Refactor by moving derivative table calculation to separate fcn

Author: kgryte

lib/node_modules/@stdlib/math/base/special/polygamma/lib/polycotpi.js

@@ -100,6 +100,44 @@ function zeros( len ) {
 	return out;
 } // end FUNCTION zeros()
 
+/**
+* Updates the derivatives table.
+*
+* @private
+* @param {PositiveInteger} n - derivative
+*/
+function calculateDerivatives( n ) {
+	var noffset; // offset for next row
+	var offset; // 1 if the first cos power is 0; otherwise 0
+	var ncols; // how many entries there are in the current row
+	var mcols; // how many entries there will be in the next row
+	var mo; // largest order of the polynomial of cos terms
+	var so; // order of the sin term
+	var co; // order of the cosine term in entry "j"
+	var i;
+	var j;
+	var k;
+
+	for ( i = table.length-1; i < n-1; i++ ) {
+		offset = ( i&1 )|0;
+		so = ( i+2 )|0;
+		mo = ( so-1 )|0;
+		ncols = ( (mo-offset)/2 )|0;
+		noffset = offset ? 0 : 1;
+		mcols = ( (mo+1-noffset)/2 )|0;
+		table.push( zeros( mcols+1 ) );
+		for ( j = 0; j <= ncols; j++ ) {
+			co = ( (2*j)+offset )|0;
+			k = ( (co+1)/2 )|0;
+			table[ i+1 ][ k ] += ((co-so)*table[i][j]) / (so-1);
+			if ( co ) {
+				k = ( (co-1)/2 )|0;
+				table[ i+1 ][ k ] += (-co*table[i][j]) / (so-1);
+			}
+		}
+	}
+}
+
 
 // MAIN //
 
@@ -125,7 +163,7 @@ function zeros( len ) {
 *
 * -   Note that there are many different ways of representing this derivative thanks to the many trigonometric identities available. In particular, the sum of powers of cosines could be replaced by a sum of cosine multiple angles, and, indeed, if you plug the derivative into Mathematica, this is the form it will give. The two forms are related via the Chebeshev polynomials of the first kind and \\( T_n(\cos(x)) = \cos(n x) \\). The polynomial form has the great advantage that all the cosine terms are zero at half integer arguments - right where this function has it's minimum - thus avoiding cancellation error in this region.
 *
-* -   And finally, since every other term in the polynomials is zero, we can save space by only storing the non-zero terms. This greatly complexifies subscripting the tables in the calculation, but halves the storage space (and complexity for that matter).
+* -   And finally, since every other term in the polynomials is zero, we can save space by only storing the non-zero terms. This greatly increases complexity when subscripting the tables in the calculation, but halves the storage space (and complexity for that matter).
 *
 *
 * @private
@@ -135,21 +173,11 @@ function zeros( len ) {
 * @returns {number} n'th derivative
 */
 function polycotpi( n, x, xc ) {
-	var nextMaxColumns;
-	var maxCosOrder;
-	var maxColumns;
-	var nextOffset;
 	var powTerms;
-	var cosOrder;
-	var sinOrder;
-	var column;
-	var offset;
-	var index;
 	var idx;
 	var out;
 	var sum;
 	var c;
-	var i;
 	var s;
 
 	s = ( abs( x ) < abs( xc ) ) ? sinpi( x ) : sinpi( xc );
@@ -180,34 +208,18 @@ function polycotpi( n, x, xc ) {
 	case 12:
 		return PI12 * c * polyval12( c*c ) / pow( s, 13.0 );
 	}
-	// We'll have to compute the coefficients up to n, complexity is O(n^2) which we don't worry about as the values are computed once and then cached. However, if the final evaluation would have too many terms just bail out right away:
-	if ( n / 2 > MAX_SERIES_ITERATIONS ) {
-		debug( 'The value of n is so large that we\'re unable to compute the result in reasonable time.' );
+	// We'll have to compute the coefficients up to `n`, complexity is O(n^2) which we don't worry about as the values are computed once and then cached. However, if the final evaluation would have too many terms just bail out right away:
+	if ( n/2 > MAX_SERIES_ITERATIONS ) {
+		debug( 'The value of `n` is so large that we\'re unable to compute the result in reasonable time.' );
 		return NaN;
 	}
-	index = n - 1;
-	if ( index >= table.length ) {
-		for ( i = table.length - 1; i < index; ++i ) {
-			offset = ( i & 1 ) | 0; // 1 if the first cos power is 0, otherwise 0.
-			sinOrder = ( i + 2 ) | 0;  // Order of the sin term
-			maxCosOrder = ( sinOrder - 1 ) | 0;  // Largest order of the polynomial of cos terms
-			maxColumns = ( ( maxCosOrder - offset ) / 2 ) | 0;  // How many entries there are in the current row.
-			nextOffset = offset ? 0 : 1;
-			nextMaxColumns = ( ( maxCosOrder + 1 - nextOffset ) / 2 ) | 0;  // How many entries there will be in the next row
-			table.push( zeros( nextMaxColumns + 1 ) );
-			for ( column = 0; column <= maxColumns; ++column ) {
-				cosOrder = ( ( 2*column ) + offset ) | 0;  // Order of the cosine term in entry "column"
-				idx = ( ( cosOrder+1 ) / 2 ) | 0;
-				table[i + 1][ idx ] += ((cosOrder - sinOrder) * table[i][column]) / (sinOrder - 1); // eslint-disable-line max-len
-				if ( cosOrder ) {
-					idx = ( ( cosOrder-1 ) / 2 ) | 0; //
-					table[i + 1][ idx ] += (-cosOrder * table[i][column]) / (sinOrder - 1); // eslint-disable-line max-len
-				}
-			}
-		}
+	idx = n - 1;
+	if ( idx >= table.length ) {
+		// Lazily calculate derivatives:
+		calculateDerivatives( n );
 	}
-	sum = evalpoly( table[ index ], c*c );
-	if ( index & 1 ) {
+	sum = evalpoly( table[ idx ], c*c );
+	if ( idx & 1 ) {
 		sum *= c; // First coefficient is order 1, and really an odd polynomial.
 	}
 	if ( sum === 0.0 ) {
@@ -216,17 +228,16 @@ function polycotpi( n, x, xc ) {
 	// The remaining terms are computed using logs since the powers and factorials get real large real quick:
 	powTerms = n * LN_PI;
 	if ( s === 0.0 ) {
-		return ( sum >= 0 ) ? PINF : NINF;
+		return ( sum >= 0.0 ) ? PINF : NINF;
 	}
-	powTerms -= ln( abs( s ) ) * ( n + 1 );
-	powTerms += gammaln( n );
-	powTerms += ln( abs(sum) );
+	powTerms -= ln( abs( s ) ) * ( n+1 );
+	powTerms += gammaln( n ) + ln( abs(sum) );
 
-	if (powTerms > MAX_LN ) {
-		return ( sum >= 0 ) ? PINF : NINF;
+	if ( powTerms > MAX_LN ) {
+		return ( sum >= 0.0 ) ? PINF : NINF;
 	}
 	out = exp( powTerms ) * signum( sum );
-	if ( s < 0 && ( (n + 1) & 1 ) ) {
+	if ( s < 0.0 && ( (n+1)&1 ) ) {
 		out *= -1;
 	}
 	return out;