).

""i"" j


6. The next player will continue to play, with his/her move (meaning to swap the node with the node) based on the current best non-zero terms pattern of the matrix.

thi thj

THE END

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�Numerical Methods�� Part: Cholesky and Decomposition��http://numericalmethods.eng.usf.eduSlide Number 2You are freeUnder the following conditionsChapter 04.09: Cholesky and� Decomposition�Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29Slide Number 30Slide Number 31Slide Number 32Slide Number 33Slide Number 34Slide Number 35Slide Number 36Slide Number 37Slide Number 38Slide Number 39Slide Number 40Slide Number 41Slide Number 42Slide Number 43Slide Number 44Slide Number 45Slide Number 46Slide Number 47Slide Number 48Slide Number 49Slide Number 50Slide Number 51Slide Number 52Slide Number 53Slide Number 54Slide Number 55Slide Number 56Slide Number 57Slide Number 58Slide Number 59Slide Number 60Slide Number 61Slide Number 62Slide Number 63Slide Number 64Slide Number 65Slide Number 66Slide Number 67Slide Number 68The EndAcknowledgementSlide Number 71The End - Really

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