diff --git a/src/NumericalAnalysis/NumericalDifferentiation/FivePointFormula.php b/src/NumericalAnalysis/NumericalDifferentiation/FivePointFormula.php index b1a409ea5..b3f7e6d8c 100644 --- a/src/NumericalAnalysis/NumericalDifferentiation/FivePointFormula.php +++ b/src/NumericalAnalysis/NumericalDifferentiation/FivePointFormula.php @@ -34,7 +34,7 @@ class FivePointFormula extends NumericalDifferentiation * - If the 3rd point is our $target, use the Midpoint Formula: * * 1 h⁴ - * f'(x₀) = - [f(x₀-2h)-8f(x₀-h)+8f(x₀+h)-f(x₀+2h)] - - f⁽⁵⁾(ζ₁) + * f′(x₀) = - [f(x₀-2h)-8f(x₀-h)+8f(x₀+h)-f(x₀+2h)] - - f⁽⁵⁾(ζ₁) * 12h 30 * * where ζ₁ lies between x₀ - 2h and x₀ + 2h @@ -43,7 +43,7 @@ class FivePointFormula extends NumericalDifferentiation * - Note that when the 3rd point is our $target, we use a negative h. * * 1 h⁴ - * f'(x₀) = - [-25f(x₀)+48f(x₀+h)-36f(x₀+2h)+16f(x₀+3h)-3f(x₀+4h)] + - f⁽⁵⁾(ζ₀) + * f′(x₀) = - [-25f(x₀)+48f(x₀+h)-36f(x₀+2h)+16f(x₀+3h)-3f(x₀+4h)] + - f⁽⁵⁾(ζ₀) * 12h 5 * * where ζ₀ lies between x₀ and x₀ + 4h @@ -88,7 +88,7 @@ public static function differentiate($target, $source, ... $args) * - If the 3rd point is our $target, use the Midpoint Formula: * * 1 h⁴ - * f'(x₀) = - [f(x₀-2h)-8f(x₀-h)+8f(x₀+h)-f(x₀+2h)] - - f⁽⁵⁾(ζ₁) + * f′(x₀) = - [f(x₀-2h)-8f(x₀-h)+8f(x₀+h)-f(x₀+2h)] - - f⁽⁵⁾(ζ₁) * 12h 30 * * where ζ₁ lies between x₀ - 2h and x₀ + 2h @@ -97,7 +97,7 @@ public static function differentiate($target, $source, ... $args) * - Note that when the 3rd point is our $target, we use a negative h. * * 1 h⁴ - * f'(x₀) = - [-25f(x₀)+48f(x₀+h)-36f(x₀+2h)+16f(x₀+3h)-3f(x₀+4h)] + - f⁽⁵⁾(ζ₀) + * f′(x₀) = - [-25f(x₀)+48f(x₀+h)-36f(x₀+2h)+16f(x₀+3h)-3f(x₀+4h)] + - f⁽⁵⁾(ζ₀) * 12h 5 * * where ζ₀ lies between x₀ and x₀ + 4h diff --git a/src/NumericalAnalysis/NumericalDifferentiation/SecondDerivativeMidpointFormula.php b/src/NumericalAnalysis/NumericalDifferentiation/SecondDerivativeMidpointFormula.php index a17900602..f2ead3dd5 100644 --- a/src/NumericalAnalysis/NumericalDifferentiation/SecondDerivativeMidpointFormula.php +++ b/src/NumericalAnalysis/NumericalDifferentiation/SecondDerivativeMidpointFormula.php @@ -31,9 +31,9 @@ class SecondDerivativeMidpointFormula extends NumericalDifferentiation * * Second Derivative Midpoint Formula: * - * 1 h² - * f''(x₀) = - [f(x₀-h) - 2f(x₀) + f(x₀+h)] - - f⁽⁴⁾(ζ) - * h² 12 + * 1 h² + * f″(x₀) = - [f(x₀-h) - 2f(x₀) + f(x₀+h)] - - f⁽⁴⁾(ζ) + * h² 12 * * where ζ lies between x₀ - h and x₀ + h * @@ -71,9 +71,9 @@ public static function differentiate($target, $source, ... $args) $h = ($sorted[2][$x] - $sorted[0][$x]) / 2; /* - * 1 h² - * f''(x₀) = - [f(x₀-h) - 2f(x₀) + f(x₀+h)] - - f⁽⁴⁾(ζ) - * h² 12 + * 1 h² + * f″(x₀) = - [f(x₀-h) - 2f(x₀) + f(x₀+h)] - - f⁽⁴⁾(ζ) + * h² 12 * * where ζ lies between x₀ - h and x₀ + h */ diff --git a/src/NumericalAnalysis/NumericalDifferentiation/ThreePointFormula.php b/src/NumericalAnalysis/NumericalDifferentiation/ThreePointFormula.php index fa4cc36ca..2d343208a 100644 --- a/src/NumericalAnalysis/NumericalDifferentiation/ThreePointFormula.php +++ b/src/NumericalAnalysis/NumericalDifferentiation/ThreePointFormula.php @@ -34,7 +34,7 @@ class ThreePointFormula extends NumericalDifferentiation * - If the 2nd point is our $target, use the Midpoint Formula: * * 1 h² - * f'(x₀) = - [f(x₀+h)-f(x₀-h)] - - f⁽³⁾(ζ₁) + * f′(x₀) = - [f(x₀+h)-f(x₀-h)] - - f⁽³⁾(ζ₁) * 2h 6 * * where ζ₁ lies between x₀ - h and x₀ + h @@ -43,7 +43,7 @@ class ThreePointFormula extends NumericalDifferentiation * - Note that when the 3rd point is our $target, we use a negative h. * * 1 h² - * f'(x₀) = - [-3f(x₀)+4f(x₀+h)-f(x₀+2h)] + - f⁽³⁾(ζ₀) + * f′(x₀) = - [-3f(x₀)+4f(x₀+h)-f(x₀+2h)] + - f⁽³⁾(ζ₀) * 2h 3 * * where ζ₀ lies between x₀ and x₀ + 2h @@ -86,7 +86,7 @@ public static function differentiate($target, $source, ... $args) * If the 2nd point is our $target, use the Midpoint Formula: * * 1 h² - * f'(x₀) = - [f(x₀+h)-f(x₀-h)] - - f⁽³⁾(ζ₁) + * f′(x₀) = - [f(x₀+h)-f(x₀-h)] - - f⁽³⁾(ζ₁) * 2h 6 * * where ζ₁ lies between x₀ - h and x₀ + h @@ -95,7 +95,7 @@ public static function differentiate($target, $source, ... $args) * Note that when the 3rd point is our $target, we use a negative h. * * 1 h² - * f'(x₀) = - [-3f(x₀)+4f(x₀+h)-f(x₀+2h)] + - f⁽³⁾(ζ₀) + * f′(x₀) = - [-3f(x₀)+4f(x₀+h)-f(x₀+2h)] + - f⁽³⁾(ζ₀) * 2h 3 * * where ζ₀ lies between x₀ and x₀ + 2h