java.lang.Object
- cds.healpix.common.math.TaylorSeries

```
public final class TaylorSeries
extends java.lang.Object
```
Utility class continaing the Taylor expansion of a few trigonometric functions. We recall that: arcos(x) = pi/2 - arcsin(x)

Author:

F.-X. Pineau

Field Summary

Fields
Modifier and Type	Field	Description
`static double`	`FIVE_OVER_112`	Constant value equal to 3/40.
`static double`	`HALF`	Constant value equal to 1/2.
`static double`	`ONE_OVER_120`	Constant value equal to 1/120 = 1/5!.
`static double`	`ONE_OVER_24`	Constant value equal to 1/24 = 1/4!.
`static double`	`ONE_OVER_5040`	Constant value equal to 1/5040 = 1/7!.
`static double`	`ONE_OVER_720`	Constant value equal to 1/720 = 1/6!.
`static double`	`ONE_SIXTH`	Constant value equal to 1/6 = 1/6.
`static double`	`ONE_THIRD`	Constant value equal to 1/3.
`static double`	`Q_17_OVER_315`	Constant value equal to 17/215.
`static double`	`THREE_OVER_40`	Constant value equal to 3/40.
`static double`	`TWO_OVER_15`	Constant value equal to 2/15.

Constructor Summary

Constructors
Constructor Description

TaylorSeries()

Method Summary

All Methods Static Methods Concrete Methods
Modifier and Type	Method	Description
`static double`	`asinO1(double x)`
`static double`	`asinO3(double x)`
`static double`	`asinO5(double x)`
`static double`	`asinO7(double x)`
`static double`	`cosO0(double a)`	Returns the Taylor serie of the trigonometric cosine around a = 0 at the 0 order (constant).
`static double`	`cosO2(double a)`	Returns the Taylor serie of the trigonometric cosine around a = 0 at the order of the 2nd degree.
`static double`	`cosO4(double a)`	Returns the Taylor serie of the trigonometric cosine around a = 0 at the order of the 4th degree.
`static double`	`cosO6(double a)`	Returns the Taylor serie of the trigonometric cosine around a = 0 at the order of the 6th degree.
`static double`	`sinO1(double a)`	Returns the Taylor serie of the trigonometric sine around a = 0 at the order of the 1st degree.
`static double`	`sinO3(double a)`	Returns the Taylor serie of the trigonometric sine around a = 0 at the order of the 3rd degree.
`static double`	`sinO5(double a)`	Returns the Taylor serie of the trigonometric sine around a = 0 at the order of the 5th degree.
`static double`	`sinO7(double a)`	Returns the Taylor serie of the trigonometric sine around a = 0 at the order of the 7th degree.
`static double`	`tanO1(double a)`	Returns the Taylor serie of the trigonometric tan around a = 0 at the order of the 1st degree.
`static double`	`tanO3(double a)`	Returns the Taylor serie of the trigonometric tan around a = 0 at the order of the 3rd degree.
`static double`	`tanO5(double a)`	Returns the Taylor serie of the trigonometric tan around a = 0 at the order of the 5th degree.
`static double`	`tanO7(double a)`	Returns the Taylor serie of the trigonometric tan around a = 0 at the order of the 7th degree.

Methods inherited from class java.lang.Object
equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

- Field Detail
  - HALF
```
public static final double HALF
```
    Constant value equal to 1/2.
    
    See Also:
    
    Constant Field Values
  - ONE_THIRD
```
public static final double ONE_THIRD
```
    Constant value equal to 1/3.
    
    See Also:
    
    Constant Field Values
  - ONE_SIXTH
```
public static final double ONE_SIXTH
```
    Constant value equal to 1/6 = 1/6.
    
    See Also:
    
    Constant Field Values
  - ONE_OVER_24
```
public static final double ONE_OVER_24
```
    Constant value equal to 1/24 = 1/4!.
    
    See Also:
    
    Constant Field Values
  - ONE_OVER_120
```
public static final double ONE_OVER_120
```
    Constant value equal to 1/120 = 1/5!.
    
    See Also:
    
    Constant Field Values
  - ONE_OVER_720
```
public static final double ONE_OVER_720
```
    Constant value equal to 1/720 = 1/6!.
    
    See Also:
    
    Constant Field Values
  - ONE_OVER_5040
```
public static final double ONE_OVER_5040
```
    Constant value equal to 1/5040 = 1/7!.
    
    See Also:
    
    Constant Field Values
  - TWO_OVER_15
```
public static final double TWO_OVER_15
```
    Constant value equal to 2/15.
    
    See Also:
    
    Constant Field Values
  - THREE_OVER_40
```
public static final double THREE_OVER_40
```
    Constant value equal to 3/40.
    
    See Also:
    
    Constant Field Values
  - FIVE_OVER_112
```
public static final double FIVE_OVER_112
```
    Constant value equal to 3/40.
    
    See Also:
    
    Constant Field Values
  - Q_17_OVER_315
```
public static final double Q_17_OVER_315
```
    Constant value equal to 17/215.
    
    See Also:
    
    Constant Field Values
- Constructor Detail
  - TaylorSeries
```
public TaylorSeries()
```
- Method Detail
  - cosO0
```
public static final double cosO0(double a)
```
    Returns the Taylor serie of the trigonometric cosine around a = 0 at the 0 order (constant). That is: 1. The error on this approximation is no more than |x|^2 / 2, i.e. ~5e-17 if x = 1e-8. We recall that the erro due to numercial approximation is 1e-16.
    
    Parameters:
    
    a - angle, in radians
    
    Returns:
    
    1, i.e. the Taylor serie of the trigonometric cosine around a = 0 at the 0 order.
  - cosO2
```
public static final double cosO2(double a)
```
    Returns the Taylor serie of the trigonometric cosine around a = 0 at the order of the 2nd degree. That is: 1 - x^2/2. The error on this approximation is no more than |x|^4 / 4!, i.e. ~4e-18 if x = 1e-4.
    
    Parameters:
    
    a - angle, in radians
    
    Returns:
    
    the Taylor serie of the trigonometric sine around a = 0 at the order of the 2nd degree.
  - cosO4
```
public static final double cosO4(double a)
```
    Returns the Taylor serie of the trigonometric cosine around a = 0 at the order of the 4th degree. That is: 1 - x^2/2 + x^4/4!. The error on this approximation is no more than |x|^6 / 6!, i.e. ~1.3e-21 if x = 1e-3.
    
    Parameters:
    
    a - angle, in radians
    
    Returns:
    
    the Taylor serie of the trigonometric sine around a = 0 at the order of the 4th degree.
  - cosO6
```
public static final double cosO6(double a)
```
    Returns the Taylor serie of the trigonometric cosine around a = 0 at the order of the 6th degree. That is: 1 - x^2/2 + x^4/4! - x^6/6!. The error on this approximation is no more than |x|^8 / 8!, i.e. ~1.7e-20 if x = 1e-2.
    
    Parameters:
    
    a - angle, in radians
    
    Returns:
    
    the Taylor serie of the trigonometric sine around a = 0 at the order of the 6th degree.
  - sinO1
```
public static final double sinO1(double a)
```
    Returns the Taylor serie of the trigonometric sine around a = 0 at the order of the 1st degree. That is: x. The error on this approximation is no more than |x|^3 / 3!, i.e. ~1.6e-19 if x = 1e-6.
    
    Parameters:
    
    a - angle, in radians
    
    Returns:
    
    the Taylor serie of the trigonometric sine around a = 0 at the order of the 1st degree.
  - sinO3
```
public static final double sinO3(double a)
```
    Returns the Taylor serie of the trigonometric sine around a = 0 at the order of the 3rd degree. That is: x - x^3/3!. The error on this approximation is no more than |x|^5 / 120, i.e. ~8.3e-18 if x = 1e-3.
    
    Parameters:
    
    a - angle, in radians
    
    Returns:
    
    the Taylor serie of the trigonometric sine around a = 0 at the order the 3rd degree.
  - sinO5
```
public static final double sinO5(double a)
```
    Returns the Taylor serie of the trigonometric sine around a = 0 at the order of the 5th degree. That is: x - x^3/3! + x^5/5!. The error on this approximation is no more than |x|^7 / 7!, i.e. ~2e-18 if x = 1e-2.
    
    Parameters:
    
    a - angle, in radians
    
    Returns:
    
    the Taylor serie of the trigonometric sine around a = 0 at the order of the 5th degree.
  - sinO7
```
public static final double sinO7(double a)
```
    Returns the Taylor serie of the trigonometric sine around a = 0 at the order of the 7th degree. That is: x - x^3/3! + x^5/5! - x^7/7!. The error on this approximation is no more than |x|^9 / 9!, i.e. ~2.8e-15 if x = 1e-1.
    
    Parameters:
    
    a - angle, in radians
    
    Returns:
    
    the Taylor serie of the trigonometric sine around a = 0 at the order of the 7th degree.
  - tanO1
```
public static final double tanO1(double a)
```
    Returns the Taylor serie of the trigonometric tan around a = 0 at the order of the 1st degree. That is: x. The error on this approximation is no more than |x|^3 / 3, i.e. ~4e-19 if x = 1e-6.
    
    Parameters:
    
    a - angle, in radians
    
    Returns:
    
    the Taylor serie of the trigonometric tan around a = 0 at the order of the 1st degree.
  - tanO3
```
public static final double tanO3(double a)
```
    Returns the Taylor serie of the trigonometric tan around a = 0 at the order of the 3rd degree. That is: x + x^3/3. The error on this approximation is no more than 2*|x|^5 / 15, i.e. ~1.3e-21 if x = 1e-4.
    
    Parameters:
    
    a - angle, in radians
    
    Returns:
    
    the Taylor serie of the trigonometric tan around a = 0 at the order of the 3rd degree.
  - tanO5
```
public static final double tanO5(double a)
```
    Returns the Taylor serie of the trigonometric tan around a = 0 at the order of the 5th degree. That is: x + x^3/3 + 2x^5/15. The error on this approximation is no more than 17|x|^7 / 315, i.e. ~5.4e-23 if x = 1e-3.
    
    Parameters:
    
    a - angle, in radians
    
    Returns:
    
    the Taylor serie of the trigonometric tan around a = 0 at the order of the 5th degree.
  - tanO7
```
public static final double tanO7(double a)
```
    Returns the Taylor serie of the trigonometric tan around a = 0 at the order of the 7th degree. That is: x + x^3/3 + 2x^5/15 + 17x^7/315. The error on this approximation is no more than 62|x|^9 / 2835, i.e. ~2.2e-20 if x = 1e-2.
    
    Parameters:
    
    a - angle, in radians
    
    Returns:
    
    the Taylor serie of the trigonometric tan around a = 0 at the order of the 7th degree.
  - asinO1
```
public static final double asinO1(double x)
```
  - asinO3
```
public static final double asinO3(double x)
```
  - asinO5
```
public static final double asinO5(double x)
```
  - asinO7
```
public static final double asinO7(double x)
```

Class TaylorSeries

Field Summary

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Field Detail

HALF

ONE_THIRD

ONE_SIXTH

ONE_OVER_24

ONE_OVER_120

ONE_OVER_720

ONE_OVER_5040

TWO_OVER_15

THREE_OVER_40

FIVE_OVER_112

Q_17_OVER_315

Constructor Detail

TaylorSeries

Method Detail

cosO0

cosO2

cosO4

cosO6

sinO1

sinO3

sinO5

sinO7

tanO1

tanO3

tanO5

tanO7

asinO1

asinO3

asinO5

asinO7