# New formulas for approximation of π and other transcendental numbers

Research output: Contribution to journalArticle

4 Citations (Scopus)

### Abstract

We derive many new formulas for the approximation of π. The formulas make use of a sequence of iteration functions called the basic family; a nontrivial determinantal generalization of Taylor's theorem; other ingredients; as well as several new results presented in the present paper. In one scheme, one evaluates members of the basic family, for an appropriately selected function, all at the same input. This scheme generates almost a fixed and preselected number of digits in each successive evaluation. The computation amounts to the evaluation of a homogeneous linear recursive formula and is equivalent to the computation of special Toeplitz matrix determinants. The approximations of π obtained via this scheme are within simple algebraic extensions of the rational field. In a second scheme, the fixed-point iteration is applied to any fixed member of the basic family, while selecting an appropriate function. In this scheme for each natural number m ≥ 2 we prove convergence of order m, starting from the initial point. We report on some preliminary computational results obtained via Maple. Analogous formulas can be used to approximate other transcendental numbers. For instance, we also give a formula for the approximation of e. In fact, our results give new formulas and arbitrary high-order methods for the approximation of roots of certain analytic functions.

Original language English (US) 59-81 23 Numerical Algorithms 24 1-2 Published - Dec 1 2000

### Fingerprint

Transcendental number
Approximation
Taylor's theorem
Iteration Function
Algebraic extension
Fixed Point Iteration
Recursive Formula
High-order Methods
Toeplitz matrix
Maple
Evaluation
Order of Convergence
Digit
Natural number
Computational Results
Analytic function
Determinant
Roots
Evaluate
Arbitrary

### All Science Journal Classification (ASJC) codes

• Applied Mathematics

### Keywords

• Algorithms
• Digits
• Iteration functions
• Order of convergence
• π

### Cite this

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title = "New formulas for approximation of π and other transcendental numbers",
abstract = "We derive many new formulas for the approximation of π. The formulas make use of a sequence of iteration functions called the basic family; a nontrivial determinantal generalization of Taylor's theorem; other ingredients; as well as several new results presented in the present paper. In one scheme, one evaluates members of the basic family, for an appropriately selected function, all at the same input. This scheme generates almost a fixed and preselected number of digits in each successive evaluation. The computation amounts to the evaluation of a homogeneous linear recursive formula and is equivalent to the computation of special Toeplitz matrix determinants. The approximations of π obtained via this scheme are within simple algebraic extensions of the rational field. In a second scheme, the fixed-point iteration is applied to any fixed member of the basic family, while selecting an appropriate function. In this scheme for each natural number m ≥ 2 we prove convergence of order m, starting from the initial point. We report on some preliminary computational results obtained via Maple. Analogous formulas can be used to approximate other transcendental numbers. For instance, we also give a formula for the approximation of e. In fact, our results give new formulas and arbitrary high-order methods for the approximation of roots of certain analytic functions.",
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author = "Bahman Kalantari",
year = "2000",
month = "12",
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In: Numerical Algorithms, Vol. 24, No. 1-2, 01.12.2000, p. 59-81.

Research output: Contribution to journalArticle

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T1 - New formulas for approximation of π and other transcendental numbers

AU - Kalantari, Bahman

PY - 2000/12/1

Y1 - 2000/12/1

N2 - We derive many new formulas for the approximation of π. The formulas make use of a sequence of iteration functions called the basic family; a nontrivial determinantal generalization of Taylor's theorem; other ingredients; as well as several new results presented in the present paper. In one scheme, one evaluates members of the basic family, for an appropriately selected function, all at the same input. This scheme generates almost a fixed and preselected number of digits in each successive evaluation. The computation amounts to the evaluation of a homogeneous linear recursive formula and is equivalent to the computation of special Toeplitz matrix determinants. The approximations of π obtained via this scheme are within simple algebraic extensions of the rational field. In a second scheme, the fixed-point iteration is applied to any fixed member of the basic family, while selecting an appropriate function. In this scheme for each natural number m ≥ 2 we prove convergence of order m, starting from the initial point. We report on some preliminary computational results obtained via Maple. Analogous formulas can be used to approximate other transcendental numbers. For instance, we also give a formula for the approximation of e. In fact, our results give new formulas and arbitrary high-order methods for the approximation of roots of certain analytic functions.

AB - We derive many new formulas for the approximation of π. The formulas make use of a sequence of iteration functions called the basic family; a nontrivial determinantal generalization of Taylor's theorem; other ingredients; as well as several new results presented in the present paper. In one scheme, one evaluates members of the basic family, for an appropriately selected function, all at the same input. This scheme generates almost a fixed and preselected number of digits in each successive evaluation. The computation amounts to the evaluation of a homogeneous linear recursive formula and is equivalent to the computation of special Toeplitz matrix determinants. The approximations of π obtained via this scheme are within simple algebraic extensions of the rational field. In a second scheme, the fixed-point iteration is applied to any fixed member of the basic family, while selecting an appropriate function. In this scheme for each natural number m ≥ 2 we prove convergence of order m, starting from the initial point. We report on some preliminary computational results obtained via Maple. Analogous formulas can be used to approximate other transcendental numbers. For instance, we also give a formula for the approximation of e. In fact, our results give new formulas and arbitrary high-order methods for the approximation of roots of certain analytic functions.

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