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Foundations of Interval Foundations of Interval ComputationComputation

Trong Wu

E-mail: twu@siue.edu

Phone: 1-618-692-4027

Department of Computer Science

Southern Illinois University Edwardsville

Edwardsville, IL 62026, U.S.A.

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AgendaAgenda

1. Introduction

2. Some Basic Algebraic Structures of Numbers

3. The Structure of Dyadic Numbers

4. The Structure of Rough Numbers

5. Hierarchical Classes and Computation

6. A Model Interval Computation

7. An Example

8. Conclusion

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Some Terminologies of NumbersSome Terminologies of Numbers

1. Real Numbers

2. Dyadic Numbers

3. Finite Dyadic Numbers

4. Limited Dyadic Numbers

5. Model Numbers

6. Rough Numbers

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1. Introduction (1)1. Introduction (1)

In 1982, Pawlak proposed a concept called rough sets, used in the theory of knowledge for the classification of features of objects [Pawlak1982].

He considered X to be a set and R to be a relation over X. The pair (X, R) is called a rough space, and R is called the rough relation.

If xRy, then one could say that x is too close to y, x and y are indiscernible, and x and y belong to the same elementary set. This set is called a rough set.

1. Introduction (2)

Later, Wu [1994, 1998] defined a new class of real numbers called rough numbers, the definition is given in Section 4, which is a one-dimensional rough set in rough set theory.

• If z1, z2 E and (z1, z2 ) R implies z1, z2 [x, y) in the rough space

(E, R) then R is an equivalence relation on E. Equivalence classes of the relation R are called basic model intervals, if the smallest interval with model number endpoints. The set of all basic model intervals in (E, R) is denoted by E/R.

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1. Introduction (3)1. Introduction (3)

Where are the problems?

(1) The real world problem is given in the real number system.

(2) The computation is done in the model number (see Section 4 for definition) system for the given machine, M.

(3) The results must be converted from model numbers into real numbers.

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1. Introduction (4)1. Introduction (4)

Machine error? VS Human mistake?

1. A problem moving from platform (1) to platform (2) for computation can induce incorrect results and so can moving from platform (2) to platform (3). The platform (2) can only provide an approximated result for platform (1).

2. These incorrect results are not machine errors, because the computer system performances exactly work as requested.

3. In fact, one can view this a human mistake by putting a real valued problem onto a model number platform for

computation.

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2.2. Some Basic Algebraic Structures of Some Basic Algebraic Structures of NumbersNumbers (1)(1)

Definition 2.1 An abelian group (G; +) is a set G together with a binary operation namely, addition, “+”, which satisfies the following conditions [Waerden 1948, Herstein 1971]:

(1) For all a, b G, such that a + b G.(2) For all a, b, c G, then (a + b) + c a + (b + c).(3) There is an identity element, e in G such that a + e = e + a, for all

a G.(4)    For all a G, there exists an inverse, (a), such that

(a) + a = a + (a) = e.(5) For all a, b G, such that a + b = b + a.

Example 2.1 The set of all integers, I, with usual addition, +, (I, +), is an abelian group.

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2.2. Some Basic Algebraic Structures of Some Basic Algebraic Structures of NumbersNumbers (2)(2)

Definition 2.2 We say that (S; , ) is a ring, if (S; +) is an abelian group, defining as a mapping from S S → S, which satisfying the following conditions:

(1) For all a, b, c S, then (a b) c a (b c).

(2) Multiplication is distributed over addition: that is for all a, b, c S, the left and right distributive laws hold:

a (b + c) = a b + a c, and (b + c) a = b a + b c.

Example 2.2 The set of all integers I with usual addition, +, and multiplication, , then (I, +, ) is a ring. This ring has no zero-divisor.

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2.2. Some Basic Algebraic Structures of Some Basic Algebraic Structures of NumbersNumbers (3)(3)

Definition 2.3 A Field (F; , ) is a ring with commutative law and multiplicative unity such that each non-zero element has a multiplicative inverse that satisfies the following conditions:

(1) For all a, b F such that a b = b a.

(2) There exists 1 such that a 1 1 a = a.

(3) For each non-zero element a in F there is an inverse (1/a) such that a (1/a) = (1/a) a = 1.

Example 2.3 The set of all rational numbers, real numbers, and complex numbers with usual addition, +, and multiplication, , are fields.

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3. The Structure of Dyadic Numbers (1)3. The Structure of Dyadic Numbers (1)

Theorem 3.1 For each real number x and B =0, 1, we have its dyadic expansion over a finite field (B; +, ) [kelley 1955]:

,sign and where, 2sign Baax ii

ii

,sign and where, 2sign Baay i

n

ni

ii

.2047,...,2,1,0

64,10232).1(

.255,...,2,1,0

,32,1272).1(

Exp

bitsforExpMsign

Exp

bitsforExpMsign

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4. The Structure of Rough Numbers (1)4. The Structure of Rough Numbers (1) The Ada programming language [Watt 1987] calls a real number a model number, if the real number can be stored exactly in a computer system with a radix of power of 2. Therefore, the set of all model numbers is a subset of dyadic numbers and we call it the set of limited dyadic numbers with respect to a given computer system.

If a real number is a model number with respect to a given machine M, then it is a special case rough number. Most computer systems use a downward rounding policy to approximate a rough number with a model number. Thus, there are infinitely many rough numbers approximated by the same model number, such that all rough numbers z [x, y) are approximated by the same value of the model number x for computation [Wu 1994, 1998].

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4. 4. The Structure of Rough Numbers (2)The Structure of Rough Numbers (2)

The Ada programming language [Watt 1987] calls a real number a model number, if the real number can be stored exactly in a computer system with a radix of power of 2. Therefore, the set of all model numbers is a subset of dyadic numbers and we call it the set of limited dyadic numbers with respect to a given computer system.

For those real numbers that cannot be stored exactly in a given computer system, we call them rough numbers with respect to the given computer system.

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4. 4. The Structure of Rough Numbers (3)The Structure of Rough Numbers (3)

Definition 4.1 Let E be the set of all real numbers within the range of a given computer system M, and let R be a relation on E. The pair (E, R) is called rough space and R is called the rough relation, if x, y E and (x, y) R, we say that x and y are indistinctive in the rough space (E, R) with respect to the given computer system M. We call x and y rough numbers and they are approximated to the same model number. If a real number is a model number with respect to a given machine M, then it is a special case rough number. Most computer systems use a downward rounding policy to approximate a rough number with a model number. Thus, there are infinitely many rough numbers approximated by the same model number, such that all rough numbers z [x, y) are approximated by the same value of the model number x for computation [Wu 1994, 1998]

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5. Hierarchical structures of dyadic numbers 5. Hierarchical structures of dyadic numbers (1)(1)

Finite dyadic numbers

Limited dyadic numbers (model numbers) (A subset of a polynomial ring)

)()(

)(

bfaf

baf

f ba

ba

Real numbers (A field) Dyadic numbers

Rough numbers

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5. Hierarchical structures of dyadic 5. Hierarchical structures of dyadic numbers (2)numbers (2)

The set of real numbers has a field structure, while the set of finite dyadic numbers is not a field; it is a subset of ring. f is not a one-to-one and onto function, f does not preserve the field structure, and cannot even be a local homomorphism between the set of real numbers within a given range of machine M and the set of limited dyadic numbers. The addition ‘+’ on the left hand side of the inequality (5.1) is a field addition and the addition ‘’ on the right hand side is a ring addition over dyadic numbers.

f(a + b) f(a) f(b) (5.1)

They are two different additions. In the inequality (5.1), computations occur over two different number systems; we need to adjust by adding an error term, Err, to form the equality:

f(a + b) f(a) f(b) + Err. (5.2)

6. Model Interval Computation (1)6. Model Interval Computation (1)Constructing a model interval

An algorithm for computing smallest model number eps w.r.t. 1.0: eps 1.0 epsp1 eps 1.0 pass = 0 while (epsp1 > 1.0)

pass = pass + 1– eps 0.5 eps– epsp1 eps 1.0

end while

For a given real number x, then the smallest model number w.r.t. x is eps(x) and the smallest model interval containing x is

[x-eps(x), x+eps(x)].

******* Computatation of Epsilon w.r.t. 1.0 *******

Pass Machine Epsilon Epsp1  1 5.000000000000000e-001 1.500000000000000e+000 2 2.500000000000000e-001 1.250000000000000e+000 3 1.250000000000000e-001 1.125000000000000e+000 4 6.250000000000000e-002 1.062500000000000e+000 5 3.125000000000000e-002 1.031250000000000e+000 6 1.562500000000000e-002 1.015625000000000e+000 7 7.812500000000000e-003 1.007812500000000e+000 8 3.906250000000000e-003 1.003906250000000e+000 9 1.953125000000000e-003 1.001953125000000e+000 10 9.765625000000000e-004 1.000976562500000e+000 11 4.882812500000000e-004 1.000488281250000e+000 12 2.441406250000000e-004 1.000244140625000e+000 13 1.220703125000000e-004 1.000122070312500e+000 14 6.103515625000000e-005 1.000061035156250e+000 15 3.051757812500000e-005 1.000030517578125e+000 16 1.525878906250000e-005 1.000015258789063e+000 17 7.629394531250000e-006 1.000007629394531e+000 18 3.814697265625000e-006 1.000003814697266e+000 19 1.907348632812500e-006 1.000001907348633e+000 20 9.536743164062500e-007 1.000000953674316e+000 21 4.768371582031250e-007 1.000000476837158e+000 22 2.384185791015625e-007 1.000000238418579e+000 23 1.192092895507813e-007 1.000000119209290e+000 24 5.960464477539063e-008 1.000000059604645e+000 25 2.980232238769531e-008 1.000000029802322e+000 26 1.490116119384766e-008 1.000000014901161e+000 27 7.450580596923828e-009 1.000000007450581e+000

Pass Machine Epsilon Epsp1

28 3.725290298461914e-009 1.000000003725290e+000 29 1.862645149230957e-009 1.000000001862645e+000 30 9.313225746154785e-010 1.000000000931323e+000 31 4.656612873077393e-010 1.000000000465661e+000 32 2.328306436538696e-010 1.000000000232831e+000 33 1.164153218269348e-010 1.000000000116415e+000 34 5.820766091346741e-011 1.000000000058208e+000 35 2.910383045673370e-011 1.000000000029104e+000 36 1.455191522836685e-011 1.000000000014552e+000 37 7.275957614183426e-012 1.000000000007276e+000 38 3.637978807091713e-012 1.000000000003638e+000 39 1.818989403545857e-012 1.000000000001819e+000 40 9.094947017729282e-013 1.000000000000910e+000 41 4.547473508864641e-013 1.000000000000455e+000 42 2.273736754432321e-013 1.000000000000227e+000 43 1.136868377216160e-013 1.000000000000114e+000 44 5.684341886080802e-014 1.000000000000057e+000 45 2.842170943040401e-014 1.000000000000028e+000 46 1.421085471520200e-014 1.000000000000014e+000 47 7.105427357601002e-015 1.000000000000007e+000 48 3.552713678800501e-015 1.000000000000004e+000 49 1.776356839400251e-015 1.000000000000002e+000 50 8.881784197001252e-016 1.000000000000001e+000 51 4.440892098500626e-016 1.000000000000000e+000 52 2.220446049250313e-016 1.000000000000000e+000 53 1.110223024625157e-016 1.000000000000000e+000 

The final Eps = 1.110223024625157e-016

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  6. 6. Model Interval Computation (2)Model Interval Computation (2)

Definition: For any real numbers x and y, there exist basic model intervals X [a1, b1] and Y [a2, b2] such that x [a1, b1) and y[a2,

b2) where a1, b1, a2, and b2 are model numbers.

Interval arithmetic:

Addition

x + y [[a1, b1] + [a2, b2] ) = [a1 + a2,, b1 + b2). (6.1)

Subtraction

x y [[a1, b1] [a2, b2] ) = [a1 b2,, b1 a2). (6.2)

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  6. 6. Model Interval Computation (3)Model Interval Computation (3)

Multiplication

x y [ min(a1a2, a1b2, b1a2, b1b2), max(a1a2, a1b2, b1a2, b1b2)). (6.3)

Division

1/y [1/b2, 1/a2), if 0 B, then we define

x/y [min(a1/b2, a1/a2, b1/b2, b1/a2), max(a1/b2, a1/a2, b1/b2, b1/a2))

(6.4)

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  6. 6. Model Interval Computation (4)Model Interval Computation (4)

Since the product of two model numbers or the inverse of a model number might not be a model number, we round the result outward to the nearest model numbers. Let m(x) be the greatest model number less than x for the lower bound and m(y) be the smallest model number greater than y for the upper bound. Therefore, we redefine multiplication and division as follows:

Multiplication

x y [m(min(a1a2, a1b2, b1a2, b1b2)), m(max(a1a2, a1b2, b1a2, b1b2))). (6.3)’

Division

1/y [1/b2, 1/a2), if 0 B, then we define

x/y [m(min(a1/b2, a1/a2, b1/b2, b1/a2)), m(max(a1/b2, a1/a2, b1/b2, b1/a2))). (6.4)’

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  6. Model Interval Computation (5)6. Model Interval Computation (5)

Theorem 6.1 The Fundamental Theorem of Interval Computation:

Let f(x1, x2, . . . , xn) be a rational function of n variables. Consider

any sequence of arithmetic steps which serve to evaluate f with given arguments x1, x2, . . . , xn. Suppose we replace the arguments xi by the

corresponding closed interval Xi (i 1, 2, . . . , n) and replace the

arithmetic steps in the sequence used to evaluate f by the corresponding interval arithmetic steps. The result will be an interval f(X1, X2, . . . , Xn). This interval contains the value of f(x1, x2, . . . , xn),

for all xi Xi (i= 1, 2, . . . , n) [Moore 1966].

7. Example (Newton Method) (1)

An example in solving non-linear equations by using interval computation is given. The Newton method is used for demonstration. The computation is done with an Ada Interval Mathematics Package.

This program uses interval method (Interval_Math package) and Newton method to solve equation:

x**3 + 2(x**2) - 7 = 0   Let f(x) = x**3 + 2(x**2) - 7 = 0

x | -3 -2 -1 0 1 2 _____|___________________________________________________ f(x) | -16 -7 -6 -7 -4 9

From the above table: f(1) = -4 f(2) = 9 For f(x) = 0, 1 < x < 2 , x lies in (1,2)

7. Example (Newton Method) (2)

Since, f(2) > 0 To obtain c that lies in (a, b) f"(x) = 6x + 4 > 0 therefore, b = 2 c = b - f(b)/f'(b) 

Approximated interval for x (Lower_bound, Upper_bound)

(1.549999999999998046007476659724E+00, 1.550000000000002042810365310288E+00)(1.435968674249482601723570951435E+00, 1.435968674249491705552372877719E+00)(1.428844709398426893187661335105E+00, 1.428844709398445989023684887798E+00)(1.428817702168641678994731591956E+00, 1.428817702168680536800593472435E+00)(1.428817701781343041389504833205E+00, 1.428817701781421423135043369257E+00)

The final interval length = 0.00000 00000 00078 37174 55385 32052

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8. Conclusion8. Conclusion

1. We have studied some basic algebraic structures

Groups, Abelian groups, Rings, and Fields2. We also have learned systems of numbers

Real numbers, Dyadic numbers, Finite dyadic numbers, Limit dyadic numbers, Rough numbers, and Model number

3. We have seen that the “Ring Computation” is not the same of“Field Computation”

4. A mapping f from a “field” into a “subset of ring” is not an isomorphism. This mapping does not preserve the the

structure. 5. Interval computation is studied, example is presented, and model interval computation is introduced. By this way, numerical

computation problem from an un-decidable turns to a decidable problem.

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Questions?Questions?

Thank You !!!