fix(docs): correct .md documentation across errors, dynamics, filters, forecasts, momentum, numerics, oscillators, reversals, statistics, trends, volatility, volume

Deep review of all indicator categories verified .md headers against .cs WarmupPeriod, parameters, inputs, and outputs. Fixes include warmup corrections, parameter documentation, output type accuracy, and Pine Script alignment.

lib/numerics/accel/accel.pine

@@ -1,83 +1,28 @@
 // Licensed under the Apache License, Version 2.0
 // © mihakralj
 //@version=6
-indicator("Acceleration (Slope of Slope) (ACCEL)", "ACCEL", overlay=false, precision=8)
+indicator("Second Derivative / Acceleration (ACCEL)", "ACCEL", overlay=false, precision=8)
 
-//@function Calculates acceleration (slope of slope)
-//@param src Source series to calculate slope from
-//@param len Lookback period for calculation
-//@returns acceleration
-accel(series float src, simple int len1) =>
-    if len1 <= 1
-        runtime.error("Length 1 for slope calculation must be greater than 1")
-    var float sumX1 = 0.0, var float sumY1 = 0.0, var float sumXY1 = 0.0, var float sumX21 = 0.0
-    var int   validCount1 = 0
-    var array<float> x_values1 = array.new_float(len1)
-    var array<float> y_values1 = array.new_float(len1)
-    var int   head1 = 0
-    var int internal_time_counter1 = 0
-    if internal_time_counter1 >= len1
-        float oldX1 = array.get(x_values1, head1)
-        float oldY1 = array.get(y_values1, head1)
-        if not na(oldY1)
-            sumX1 := sumX1 - oldX1, sumY1 := sumY1 - oldY1
-            sumXY1 := sumXY1 - oldX1 * oldY1, sumX21 := sumX21 - oldX1 * oldX1
-            validCount1 := validCount1 - 1
-    float currentX1 = internal_time_counter1
-    float currentY1 = src
-    array.set(x_values1, head1, currentX1)
-    array.set(y_values1, head1, currentY1)
-    if not na(currentY1)
-        sumX1 := sumX1 + currentX1, sumY1 := sumY1 + currentY1
-        sumXY1 := sumXY1 + currentX1 * currentY1, sumX21 := sumX21 + currentX1 * currentX1
-        validCount1 := validCount1 + 1
-    head1 := (head1 + 1) % len1
-    internal_time_counter1 := internal_time_counter1 + 1
-    float current_slope = na
-    if validCount1 >= 2
-        float n1 = validCount1
-        float divisor1 = n1 * sumX21 - sumX1 * sumX1
-        if divisor1 != 0.0
-            current_slope := (n1 * sumXY1 - sumX1 * sumY1) / divisor1
-    var float sumX2 = 0.0, var float sumY2 = 0.0, var float sumXY2 = 0.0, var float sumX22 = 0.0
-    var int   validCount2 = 0
-    var array<float> x_values2 = array.new_float(len1)
-    var array<float> y_values2 = array.new_float(len1)
-    var int   head2 = 0
-    var int internal_time_counter2 = 0
-    if internal_time_counter2 >= len1
-        float oldX2 = array.get(x_values2, head2)
-        float oldY2 = array.get(y_values2, head2)
-        if not na(oldY2)
-            sumX2 := sumX2 - oldX2, sumY2 := sumY2 - oldY2
-            sumXY2 := sumXY2 - oldX2 * oldY2, sumX22 := sumX22 - oldX2 * oldX2
-            validCount2 := validCount2 - 1
-    float currentX2 = internal_time_counter2
-    float currentY2 = current_slope
-    array.set(x_values2, head2, currentX2)
-    array.set(y_values2, head2, currentY2)
-    if not na(currentY2)
-        sumX2 := sumX2 + currentX2, sumY2 := sumY2 + currentY2
-        sumXY2 := sumXY2 + currentX2 * currentY2, sumX22 := sumX22 + currentX2 * currentX2
-        validCount2 := validCount2 + 1
-    head2 := (head2 + 1) % len1
-    internal_time_counter2 := internal_time_counter2 + 1
-    float calculatedAccel = na
-    if validCount2 >= 2
-        float n2 = validCount2
-        float divisor2 = n2 * sumX22 - sumX2 * sumX2
-        if divisor2 != 0.0
-            calculatedAccel := (n2 * sumXY2 - sumX2 * sumY2) / divisor2
-    calculatedAccel
+//@function Calculates the second finite difference (acceleration): Accel = V[t] - 2*V[t-1] + V[t-2]
+//@param src Source series
+//@returns Second difference. Returns 0.0 until 3 bars are available.
+//@optimized Uses direct history access and FMA-equivalent for zero-allocation streaming.
+accel(series float src) =>
+    float v0 = src
+    float v1 = src[1]
+    float v2 = src[2]
+    if na(v1) or na(v2)
+        0.0
+    else
+        v0 - 2.0 * v1 + v2
 
 // ---------- Main loop ----------
 
 // Inputs
-i_period = input.int(14, "Period", minval=2)
 i_source = input.source(close, "Source")
 
 // Calculation
-a = accel(i_source, i_period)
+a = accel(i_source)
 
 // Plot
 plot(a, "Accel", color=color.yellow, linewidth=2)