NEW APPROACH TO CALCULATION OF RANGE OF

POLYNOMIALS USING BERNSTEIN FORMS

2

SCOPE OF THE PRESENTATION INTRODUCTION BERNSTEIN FORMS :

1) DEFINITION2) PROPERTIES3) BASIS CONVERSION

RANGE CALCULATIONS1) IMPORTANT THEOREMS

2) SUBDIVISION NEW PROPOSITIONS EXAMPLES AND CONCLUSION FUTURE WORK

3

INTRODUCTION

POLYNOMIALS ARE USEFUL MATHEMATICAL TOOLS

IMPLICIT POWER FORM – LINEAR COMBINATION OF POWER BASIS

REPRESENTATION IN BERNSTEIN FORM – LINEAR COMBINATION OF BERNSTEIN BASIS

4

ADVANTAGES OF BERNSTEIN FORM AVOIDS FUNCTION EVALUATION –

COSTLY IMPROVEMENT OVER TAYLOR FORM

INFORMATION ABOUT SHARPNESS OF BOUNDS- TOLERANCE CAN BE

SPECIFIED FASTER RATE OF CONVERGENCE CAPABLE OF GIVING EXACT RANGE

5

BERNSTEIN FORM………….

DEFINTION – POLYNOMIAL OF DEGREE n

B nk ( x ) = ( n

k ) x k ( 1 - x ) n-k

k = 0 , 1 , ……….. , n

V = x ∈ [ 0 , 1 ]

6

BERNSTEIN FORM………. ANY POLYNOMIAL IN POWER FORM IS REPRESENTED AS

BERNSTEIN FORM OF REPRESENTATION IS

NnRaxaxp i ni

n

ii , , )(

0

[0,1] )( )(0

xxBbxp ni

n

i

ni

ininij

n

j

ni x)(x

i

n(x)B a

j

n

j

i

b

1,0

7

BERNSTEIN FORM…………

MULTIVARIATE CASE – POLYNOMIAL OF DEGREE n

p ni ( x ) = ∑ i ∈ K ∏ q

j = 1 ( nij

) x i j j ( 1 - x j ) n –

i j

i = ( i1 ……….. i q ) ∈ K

= ∑ i∈ K b ni p n

i ( x )

8

IMPORTANT PROPERTIES OF BERNSTEIN POLYNOMIALS INVARIANT UNDER AFFINE

TRANSFORMATIONS RECURSIVE GENERATION OF nth

ORDER BASIS FROM (n – 1)th ORDER BASIS IS POSSIBLE

ALL TERMS OF BERNSTEIN BASIS ARE POSITIVE AND THEIR SUM EQUALS 1

9

IMPORTANT PROPERTIES OF BERNSTEIN POLYNOMIAL

BETTER CONDITIONED AND BETTER NUMERICAL STABILITY

DEGREE ELEVATIONb n+1

k = k / n+1 * b nk-1 + ( 1 - k /

n+1 ) b nk

10

BASIS CONVERSION

p (x) = ∑ n k = 0 a k x k

= ∑ n k = 0 b n k B n

k ( x )

CONVERSION OF ONE SET OF COEFFICIENTS TO OTHER :-

a k = ∑ k j = 0 ( -1 ) k – j ( n k ) ( k j ) b

nj

b n k = ∑ k j = 0 ( k j ) / ( n

j ) a j

11

BASIS CONVERSION………MATRIX METHOD

•WE CAN WRITE P(X) = XA = BXB ,WHERE X IS THE VARIABLE(ROW) MATRIX

• A IS THE COEFFICIENT (COLUMN) MATRIX

• BX IS THE BERNSTEIN BASIS(ROW) MATRIX AND

• B IS THE BERNSTEIN COEFFICIENT(COLUMN) MATRIX.

• AFTER CERTAIN COMPUTATIONS,

BXB = XUXB

WHERE UX IS A LOWER TRIANGULAR MATRIX.

12

• BASIS CONVERSION……

FOR A GENERAL INTERVAL

BX = XWXVXUX

WHERE WX IS AN UPPER TRIANGULAR MATRIX AND VX IS A DIAGONAL MATRIX.XA = BXBTHUS, AND

BUVWXXA XXX

AWVUB XXX111

13

BASIS CONVERSION…….. FOR A BIVARIATE CASE, BY ANALOGY WITH UNIVARIATE CASE

TXXTXX BBBBBBXXp

I221),( 21

TT AXXAXXXXp 122121 ),(

1111 1 XXXX UVWXB

2222 2 XXXX UVWXB

14

BASIS CONVERSION……. USING PROPERTIES OF MATRICES

THE SAME LOGIC CAN BE EXTENDED TO AN L-VARIATE CASE

WE EXPLAIN WITH AN ILLUSTRATION FOR TRIVARIATE CASE

TT1X

1X

1X

1X

1X

1X AWVUWVUB

111222

TTAMM 12

15

BASIS CONVERSION……. FOR A TRIVARIATE CASE

HERE TRANSPOSE MEANS CONVERTING SECOND CO-ORDINATE DIRECTION TO FIRST, THIRD TO SECOND AND FIRST TO THIRD

SAME ANALOGY EXTENDS TO L-VARIATE CASE

TTTAMMMB321123

16

17

PROPOSED (MATRIX) METHOD

A NEW METHOD IS PROPOSED WHERE THE POLYNOMIAL COEFFICIENTS ARE INPUTTED IN A MATRIX FORM ‘A’ AND THE RESULTING BERNSTEIN COEFFICIENTS ARE COMPUTED IN MATRIX FORM ‘B’

FOR A 3-D CASE, INSTEAD OF CONSIDERING THE POLYNOMIAL COEFFICIENT MATRIX ‘A’ AS A 3-D ARRAY, IT CAN BE CONSIDERED AS A MATRIX WITH 0 TO n1 ROWS AND 0 TO (n2+1)(n3+1) –1 COLUMNS

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MATRIX METHOD…… ORIGINAL ARRAY

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THE 3-D ARRAY IN MATRIX FORM

AFTER FIRST TRANSPOSE AND RESHAPE

O TO n2 ROWS; 0 TO COLUMNS

32131

3

323

2111

2

2111

2

.............0

10

0................00

......................

1......1101

:

:

:

:.......111101

10......011001

0......1000

:

:

200

:.......110100

00......010000

nnnnn

n

nnn

nnnn

n

nnnn

n

A

32121

1

1

311

3112

3

3222

3

3

...............0

20

10

0...................00

......................

1......1101

:

:

120

:.......111110

10......101100

0......1000

:

:

020

01......011010

00......001000

nnnnn

n

n

nnn

nnnn

n

nnnn

n

n

A

111 13 nn

20

SIMILARLY AFTER SECOND AND THIRD TRANSPOSE AND RESHAPE, WE GET THE ORIGINAL MATRIX

32132

2

212

313

1

3133

1

1

...............0

:

:

:

10

0...................00

......................

1...........01

:

:

:012

:..................011

10................010

0......1000

::

::

:102002

01........101001

00........100000

nnnnn

n

nnn

nnn

n

nnnn

n

n

A

32131

3

323

2111

2

2111

2

.............0

10

0................00

......................

1......1101

:

:

:

:.......111101

10......011001

0......1000

:

:

200

:.......110100

00......010000

nnnnn

n

nnn

nnnn

n

nnnn

n

A

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RANGE CALCULATION IMPORTANT THEOREMS

THEOREM 1 : THE MINIMUM AND MAXIMUM BERNSTEIN COEFFICIENT GIVE AN ENCLOSURE OF THE RANGE OF POLYNOMIAL IN THE GIVEN INTERVAL.

THEOREM 2 : VERTEX CONDITION : IF THE MINIMUM AND MAXIMUM BERNSTEIN COEFFICIENT OF POLYNOMIAL IN BERNSTEIN FORM OCCUR AT THE VERTICES OF THE BERNSTEIN COEFFICIENT ARRAY, THEN THE ENCLOSURE IS THE EXACT RANGE.

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IMPORTANT THEOREMS

THEOREM 3 : BERNSTEIN APPROXIMATIONS CONVERGE TO THE RANGE AND THE CONVERGENCE IS AT LEAST LINEAR IN THE ORDER OF APPROXIMATIONS.

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SUBDIVISION

VERTEX CONDITION NOT SATISFIED – SUBDIVIDE

UNIT BOX I INTO 2q SUBBOXES OF EDGE LENGTH ½

CALCULATE BERNSTEIN COEFFICIENT OF p(x) ON THESE SUBBOXES

CHECK VERTEX CONDITION ON EACH SUBBOX

FIND RANGE OF POLYNOMIAL

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SWEEP PROCEDURE

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DISADVANTAGES OF THE EXISTING ALGORITHM

FORTRAN 95 CAN NOT CATER TO MORE THAN SIX DIMENSIONAL ARRAYS

BERNSTEIN COEFFICIENT GENERATED ARE STORED IN MULTIDIMENSIONAL ARRAYS

FOR SHARP ENCLOSURES, SUBDIVISION CREATES LARGE DATA

SLOWS DOWN COMPUTATIONS

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PROPOSITIONS

PROPOSE NEW METHODS FOR

ACCELERATION OF ALGORITHM

FASTER TERMINATION

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PROPOSITION 1 : MATRIX METHOD

STORE BERNSTEIN COEFFICIENT IN SINGLE VECTOR, 2 DIMENSIONAL

NUMBER OF ELEMENTS OF VECTOR DEPEND ON NUMBER OF VARIABLES MAXIMUM POWER OF EACH VARIABLE

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MATRIX METHOD

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EXAMPLE : 3-D POLYNOMIAL

P(x)=2+4x1+5x12-x2+2x1x2+x1x3-

x2x3+6x12x2x3+2x3

2-x1x32+x1

2x2x32

n1=2 ; n2=1 ; n3=2 COEFFICIENT MATRIX

000 010 001 011 002 012

100 110 101 111 102 112

200 210 201 211 202 212

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MATRIX METHOD……contd

BERNSTEIN COEFFICIENT ALSO STORED IN 3 6 MATRIX, NOT 3-D ARRAY

ALL OPERATIONS AND SUBDIVISION CARRIED OUT ON 2-D MATRICES

FASTER COMPUTATIONS NO RESTRICTION ON THE DIMENSION

OF POLYNOMIAL

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PROPOSITION 2 : CUT OFF TEST AVOIDS UNNECCESARY SUBDIVISIONS FASTER TERMINATION OF ALGORITHM CHECK IF RANGE IN EACH NEW PATCH

INCLUDED IN ACTUAL RANGE STORED REJECT PATCH IF YES

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PROPOSITION 3 : MONOTONICITY TEST

IF THE POLYNOMIAL IS MONOTONIC W.R.T ANY DIRECTION ON A BOX, AND IF THE BOX HAS NO EDGE IN COMMON WITH THE INITIAL INTERVAL, THEN THIS BOX CAN BE REJECTED

AVOIDS UNNECCESARY SUBDIVISIONS FASTER TERMINATION OF ALGORITHM

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PROPOSITION 4 : NEW CONDITION

IF THE MIN BERNSTEIN COEFF OF A PATCH IS MINIMUM OF ALL THE COEFFS IN UNTESTED PATCHES AND APPEARS AT A VERTEX, THEN CHECK THE FOLLOWING

IF MAX BERNSTEIN COEFF IN THAT PATCH IS LESS THAN SUPREMUM OF SOLUTIONS EVALUATED SO FAR, THEN THE PATCH IS A SOLUTION

THIS PATCH NEED NOT BE SUBDIVIDED FURTHER

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PROPOSED ALGORITHM1. Read the initial intervals & the

maximum degree for each variable in the polynomial

2. Read the Bernstein coefficients in matrix form.

3. Initialise list 'l' which contains all the patches to be tested; and list 'lsol' which consists the number of solutions i.e. the patches where vertex condition is satisfied. Solution patch contains min B (D) and max B(D).

4.Take the first patch from list 'l '

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PROPOSED ALGORITHM5.Check the vertex condition. If 'true' then lsol = lsol +1 p_bar(lsol)= interval (minB(D), max

B(D)) delete the patch from 'l' else subdivide the patch in 1st

direction into two patches, each of which is a matrix and add the new entries at the end of the list 'l'. Delete the tested patch.

6. If 'l´ is empty go to step 12 else pick the first patch from 'l' and go to step 7.

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PROPOSED ALGORITHM

7. Check the vertex condition. If 'true' then update 'lsol' and

'p_bar(lsol)' and delete the patch from 'l’ go to step 5

else go to step 8.8. Check the new condition If 'true' then update 'lsol' and

'p_bar(lsol)' and delete the patch from 'l‘ & go to step 5

else go to step 9.

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PROPOSED ALGORITHM

9. Monotonicity test: Check for common edge with the original box. If no edge in common, test for monotonicity in all directions and delete the monotonic patch and go to step 5.

If common edge, test monotonicity in that direction-

If monotonic, retain patch; else go to step 1010. Reshape and then subdivide the patch

in next cyclic direction into two patches, each of which is a matrix.

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PROPOSED ALGORITHM

11. Add the new entries at the end of the list 'l'. Delete the tested patch.

12. Carry out the cut off test and go to step 5.

13. Compute the exact range p(X) p(X)=interval(minval(inf(p_bar(1:lsol))),

maxval(sup(p_bar(1:lsol)))14. Output p(X)End

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EXAMPLES AND RESULTS EXAMPLE 1 : 3-D POLYNOMIAL

Initial box [0,1] , [0,10] , [-1,1]USING THE MATRIX METHOD Total no. of solutions = 27 Total no. of subdivisions = 246 Range of function =

[1.8567633650742547,2672.0] cpu time = 0.18578312 sec

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EXAMPLE 1……contd

RANGE OF THE POLYNOMIAL , USING VECTORIZED MOORE SKELBOE ALGORITHM

RANGE OF FUNCTION = [1.8567633669124248,2672]

CPU TIME = 258.01999 sec

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EXAMPLE 2 : 4-D POLYNOMIAL

Initial box [-1,1], [0,1], [0,1], [-2,0]USING MATRIX METHOD Total no. of solutions = 1 Total no. of subdivisions = 0 Range of function = [-1.6666667,4.3333333999999999] cpu time = 9.51403E-4sec

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EXAMPLE 2 ……..contd

RANGE OF THE POLYNOMIAL , USING VECTORIZED MOORE SKELBOE ALGORITHM

RANGE OF FUNCTION = [-1.6666666670000002,

4.3333333329999996]

CPU TIME = 4.2621371 sec

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EXAMPLE 3 : 5 -D POLYNOMIAL

Initial box [-1,1], [0,1], [-1,1], [0,1], [-1,1]USING MATRIX METHOD Total no. of solutions = 12 Total no. of subdivisions = 27 Range of function = [-14.0,23.0] cpu time = 0.03255 sec

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EXAMPLE 3 ……..contd

RANGE OF THE POLYNOMIAL , USING VECTORIZED MOORE SKELBOE ALGORITHM

RANGE OF FUNCTION = [-14.0,23.0]CPU TIME = 44.351025 sec

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CONCLUSIONS

THE PROPOSED ALGORITHM CAN THEORETICALLY SOLVE PROBLEMS OF RANGE FINDING FOR ANY DIMENSION POLYNOMIAL.

THE PROPOSED ‘MATRIX METHOD’ ALONG WITH THE CUT- OFF TEST, MONOTONICITY TEST AND THE NEW CONDITION, CONSIDERABLY SPEEDS UP THE ALGORITHM

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CONCLUSIONS

THE RANGE COMPUTED BY THE PROPOSED METHOD CAN BE MADE AS ACCURATE AS DESIRED, BY SPECIFYING THE TOLERANCE

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FUTURE WORK DEVELOP CODE TO EXTEND TO 7-D

AND HIGHER INTEGRATE THE CODE WITH ‘COSY’

PACKAGE INTRODUCE MORE EFFICIENT

SUBDIVISION STRATEGY TO FURTHER SPEED UP THE ALGORITHM

APPLY THE METHOD TO CONTROL PROBLEMS

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THANK YOU !