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Sep 05, 2020

Computing special functions with the trapezoidal rule

Javier Segura

Departamento de Matemáticas, Estadística y Computación Universidad de Cantabria, Spain

January 15, 2010

J. Segura (Universidad de Cantabria) The trapezoidal rule and SF January 15, 2010 1 / 25

Contents:

1 What is a Special Function? Why compute them? What is it? Why compute them?

2 A case study: Airy functions Convergent and divergent series Methods for intermediate regions. Numerical quadrature

3 Why the trapezoidal rule? Trapezoidal rule: simple and sometimes optimal Steepest descent and the trapezoidal rule Additional examples where saddle point analysis is fruitful Trapezoidal rules with changes of variable

J. Segura (Universidad de Cantabria) The trapezoidal rule and SF January 15, 2010 2 / 25

Contents:

1 What is a Special Function? Why compute them? What is it? Why compute them?

2 A case study: Airy functions Convergent and divergent series Methods for intermediate regions. Numerical quadrature

3 Why the trapezoidal rule? Trapezoidal rule: simple and sometimes optimal Steepest descent and the trapezoidal rule Additional examples where saddle point analysis is fruitful Trapezoidal rules with changes of variable

J. Segura (Universidad de Cantabria) The trapezoidal rule and SF January 15, 2010 2 / 25

Contents:

1 What is a Special Function? Why compute them? What is it? Why compute them?

2 A case study: Airy functions Convergent and divergent series Methods for intermediate regions. Numerical quadrature

3 Why the trapezoidal rule? Trapezoidal rule: simple and sometimes optimal Steepest descent and the trapezoidal rule Additional examples where saddle point analysis is fruitful Trapezoidal rules with changes of variable

J. Segura (Universidad de Cantabria) The trapezoidal rule and SF January 15, 2010 2 / 25

What is a Special Function? Why compute them? What is it?

What is a Special Function?

What is an elementary function? Everybody knows Elementary functions and operations:

1 +,-,*,/ 2 Polynomials 3 Trigonometric 4 Exponential and logarithm

Elementary functions (trigonometric functions, exponential, log): algorithms based on polynomial approximation and/or table lookup; Shift-and-Add algorithms.

Enough? Of course, not. There is "a bunch" of useful functions which do not fall inside this narrow category⇒ SPECIAL FUNCTIONS

J. Segura (Universidad de Cantabria) The trapezoidal rule and SF January 15, 2010 3 / 25

What is a Special Function? Why compute them? What is it?

What is a Special Function? What is an elementary function?

Everybody knows Elementary functions and operations:

1 +,-,*,/ 2 Polynomials 3 Trigonometric 4 Exponential and logarithm

Elementary functions (trigonometric functions, exponential, log): algorithms based on polynomial approximation and/or table lookup; Shift-and-Add algorithms.

Enough? Of course, not. There is "a bunch" of useful functions which do not fall inside this narrow category⇒ SPECIAL FUNCTIONS

J. Segura (Universidad de Cantabria) The trapezoidal rule and SF January 15, 2010 3 / 25

What is a Special Function? Why compute them? What is it?

What is a Special Function? What is an elementary function? Everybody knows Elementary functions and operations:

1 +,-,*,/ 2 Polynomials 3 Trigonometric 4 Exponential and logarithm

Elementary functions (trigonometric functions, exponential, log): algorithms based on polynomial approximation and/or table lookup; Shift-and-Add algorithms.

Enough? Of course, not. There is "a bunch" of useful functions which do not fall inside this narrow category⇒ SPECIAL FUNCTIONS

J. Segura (Universidad de Cantabria) The trapezoidal rule and SF January 15, 2010 3 / 25

What is a Special Function? Why compute them? What is it?

What is a Special Function? What is an elementary function? Everybody knows Elementary functions and operations:

1 +,-,*,/ 2 Polynomials 3 Trigonometric 4 Exponential and logarithm

Enough?

Of course, not. There is "a bunch" of useful functions which do not fall inside this narrow category⇒ SPECIAL FUNCTIONS

J. Segura (Universidad de Cantabria) The trapezoidal rule and SF January 15, 2010 3 / 25

What is a Special Function? Why compute them? What is it?

What is a Special Function? What is an elementary function? Everybody knows Elementary functions and operations:

1 +,-,*,/ 2 Polynomials 3 Trigonometric 4 Exponential and logarithm

Enough? Of course, not. There is "a bunch" of useful functions which do not fall inside this narrow category

⇒ SPECIAL FUNCTIONS

J. Segura (Universidad de Cantabria) The trapezoidal rule and SF January 15, 2010 3 / 25

What is a Special Function? Why compute them? What is it?

1 +,-,*,/ 2 Polynomials 3 Trigonometric 4 Exponential and logarithm

J. Segura (Universidad de Cantabria) The trapezoidal rule and SF January 15, 2010 3 / 25

What is a Special Function? Why compute them? What is it?

1 The error function: erf(x) = 2√ π

∫ x 0

e−t 2 dt

2 The gamma function: Γ(α) = ∫ +∞

0 xαe−xdx

3 The Airy functions: solutions of y ′′(z)− zy(z) = 0 4 And many more, some of them depending on several parameters

(hypergeometric functions among them)

J. Segura (Universidad de Cantabria) The trapezoidal rule and SF January 15, 2010 4 / 25

What is a Special Function? Why compute them? What is it?

1 The error function: erf(x) = 2√ π

∫ x 0

e−t 2 dt

2 The gamma function: Γ(α) = ∫ +∞

0 xαe−xdx

3 The Airy functions: solutions of y ′′(z)− zy(z) = 0 4 And many more, some of them depending on several parameters

(hypergeometric functions among them)

J. Segura (Universidad de Cantabria) The trapezoidal rule and SF January 15, 2010 4 / 25

What is a Special Function? Why compute them? What is it?

1 The error function: erf(x) = 2√ π

∫ x 0

e−t 2 dt

2 The gamma function: Γ(α) = ∫ +∞

0 xαe−xdx

3 The Airy functions: solutions of y ′′(z)− zy(z) = 0

4 And many more, some of them depending on several parameters (hypergeometric functions among them)

J. Segura (Universidad de Cantabria) The trapezoidal rule and SF January 15, 2010 4 / 25

What is a Special Function? Why compute them? What is it?

1 The error function: erf(x) = 2√ π

∫ x 0

e−t 2 dt

2 The gamma function: Γ(α) = ∫ +∞

0 xαe−xdx

3 The Airy functions: solutions of y ′′(z)− zy(z) = 0 4 And many more, some of them depending on several parameters

(hypergeometric functions among them)

J. Segura (Universidad de Cantabria) The trapezoidal rule and SF January 15, 2010 4 / 25

What is a Special Function? Why compute them? Why compute them?

Why compute special functions?

Some reasons: 1 For the same reason we compute elementary functions, because

they are needed.

2 When additional functions can be computed, the toolbox of available functions becomes richer and more powerful

3 In particular, some of these non-elementary functions can be used as approximation tools.

J. Segura (Universidad de Cantabria) The trapezoidal rule and SF January 15, 2010 5 / 25

What is a Special Function? Why compute them? Why compute them?

Why compute special functions?

Some reasons: 1 For the same reason we compute elementary functions, because

they are needed. 2 When additional functions can be computed, the toolbox of

available functions becomes richer and more powerful

3 In particular, some of these non-elementary functions can be used as approximation tools.

J. Segura (Universidad de Cantabria) The trapezoidal rule and SF January 15, 2010 5 / 25

What is a Special Function? Why compute them? Why compute them?

Why compute special functions?

Some reasons: 1 For the same reason we compute elementary functions, because

they are needed. 2 When additional functions can be computed, the toolbox of

available functions becomes richer and more powerful 3 In particular, some of these non-elementary functions can be used

as approximation tools.

J. Segura (Universidad de Cantabria) The trapezoidal rule and SF January 15, 2010 5 / 25

A case study: Airy functions

Airy functions are the solution of the ODE:

The Airy equation

y

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