A NEW APPROACH TO COMPUTING RANGES OF POLYNOMIALS USING BERNSTEIN FORM

1
A NEW APPROACH TO COMPUTING RANGES OF
POLYNOMIALS USING BERNSTEIN FORM

• Lt Vikas Chawla

2
SCOPE OF THE PRESENTATION

• INTRODUCTION
• BERNSTEIN FORMS
• 1) DEFINITION
• 2) PROPERTIES
• 3) BASIS CONVERSION
• RANGE CALCULATIONS
• 1) IMPORTANT THEOREMS
• 2) SUBDIVISION
• MOTIVATION- APPLICATION TO CONTROL PROBLEMS
• NEW PROPOSITIONS
• RESULTS AND DISCUSSION
• CONCLUSIONS

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
• FAST RATE OF CONVERGENCE
• CAPABLE OF GIVING EXACT RANGE

5
BERNSTEIN FORM

• DEFINITION POLYNOMIAL OF DEGREE n 1

6
BERNSTEIN FORM(contd.)

• ANY POLYNOMIAL IN POWER FORM IS REPRESENTED AS
• BERNSTEIN FORM OF REPRESENTATION IS
• ARE THE BERNSTEIN COEFFICIENTS
  CORRESPONDING TO THE DEGREE n BASIS
• 
  
• 
  

7
BERNSTEIN FORM(contd.)

• MULTIVARIATE CASE BERNSTEIN POLYNOMIAL OF
  DEGREE k

8
IMPORTANT PROPERTIES OF BERNSTEIN POLYNOMIALS 1

• 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
  (FAROUKI et. al2)
• DEGREE ELEVATION- A POLYNOMIAL OF DEGREE n CAN BE
  REPRESENTED IN TERMS OF BERNSTEIN BASIS OF DEGREE
  n1

10
BASIS CONVERSION 1

• CONVERSION OF ONE SET OF COEFFICIENTS TO OTHER -

11

• BASIS CONVERSION(contd.)
• MATRIX METHOD (BERCHTOLD et al.1)
• 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(contd.)FOR A GENERAL
INTERVAL BX XWXVXUX
WHERE WX IS AN UPPER TRIANGULAR MATRIX AND VX
IS A DIAGONAL MATRIX. XA
BXBTHUS,
AND
13
BASIS CONVERSION(contd.)BIVARIATE CASE

• BY ANALOGY WITH UNIVARIATE CASE
• THE BERNSTEIN COEFFICIENTS

14
COMPUTATION OF RANGES IMPORTANT THEOREMS

• THEOREM 1 THE MINIMUM AND MAXIMUM BERNSTEIN
  COEFFICIENT GIVE AN ENCLOSURE OF THE RANGE OF
  POLYNOMIAL IN THE GIVEN INTERVAL.(CARGO
  et.al.3)
• 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.

15
SUBDIVISION

• VERTEX CONDITION NOT SATISFIED SUBDIVIDE
  (GARLOFF 4)
• UNIT BOX I INTO 2q SUBBOXES OF EDGE LENGTH ½
• CALCULATE BERNSTEIN COEFFICIENT OF p(x) ON THESE
  SUBBOXES
• BERNSTEIN COEFFICIENTS AT THE FIRST SUBDIVISION
  LEVEL COMPUTED FROM COEFFICIENTS OF p ON I

16
SUBDIVISION(contd.)

• SWEEP IN A PARTICULAR COORDINATE DIRECTION GIVES
  THE BERNSTEIN COEFFICIENTS ON THE NEIGHBORING
  SUBBOX AS INTERMEDIATE VALUES
• COMPUTATIONAL EFFORT REDUCES CONSIDERABLY
• COMBINE THE TOOLS OF SUBDIVISION AND VERTEX
  CONDITION TO IMPROVE THE BOUNDS TILL THEY ARE
  EXACT

17
SWEEP PROCEDURE
18
APPLICATION OF BERNSTEIN EXPANSION TO CONTROL
PROBLEMS

• BERNSTEIN EXPANSION CAN BE APPLIED TO STRICT
  INEQUALITIES /EQUATIONS WITH MULTIVARIATE
  POLYNOMIALS
• BETTER THAN INTERVAL METHODS IN REGARD TO
  COMPUTING TIME AND TIGHTNESS OF BOUNDS
• CAN BE APPLIED TO ARBITRARY FUNCTIONS

19
APPLICATION TO CONTROLS(contd.)IMPORTANT
APPLICATIONS(GARLOFF5)

• ROBUSTNESS ANALYSIS OF POLYNOMIALS WITH
  POLYNOMIAL PARAMETER DEPENDENCY (HURWITZ
  STABILITY)
• COMPUTATION OF STABILITY RADII FOR D-STABLE
  SYSTEMS
• SOLVING SYSTEMS OF STRICT POLYNOMIAL
  INEQUALITIES IN ROBUST FEEDBACK DESIGN

20
APPLICATION TO CONTROLS(contd.) CHECKING ROBUST
HURWITZ STABILITY

• CONSIDER FAMILY OF POLYNOMIALS-
• COEFFICIENTS DEPEND POLYNOMIALLY ON PARAMETERS
• AN m?m HURWITZ MATRIX IS ASSOCIATED WITH
  POLYNOMIAL FAMILY

21
APPLICATION TO CONTROLS(contd.) CHECKING ROBUST
HURWITZ STABILITY

• DETERMINANT OF HURWITZ MATRIX IS CALLED HURWITZ
  DETERMINANT
• FOR ROBUST STABILITY OF POLYNOMIAL FAMILY ALL
  PRINCIPAL MINORS OF D 0

22
APPLICATION TO CONTROLS(contd.)

• CHECK FOR AT LEAST ONE STABLE POLYNOMIAL (PLANT)
  IN THE FAMILY
• TO TEST HURWITZ DETERMINANT FOR POSITIVITY,
  CARRY OUT THE BERNSTEIN EXPANSION (l -VARIATE
  POLYNOMIAL IN q)
• IF MINIMUM OF BERNSTEIN COEFFICIENTS IS
  POSITIVE, POLYNOMIAL FAMILY IS STABLE
• RESTRICTED TO MODERATE NO. OF PARAMETERS AND LOW
  DEGREE POLYNOMIALS

23
APPLICATION TO CONTROLS(contd.)INSPECTION OF
THE VALUE SET

• FOR LARGE ROBUST STABILITY PROBLEMS, WE EXPLORE
  THE VALUE SET OF THE POLYNOMIAL FAMILY
  (TEMPLATES)
• SPLIT THE POLYNOMIAL FAMILY INTO EVEN AND ODD
  PARTS
• CHECK FOR AT LEAST ONE STABLE POLYNOMIAL IN THE
  FAMILY

24
APPLICATION TO CONTROLS(contd.)INSPECTION OF
THE VALUE SET

• EXPAND BOTH EVEN AND ODD POLYNOMIALS
  SIMULTANEOUSLY INTO THEIR BERNSTEIN FORMS, AT
  EACH FREQUENCY
• THE FAMILY IS ROBUSTLY STABLE IF EVEN AND ODD
  POLYNOMIALS DO NOT HAVE A REAL ZERO IN COMMON AT
  ANY FREQUENCY 0,8), i.e. RANGES DO NOT CONTAIN
  ORIGIN
• SEARCH REGION FOR ? CAN BE REDUCED USING
  SUBDIVISION

25
APPLICATION TO CONTROLS(contd.)COMPUTATION OF
STABILITY RADII

• TO FIND THE SMALLEST DESTABILISING PERTURBATION
  OF A D-STABLE SYSTEM
• FOR A D-STABLE POLYNOMIAL p, FIND THE LARGEST ?
  (STABILITY RADIUS) SUCH THAT THE FAMILY IS
  D-STABLE FOR ALL q WITH
• STABILITY RADIUS IS COMPUTED BY BISECTION SEARCH
  OVER ? WITH STABILITY CHECK AT EACH STEP

26
APPLICATION TO CONTROLS(contd.)SOLVING STRICT
POLYNOMIAL INEQUALITIES

• THE SOLUTION OF MANY CONTROL SYSTEM DESIGN AND
  ANALYSIS PROBLEMS CAN BE RECAST AS SYSTEM OF
  INEQUALITIES
• NOMINAL CONTROLLER USUALLY DESIGNED FOR CLOSED
  LOOP STABILITY, DISTURBANCE REJECTION, TIME
  RESPONSE OVERSHOOT, REFERENCE INPUT TRACKING etc.
• THESE PERFORMANCE SPECS ARE FORMULATED AS
  POLYNOMIAL INEQUALITIES IN FREQUENCY DOMAIN

27
APPLICATION TO CONTROLS(contd.)SOLVING STRICT
POLYNOMIAL INEQUALITIES

• THE INEQUALITIES ARE IN TERMS OF CONTROLLER
  PARAMETERS, WHICH BELONG TO SOME INITIAL
  INTERVALS
• LET THE POLYNOMIAL INEQUALITIES IN TERMS OF
  CONTROLLER PARAMETERS A,B AND D BE-

28
APPLICATION TO CONTROLS(contd.)SOLVING STRICT
POLYNOMIAL INEQUALITIES

• THE CONTROLLER DESIGN PROBLEM REDUCES TO SHOWING
  THAT THERE IS A SOLUTION TO THIS SYSTEM OF
  INEQUALITIES
• USING BERNSTEIN EXPANSION OF THE POLYNOMIALS,
  INSPECT WHETHER MINIMUM BERNSTEIN COEFFICIENT OF
  EACH POLYNOMIAL IS GREATER THAN ZERO
• ALGORITHM TERMINATES IF ANY INEQUALITY NOT
  SATISFIED

29
APPLICATION TO CONTROLS(contd.)SOLVING STRICT
POLYNOMIAL INEQUALITIES

• BISECT THE INITIAL BOX AND PROCEED AS BEFORE
• THE PROCEDURE CONTINUES UNTILL ALL THE
  POLYNOMIAL INEQUALITIES ARE SATISFIED FOR A
  SUBDIVIDED BOX
• THE FINAL BOX GIVES THE ACCEPTABLE CONTROLLER
  PARAMETERS

30
NEW PROPOSITIONS

• COMPUTATION OF BERNSTEIN COEFFICIENTS
• COMPUTATION OF RANGE OF POLYNOMIALS

31
NEW PROPOSITIONS(contd.)1. BERNSTEIN
COEFFICIENTS

• FOR A BIVARIATE CASE, BY ANALOGY WITH UNIVARIATE
  CASE

32
NEW PROPOSITIONS(contd.)BERNSTEIN COEFFICIENTS

• USING PROPERTIES OF MATRICES
• THE SAME LOGIC CAN BE EXTENDED TO AN L-VARIATE
  CASE
• WE EXPLAIN WITH AN ILLUSTRATION FOR TRIVARIATE
  CASE

33
NEW PROPOSITIONS(contd.)BERNSTEIN COEFFICIENTS

• 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

34
ROTATION OF AXES IN 3-D ARRAY
35
ROTATION OF AXES IN 3-D ARRAY
36
NEW PROPOSITIONS(contd.)BERNSTEIN COEFFICIENTS

• A NEW METHOD IS PROPOSED WHERE THE POLYNOMIAL
  COEFFICIENTS ARE INPUTTED IN A 2-DIMENSIONAL
  MATRIX FORM A AND THE RESULTING BERNSTEIN
  COEFFICIENTS ARE COMPUTED IN 2-D 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 2-D MATRIX WITH 0 TO n1
  ROWS AND 0 TO (n21)(n31) 1 COLUMNS

37
NEW PROPOSITIONS(contd.)

• ORIGINAL ARRAY

38
NEW PROPOSITIONS(contd.)BERNSTEIN
COEFFICIENTS
39

• THE 3-D ARRAY IN MATRIX FORM
• AFTER FIRST TRANSPOSE AND RESHAPE
• O TO n2 ROWS 0 TO COLUMNS

40

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

41
NEW PROPOSITIONS(contd.)2.COMPUTATION OF RANGE
OF POLYNOMIALS

• FORTRAN 95 CAN NOT CATER TO MORE THAN SEVEN
  DIMENSIONAL ARRAYS
• BERNSTEIN COEFFICIENT GENERATED ARE STORED IN
  MULTIDIMENSIONAL ARRAYS
• FOR SHARP ENCLOSURES, SUBDIVISION CREATES LARGE
  DATA
• SLOWS DOWN COMPUTATIONS

42
NEW PROPOSITIONS(contd.)COMPUTATION OF RANGE OF
POLYNOMIALS

• PROPOSE NEW METHODS FOR
• ACCELERATION OF ALGORITHM
• FASTER TERMINATION

43
PROPOSITION 1 2-DIMENSIONAL 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

44
EXAMPLE 3-D POLYNOMIAL

• P(x)24x15x12-x22x1x2x1x3-
  x2x36x12x2x32x32-x1x32x12x2x32
• n12 n21 n32
• POWER COEFFICIENTS MATRIX

45
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

46
PROPOSITION 2 CUT OFF TEST

• AT ANY SUBDIVISION LEVEL, CHECK IF RANGE IN EACH
  NEW PATCH IS ALREADY INCLUDED IN ACTUAL RANGE
  STORED (PATCHES FOR WHICH VERTEX CONDITION IS
  SATISFIED)
• REJECT PATCH IF YES
• AVOIDS UNNECESSARY SUBDIVISIONS
• FASTER TERMINATION OF ALGORITHM

47
PROPOSITION 3 MONOTONICITY TEST

• IF THE POLYNOMIAL IS MONOTONIC W.R.T ANY
  DIRECTION ON A BOX, THEN THE INTERIOR OF THE BOX
  CANNOT CONTAIN A GLOBAL MINIMA/ MAXIMA
• IF THIS BOX HAS NO EDGE IN COMMON WITH THE
  INITIAL INTERVAL, THEN THIS BOX CAN BE REJECTED
• AVOIDS UNNECESSARY SUBDIVISIONS
• FASTER TERMINATION OF ALGORITHM

48
PROPOSITION 4 SIMPLIFIED VERTEX 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

49
PROPOSITION 4 ..(contd.) SIMPLIFIED
VERTEX CONDITION

• SIMILARLY, IF THE MAX BERNSTEIN COEFF OF A PATCH
  IS MAXIMUM OF ALL THE COEFFS IN UNTESTED PATCHES
  AND APPEARS AT A VERTEX, THEN CHECK THE FOLLOWING
• IF MIN BERNSTEIN COEFF IN THAT PATCH IS GREATER
  THAN INFIMUM OF SOLUTIONS EVALUATED SO FAR, THEN
  THE PATCH IS A SOLUTION
• THIS PATCH NEED NOT BE SUBDIVIDED FURTHER

50
PROPOSED ALGORITHM

• 1. 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 (working list) and list
  'lsol' which consists the number of solutions
  i.e. the patches where vertex condition is
  satisfied (solution list). Solution patch
  contains min B (D) and max B(D).
• 4. Take the first patch from working list

51
PROPOSED ALGORITHM

• 5.Check the vertex condition.
• If 'true' then
• update solution list by 1
• b_range 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 working
  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.

52
PROPOSED ALGORITHM

• 7. Check the vertex condition.
• If 'true' then update solution list'
  and b_range and delete the patch from
  working list go to step 6
• else go to step 8.
• 8. Check the simplified vertex condition
• If 'true' then update solution list'
  and b_range and delete the patch from working
  list go to step 6
• else go to step 9.

53
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 6.
• If common edge, test monotonicity in
  that direction-
• If monotonic, retain patch
• else go to step 10
• 10. Reshape and then subdivide the patch in next
  cyclic direction into two patches, each of which
  is a matrix.

54
PROPOSED ALGORITHM

• 11. Add the new entries at the end of the working
  list. Delete the tested patch.
• 12. Carry out the cut off test and go to step 6.
• 13. Compute the exact range p(X)
• p(X)interval(minval(inf(b_range (1lsol))),
• maxval(sup(b_range(1l
  sol)))
• 14. Output p(X) (Range of the polynomial)
• End

55
RESULTS AND DISCUSSION

• TEST PROBLEM 1 3-D POLYNOMIAL
• Initial box 0,1 , 0,10 , -1,1
• USING THE PROPOSED METHOD
• Total no. of subdivisions 246
• Range of function 1.8567633650742547,2672.0
• cpu time 0.0348138 sec

56
TEST PROBLEM 1(contd.)

• RANGE OF THE POLYNOMIAL , USING VECTORISED MOORE
  SKELBOE ALGORITHM
• range of function 1.8567633669124248,2672
• cpu time 11.0781678 sec

57
TEST PROBLEM 2 4-D POLYNOMIAL

• Initial box -1,1, 0,1, 0,1, -2,0
• USING PROPOSED METHOD
• Total no. of solutions 1
• Total no. of subdivisions 0
• Range of function
• -1.6666667,4.3333333999999999
  
• cpu time 6.288E-4 sec

58
TEST PROBLEM 2 (contd.)

• RANGE OF THE POLYNOMIAL , USING VECTORIZED MOORE
  SKELBOE ALGORITHM
• range of function
• -1.6666666670000002, 4.3333333330000006
• cpu time 0.009336 sec

59
TEST PROBLEM 3 4-D POLYNOMIAL

• Initial box -47,39,0,98,-15,75,-50,50
• USING PROPOSED METHOD
• Total no. of solutions 8
• Total no. of subdivisions 23
• Range of function
• -264375.66666666698,464374.33333333303
• cpu time 3.5858E-3 sec

60
TEST PROBLEM 3 (contd.)

• RANGE OF THE POLYNOMIAL , USING VECTORIZED MOORE
  SKELBOE ALGORITHM
• range of function
• -264375.66666666704,464374.33333333303
• cpu time 0.0101686 sec

61
TEST PROBLEM 4 5 -D POLYNOMIAL

• Initial box -1,1, 0,1, -1,1, 0,1,
  -1,1
• USING PROPOSED METHOD
• Total no. of solutions 2
• Total no. of subdivisions 0
• Range of function -14.0,23.0
• cpu time 6.634E-4 sec

62
TEST PROBLEM 4 (contd.)

• RANGE OF THE POLYNOMIAL , USING VECTORIZED MOORE
  SKELBOE ALGORITHM
• Range of function -14.0,23.0
• cpu time 0.0738324 sec

63
TEST PROBLEM 5 5 -D POLYNOMIAL

• Initial box -7,14,-1,20,-14,14,0,5,-20,1
  5
• USING PROPOSED METHOD
• Total no. of solutions 21
• Total no. of subdivisions 42
• Range of function
• -39405523.0,19874467.0
• cpu time 0.0116186 sec

64
TEST PROBLEM 5 (contd.)

• RANGE OF THE POLYNOMIAL , USING VECTORIZED MOORE
  SKELBOE ALGORITHM
• Range of function -39405523.0,19874467.0
• cpu time 0.1342708 sec

65
TEST PROBLEM 6 5 -D POLYNOMIAL

• Initial box -0.9,-0.3,-0.8,-0.3,-0.8,-0.3,
• -0.8,-0.3,-0.8,2
• USING PROPOSED METHOD
• Total no. of solutions 14
• Total no. of subdivisions 35
• Range of function
• -3.3380000000000001,1.7470000032186515
• cpu time 0.0238878 sec

66
TEST PROBLEM 6(contd.)

• RANGE OF THE POLYNOMIAL , USING VECTORIZED MOORE
  SKELBOE ALGORITHM
• Range of function
• -3.3380000000000019,1.7470000000000004
• cpu time 0.0143822 sec

67
TEST PROBLEM 7 6 -D POLYNOMIAL

• Initial box -3.5,0.3 -3.5,0.4 -1.9,0
  
• -7,0.1 -0.1,5 -0.1,0.8
• USING PROPOSED METHOD
• Total no. of solutions 1
• Total no. of subdivisions 0
• Range of function
• -636.94994999999995,594.25
• cpu time 7.016E-4 sec

68
TEST PROBLEM 7.(contd.)

• RANGE OF THE POLYNOMIAL , USING VECTORIZED MOORE
  SKELBOE ALGORITHM
• range of function
• -636.95000000000062, 594.25000000000012
• cpu time 0.0164482 sec

69
TEST PROBLEM 8 7 -D POLYNOMIAL

• Initial box -5,-0.1,-1.1,-0.8,-1.1,-0.699,
• -1,-0.2,-1,-0.2,0.5,1.1,-0.5,
  0.5
• USING PROPOSED METHOD
• Total no. of solutions 1
• Total no. of subdivisions 0
• Range of function
• -3.9715400000000001,4.3395000000000011
• cpu time 1.038E-3 sec

70
TEST PROBLEM 8(contd.)

• RANGE OF THE POLYNOMIAL , USING VECTORIZED MOORE
  SKELBOE ALGORITHM
• Range of function
• -3.9719000000000007,4.3395000000000028
• cpu time 0.004754 sec

71
TEST PROBLEM 9 7 -D POLYNOMIAL

• Initial box -15,0,-10,1,-5,0,-10,2,
• -10,1,-1,5,-5,4
• USING PROPOSED METHOD
• Total no. of solutions 1
• Total no. of subdivisions 0
• Range of function
• -311.91149999999999,1323.0885000000001
• cpu time 6.852E-3 sec

72
TEST PROBLEM 9(contd.)

• RANGE OF THE POLYNOMIAL , USING VECTORIZED MOORE
  SKELBOE ALGORITHM
• Range of function
• -311.91150000000005,1323.0885000000001
• cpu time 0.0121438 sec

73
COMPARISON OF RESULTS
74
DISCUSSION OF RESULTS

• THE PROPOSED ALGORITHM IS CONSIDERABLY FASTER
  THAN THE VECTORISED MOORE SKELBOE ALGORITHM,
  EXCEPT IN ONE CASE
• THE MAXIMUM SPEEDING UP FACTOR ACHIEVED IS
  318.2119 AND MINIMUM SPEEDING UP IS 1.7723 FOR
  THE TEST PROBLEMS CONSIDERED
• THE ALGORITHM GIVES THE EXACT RANGE (TO THE
  TOLERANCE SPECIFIED) IN ALL CASES CONSIDERED

75
DISCUSSION OF RESULTS

• THE PROPOSED ALGORITHM HAS BEEN USED TO COMPUTE
  RANGES OF 7-DIMENSIONAL POLYNOMIAL ALSO, AND IT
  RETURNS THE CORRECT RANGE
• THE PROPOSED ALGORITHM IS THEORETICALLY CAPABLE
  OF HANDLING MULTIVARIATE POLYNOMIALS OF ANY
  DIMENSION

76
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 SIMPLIFIED
  VERTEX CONDITION, CONSIDERABLY SPEEDS UP THE
  ALGORITHM

77
CONCLUSIONS

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

78
SUGGESTIONS FOR FUTURE WORK

• DEVELOP CODE TO EXTEND TO 8-D AND HIGHER
• INTEGRATE THE CODE WITH COSY 6 PACKAGE TO
  ENABLE HANDLING OF ANY FUNCTION
• INTRODUCE MORE EFFICIENT SUBDIVISION STRATEGY TO
  FURTHER SPEED UP THE ALGORITHM
• APPLY THE METHOD TO CONTROL PROBLEMS

79
REFERENCES

• 1. J. Berchtold and I. Voiculescu and A. Bowyer.
  Multivariate Bernstein form polynomials.
  Technical report 31/98 School of Mechanical
  Engineering, University of Bath, 1998.
• 2. R. T. Farouki and V. T. Rajan. Algorithms for
  polynomials in Bernstein form. Computer Aided
  Geometric Design,51-26,1998.
• 3. G. T. Cargo and O. Shisha. The Bernstein form
  of a polynomial. Jl. of research of
  NBS,70B79-81,1966.
• 4. J. Garloff. The Bernstein Algorithm. Interval
  Computations,(2)155-168,1993

80
REFERENCES

• 5. J. Garloff. Application of Bernstein Expansion
  to the Solution of Control Problems. Proc. of
  MISC'99- Workshop on Applications of Interval
  Analysis to Systems and Control, Girona,
  Spain,1999.
• 6. M. Berz and J. Hoefkens. COSY INFINITY Version
  8.1 Programming Manual. Technical report
  MSUCL-1196, National Superconducting Cyclotron
  Laboratory, Michigan State University, East
  Lansing, MI 48824, 2001.

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THANK YOU !