1. Ramanujan–Sato series
    Class number 2 (1989)  Class number 4 (1993) 

2. Iterative algorithms
    Quadratic convergence (1984)  Cubic convergence (1991)  Quartic convergence (1985)  Quintic convergence  Nonic convergence 

3. See also

4. References

5. External links

In mathematics, Borwein's algorithm is an algorithm devised by Jonathan and Peter Borwein to calculate the value of 1/{{pi}}. They devised several other algorithms. They published the book Pi and the AGM – A Study in Analytic Number Theory and Computational Complexity.^[1]

Ramanujan–Sato series

These two are examples of a Ramanujan–Sato series. The related Chudnovsky algorithm uses a discriminant with class number 1.

Class number 2 (1989)

Start by setting{{Citation needed|date=June 2011}}

Then

Each additional term of the partial sum yields approximately 25 digits.

Class number 4 (1993)

Start by setting{{Citation needed|date=June 2011}}

Then

Each additional term of the series yields approximately 50 digits.

Iterative algorithms

Quadratic convergence (1984)

Start by setting^[2]

Then iterate

Then p_k converges quadratically to {{pi}}; that is, each iteration approximately doubles the number of correct digits. The algorithm is not self-correcting; each iteration must be performed with the desired number of correct digits for {{pi}}'s final result.

Cubic convergence (1991)

Start by setting

Then iterate

Then a_k converges cubically to 1/{{pi}}; that is, each iteration approximately triples the number of correct digits.

Quartic convergence (1985)

Start by setting^[3]

Then iterate

Then a_k converges quartically against 1/{{pi}}; that is, each iteration approximately quadruples the number of correct digits. The algorithm is not self-correcting; each iteration must be performed with the desired number of correct digits for {{pi}}'s final result.

Quintic convergence

Start by setting

Then iterate

Then a_k converges quintically to 1/{{pi}} (that is, each iteration approximately quintuples the number of correct digits), and the following condition holds:

Nonic convergence

Start by setting

Then iterate

Then a_k converges nonically to 1/{{pi}}; that is, each iteration approximately multiplies the number of correct digits by nine.^[4]

See also

• Bailey–Borwein–Plouffe formula
• Chudnovsky algorithm
• Gauss–Legendre algorithm
• Ramanujan–Sato series

References

External links

• Pi Formulas from Wolfram MathWorld

• GA Hick
• Gahiga
• Gahinasazi
• Gahini
• Gahise
• Gahkuch
• Gahkyeng
• Gahlai
• Gahlbee
• Gahling
• Gahm
• Gahnia
• Gahnia aspera
• Gahnia filum
• Gahnia grandis
• Gahnia melanocarpa
• Gahnia trifida
• Gahnpa
• Gahoe-dong
• Gahoe Museum
• Ga-Hon-Ga Lodge
• Gahora
• Gahoun Georges Hegbor
• Gahoun Georges Hégbor
• Gahoun Hegbor