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Constructive Analysis with Witnesses
"... Contents 1. Real Numbers 3 2 3 1.2. Reals, Equality of Reals 5 1.3. The Archimedian Axiom 6 1.4. Nonnegative and Positive Reals 6 1.5. Arithmetical Functions 7 1.6. Comparison of Reals 8 1.7. NonCountability 10 1.8. Cleaning of Reals 11 2. Sequences and Series of Real Numbers 11 2.1. Completenes ..."
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Contents 1. Real Numbers 3 2 3 1.2. Reals, Equality of Reals 5 1.3. The Archimedian Axiom 6 1.4. Nonnegative and Positive Reals 6 1.5. Arithmetical Functions 7 1.6. Comparison of Reals 8 1.7. NonCountability 10 1.8. Cleaning of Reals 11 2. Sequences and Series of Real Numbers 11 2.1. Completeness 11 2.2. Limits and Inequalities 13 2.3. Series 13 2.4. Redundant Dyadic Representation of Reals 14 2.5. Convergence Tests 15 2.6. Reordering Theorem 17 2.7. The Exponential Series 18 3. The Exponential Function for Complex Numbers 21 4. Continuous Functions 23 4.1. Suprema and In ma 24 4.2. Continuous Functions 25 4.3. Application of a Continuous Function to a Real 27 4.4. Continuous Functions and Limits 28 4.5. Composition of Continuous Functions 28 4.6. Properties of Continuous Functions 29 4.7. Intermediate Value Theorem 30 4.8. Continuity of Functions with More Than One Variable 32 5. Dierentiation 33 5.1. Derivatives 33 5.2. Bounds on the Slope 33 5.3. Properties of Derivatives 34 5
Two Algorithms for Root Finding in Exact Real Arithmetic
, 1998
"... We present two algorithms for computing the root, or equivalently the fixed point, of a function in exact real arithmetic. The first algorithm uses the iteration of the expression tree representing the function in real arithmetic based on linear fractional transformations and exact floating point. T ..."
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We present two algorithms for computing the root, or equivalently the fixed point, of a function in exact real arithmetic. The first algorithm uses the iteration of the expression tree representing the function in real arithmetic based on linear fractional transformations and exact floating point. The second and more general algorithm is based on a trisection of intervals and can be compared with the wellknown bisection method in numerical analysis. It can be applied to any representation for exact real numbers; here it is described for the sign binary system in [\Gamma1; 1] which is equivalent to the exact floating point with linear fractional transformations. Keywords : Shrinking intervals, Normal products, Exact floating point, Expression trees, Sign Binary System, Iterative method, Trisection. 1 Introduction In the past few years, continued fractions and linear fractional transformations (lft), also called homographies or Mobius transformations, have been used to develop various...
Exact Real Number Computation Using Linear Fractional Transformations
"... which has provided the first proper data type for solving ordinary differential equations up to any degree of accuracy. ..."
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which has provided the first proper data type for solving ordinary differential equations up to any degree of accuracy.
Exact Arithmetic Using the Golden Ratio
, 1999
"... : The usual approach to real arithmetic on computers consists of using oating point approximations. Unfortunately, oating point arithmetic can sometimes produce wildly erroneous results. One alternative approach is to use exact real arithmetic. Exact real arithmetic allows exact real number computat ..."
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: The usual approach to real arithmetic on computers consists of using oating point approximations. Unfortunately, oating point arithmetic can sometimes produce wildly erroneous results. One alternative approach is to use exact real arithmetic. Exact real arithmetic allows exact real number computation to be performed without the roundo errors characteristic of other methods. Conventional representations such as decimal and binary notation are inadequate for this purpose. We consider an alternative representation of reals, using the golden ratio. Firstly we look at the golden ratio and its relation to the Fibonacci series, nding some interesting identities. Then we implement algorithms for basic arithmetic operations, trigonometric and logarithmic functions, conversion and integration. These include new algorithms for addition, multiplication, multiplication by 2, division by 2 and manipulating nite and innite streams. Acknowledgements I would especially like to than my supe...
Admissible Digit Sets and a Modified SternBrocot Representation
, 2004
"... We examine a special case of admissible representations of the closed interval, namely those which arise via sequences of a nite number of Möbius transformations. We regard certain sets of Möbius transformations as a generalized notion of digits and introduce sucient conditions that such a "dig ..."
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We examine a special case of admissible representations of the closed interval, namely those which arise via sequences of a nite number of Möbius transformations. We regard certain sets of Möbius transformations as a generalized notion of digits and introduce sucient conditions that such a "digit set" yields an admissible representation of [0; +1]. Furthermore we establish the productivity and correctness of the homographic algorithm for such "admissible" digit sets. In the second part of the paper we discuss representation of positive real numbers based on the SternBrocot tree. We show how we can modify the usual SternBrocot representation to yield a ternary admissible digit set.
CO620
"... When a number is represented as a continued fraction, then it comes with a natural error bound. Continued fractions can be expressed as digit streams. Arbitrary precision can be achieved by truncating the stream appropriately. Introducing more terms will refine the representation whilst preserving t ..."
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When a number is represented as a continued fraction, then it comes with a natural error bound. Continued fractions can be expressed as digit streams. Arbitrary precision can be achieved by truncating the stream appropriately. Introducing more terms will refine the representation whilst preserving the ability for further refinement. The value of continued fraction arithmetic has been recognized by the functional programming community, because continued fractions can be naturally implemented as lazy streams, but is not as widely known in logic programming. Delay declarations can be used to orchestrate the control needed to compute numeric results lazily to the required degree of precision. Irrational numbers can be represented by infinite continued fractions, which, if they have recurring patterns, can be represented exactly by rational trees. This project demonstrates how continued fraction arithmetic works and how it can be implemented using logic programming features to achieve the desired precision of a result. 1.
Admissible Digit Sets and a Modified Stern–Brocot Representation
"... We examine a special case of admissible representations of the closed interval, namely those which arise via sequences of a finite number of Möbius transformations. We regard certain sets of Möbius transformations as a generalized notion of digits and introduce sufficient conditions that such a “dig ..."
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We examine a special case of admissible representations of the closed interval, namely those which arise via sequences of a finite number of Möbius transformations. We regard certain sets of Möbius transformations as a generalized notion of digits and introduce sufficient conditions that such a “digit set ” yields an admissible representation of [0, +∞]. Furthermore we establish the productivity and correctness of the homographic algorithm for such “admissible” digit sets. In the second part of the paper we discuss representation of positive real numbers based on the Stern–Brocot tree. We show how we can modify the usual Stern–Brocot representation to yield a ternary admissible digit set.