Package cds.healpix.common.math
Class TaylorSeries
- java.lang.Object
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- cds.healpix.common.math.TaylorSeries
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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
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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.
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Constructor Summary
Constructors Constructor Description TaylorSeries()
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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.
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Field Detail
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HALF
public static final double HALF
Constant value equal to 1/2.- See Also:
- Constant Field Values
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ONE_THIRD
public static final double ONE_THIRD
Constant value equal to 1/3.- See Also:
- Constant Field Values
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ONE_SIXTH
public static final double ONE_SIXTH
Constant value equal to 1/6 = 1/6.- See Also:
- Constant Field Values
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ONE_OVER_24
public static final double ONE_OVER_24
Constant value equal to 1/24 = 1/4!.- See Also:
- Constant Field Values
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ONE_OVER_120
public static final double ONE_OVER_120
Constant value equal to 1/120 = 1/5!.- See Also:
- Constant Field Values
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ONE_OVER_720
public static final double ONE_OVER_720
Constant value equal to 1/720 = 1/6!.- See Also:
- Constant Field Values
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ONE_OVER_5040
public static final double ONE_OVER_5040
Constant value equal to 1/5040 = 1/7!.- See Also:
- Constant Field Values
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TWO_OVER_15
public static final double TWO_OVER_15
Constant value equal to 2/15.- See Also:
- Constant Field Values
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THREE_OVER_40
public static final double THREE_OVER_40
Constant value equal to 3/40.- See Also:
- Constant Field Values
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FIVE_OVER_112
public static final double FIVE_OVER_112
Constant value equal to 3/40.- See Also:
- Constant Field Values
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Q_17_OVER_315
public static final double Q_17_OVER_315
Constant value equal to 17/215.- See Also:
- Constant Field Values
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Method Detail
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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 than2*|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.
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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 than17|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.
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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 than62|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.
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asinO1
public static final double asinO1(double x)
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asinO3
public static final double asinO3(double x)
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asinO5
public static final double asinO5(double x)
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asinO7
public static final double asinO7(double x)
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