Merge pull request #216 from RonaldoCMP/master

Extensive changes in arrays.scad, vectors.scad, common.scad and their regression test codes

common.scad

@@ -15,7 +15,8 @@
 // Usage:
 //   typ = typeof(x);
 // Description:
-//   Returns a string representing the type of the value.  One of "undef", "boolean", "number", "nan", "string", "list", or "range"
+//   Returns a string representing the type of the value.  One of "undef", "boolean", "number", "nan", "string", "list", "range" or "invalid".
+//   Some malformed "ranges", like '[0:NAN:INF]' and '[0:"a":INF]', may be classified as "undef" or "invalid".
 function typeof(x) =
     is_undef(x)? "undef" :
     is_bool(x)? "boolean" :
@@ -23,7 +24,9 @@ function typeof(x) =
     is_nan(x)? "nan" :
     is_string(x)? "string" :
     is_list(x)? "list" :
-    "range";
+    is_range(x) ? "range" :
+    "invalid";
+
 
 
 // Function: is_type()
@@ -70,8 +73,8 @@ function is_str(x) = is_string(x);
 //   is_int(n)
 // Description:
 //   Returns true if the given value is an integer (it is a number and it rounds to itself).  
-function is_int(n) = is_num(n) && n == round(n);
+function is_int(n) = is_finite(n) && n == round(n);
-function is_integer(n) = is_num(n) && n == round(n);
+function is_integer(n) = is_finite(n) && n == round(n);
 
 
 // Function: is_nan()
@@ -93,7 +96,17 @@ function is_finite(v) = is_num(0*v);
 // Function: is_range()
 // Description:
 //   Returns true if its argument is a range
-function is_range(x) = is_num(x[0]) && !is_list(x);
+function is_range(x) = !is_list(x) && is_finite(x[0]+x[1]+x[2]) ;
+
+
+// Function: valid_range()
+// Description:
+//   Returns true if its argument is a valid range (deprecated ranges excluded).
+function valid_range(x) = 
+    is_range(x) 
+    && ( x[1]>0 
+         ? x[0]<=x[2]
+         : ( x[1]<0 && x[0]>=x[2] ) );
 
 
 // Function: is_list_of()
@@ -106,13 +119,15 @@ function is_range(x) = is_num(x[0]) && !is_list(x);
 //   is_list_of([3,4,5], 0);            // Returns true
 //   is_list_of([3,4,undef], 0);        // Returns false
 //   is_list_of([[3,4],[4,5]], [1,1]);  // Returns true
+//   is_list_of([[3,"a"],[4,true]], [1,undef]);  // Returns true
 //   is_list_of([[3,4], 6, [4,5]], [1,1]);  // Returns false
-//   is_list_of([[1,[3,4]], [4,[5,6]]], [1,[2,3]]);    // Returne true
+//   is_list_of([[1,[3,4]], [4,[5,6]]], [1,[2,3]]);    // Returns true
-//   is_list_of([[1,[3,INF]], [4,[5,6]]], [1,[2,3]]);  // Returne false
+//   is_list_of([[1,[3,INF]], [4,[5,6]]], [1,[2,3]]);  // Returns false
+//   is_list_of([], [1,[2,3]]);                        // Returns true
 function is_list_of(list,pattern) =
     let(pattern = 0*pattern)
     is_list(list) &&
-    []==[for(entry=list) if (entry*0 != pattern) entry];
+    []==[for(entry=0*list) if (entry != pattern) entry];
 
 
 // Function: is_consistent()
@@ -128,7 +143,15 @@ function is_list_of(list,pattern) =
 //   is_consistent([[3,[3,4,[5]]], [5,[2,9,[9]]]]); // Returns true
 //   is_consistent([[3,[3,4,[5]]], [5,[2,9,9]]]);   // Returns false
 function is_consistent(list) =
-  is_list(list) && is_list_of(list, list[0]);
+  is_list(list) && is_list_of(list, _list_pattern(list[0]));
+
+
+//Internal function
+//Creates a list with the same structure of `list` with each of its elements substituted by 0.
+function _list_pattern(list) =
+  is_list(list) 
+  ? [for(entry=list) is_list(entry) ? _list_pattern(entry) : 0]
+  : 0;
 
 
 // Function: same_shape()
@@ -139,7 +162,7 @@ function is_consistent(list) =
 // Example:
 //   same_shape([3,[4,5]],[7,[3,4]]);   // Returns true
 //   same_shape([3,4,5], [7,[3,4]]);    // Returns false
-function same_shape(a,b) = a*0 == b*0;
+function same_shape(a,b) = _list_pattern(a) == b*0;
 
 
 // Section: Handling `undef`s.
@@ -313,6 +336,7 @@ function scalar_vec3(v, dflt=undef) =
 //   r = Radius of circle to get the number of segments for.
 function segs(r) = 
     $fn>0? ($fn>3? $fn : 3) :
+    let( r = is_finite(r)? r: 0 ) 
     ceil(max(5, min(360/$fa, abs(r)*2*PI/$fs))) ;
 
 
@@ -322,7 +346,7 @@ function segs(r) =
 
 function _valstr(x) =
     is_list(x)? str("[",str_join([for (xx=x) _valstr(xx)],","),"]") :
-    is_num(x)? fmt_float(x,12) : x;
+    is_finite(x)? fmt_float(x,12) : x;
 
 
 // Module: assert_approx()

math.scad

@@ -33,7 +33,10 @@ NAN = acos(2);  // The value `nan`, useful for comparisons.
 //   sqr([3,4]); // Returns: [9,16]
 //   sqr([[1,2],[3,4]]);  // Returns [[1,4],[9,16]]
 //   sqr([[1,2],3]);      // Returns [[1,4],9]
-function sqr(x) = is_list(x) ? [for(val=x) sqr(val)] : x*x;
+function sqr(x) = 
+    is_list(x) ? [for(val=x) sqr(val)] : 
+    is_finite(x) ? x*x :
+    assert(is_finite(x) || is_vector(x), "Input is not neither a number nor a list of numbers.");
 
 
 // Function: log2()
@@ -45,8 +48,11 @@ function sqr(x) = is_list(x) ? [for(val=x) sqr(val)] : x*x;
 //   log2(0.125);  // Returns: -3
 //   log2(16);     // Returns: 4
 //   log2(256);    // Returns: 8
-function log2(x) = ln(x)/ln(2);
+function log2(x) = 
+    assert( is_finite(x), "Input is not a number.")
+    ln(x)/ln(2);
 
+// this may return NAN or INF; should it check x>0 ?
 
 // Function: hypot()
 // Usage:
@@ -60,7 +66,9 @@ function log2(x) = ln(x)/ln(2);
 // Example:
 //   l = hypot(3,4);  // Returns: 5
 //   l = hypot(3,4,5);  // Returns: ~7.0710678119
-function hypot(x,y,z=0) = norm([x,y,z]);
+function hypot(x,y,z=0) = 
+    assert( is_vector([x,y,z]), "Improper number(s).")
+    norm([x,y,z]);
 
 
 // Function: factorial()
@@ -76,11 +84,53 @@ function hypot(x,y,z=0) = norm([x,y,z]);
 //   y = factorial(6);  // Returns: 720
 //   z = factorial(9);  // Returns: 362880
 function factorial(n,d=0) =
-    assert(n>=0 && d>=0, "Factorial is not defined for negative numbers")
+    assert(is_int(n) && is_int(d) && n>=0 && d>=0, "Factorial is not defined for negative numbers")
     assert(d<=n, "d cannot be larger than n")
     product([1,for (i=[n:-1:d+1]) i]);
 
 
+// Function: binomial()
+// Usage:
+//   x = binomial(n);
+// Description:
+//   Returns the binomial coefficients of the integer `n`.  
+// Arguments:
+//   n = The integer to get the binomial coefficients of
+// Example:
+//   x = binomial(3);  // Returns: [1,3,3,1]
+//   y = binomial(4);  // Returns: [1,4,6,4,1]
+//   z = binomial(6);  // Returns: [1,6,15,20,15,6,1]
+function binomial(n) =
+    assert( is_int(n) && n>0, "Input is not an integer greater than 0.")
+    [for( c = 1, i = 0; 
+        i<=n; 
+         c = c*(n-i)/(i+1), i = i+1
+        ) c ] ;
+
+
+// Function: binomial_coefficient()
+// Usage:
+//   x = binomial_coefficient(n,k);
+// Description:
+//   Returns the k-th binomial coefficient of the integer `n`.  
+// Arguments:
+//   n = The integer to get the binomial coefficient of
+//   k = The binomial coefficient index
+// Example:
+//   x = binomial_coefficient(3,2);  // Returns: 3
+//   y = binomial_coefficient(10,6); // Returns: 210
+function binomial_coefficient(n,k) =
+    assert( is_int(n) && is_int(k), "Some input is not a number.")
+    k < 0 || k > n ? 0 :
+    k ==0 || k ==n ? 1 :
+    let( k = min(k, n-k),
+         b = [for( c = 1, i = 0; 
+                   i<=k; 
+                   c = c*(n-i)/(i+1), i = i+1
+                 ) c] )
+    b[len(b)-1];
+
+
 // Function: lerp()
 // Usage:
 //   x = lerp(a, b, u);
@@ -91,8 +141,8 @@ function factorial(n,d=0) =
 //   If `u` is 0.0, then the value of `a` is returned.
 //   If `u` is 1.0, then the value of `b` is returned.
 //   If `u` is a range, or list of numbers, returns a list of interpolated values.
-//   It is valid to use a `u` value outside the range 0 to 1.  The result will be a predicted
+//   It is valid to use a `u` value outside the range 0 to 1.  The result will be an extrapolation
-//   value along the slope formed by `a` and `b`, but not between those two values.
+//   along the slope formed by `a` and `b`.
 // Arguments:
 //   a = First value or vector.
 //   b = Second value or vector.
@@ -113,9 +163,9 @@ function factorial(n,d=0) =
 //   rainbow(pts) translate($item) circle(d=3,$fn=8);
 function lerp(a,b,u) =
     assert(same_shape(a,b), "Bad or inconsistent inputs to lerp")
-    is_num(u)? (1-u)*a + u*b :
+    is_finite(u)? (1-u)*a + u*b :
-    assert(!is_undef(u)&&!is_bool(u)&&!is_string(u), "Input u to lerp must be a number, vector, or range.")
+    assert(is_finite(u) || is_vector(u) || valid_range(u), "Input u to lerp must be a number, vector, or range.")
-    [for (v = u) lerp(a,b,v)];
+    [for (v = u) (1-v)*a + v*b ];
 
 
 
@@ -124,40 +174,45 @@ function lerp(a,b,u) =
 // Function: sinh()
 // Description: Takes a value `x`, and returns the hyperbolic sine of it.
 function sinh(x) =
+    assert(is_finite(x), "The input must be a finite number.")
     (exp(x)-exp(-x))/2;
 
 
 // Function: cosh()
 // Description: Takes a value `x`, and returns the hyperbolic cosine of it.
 function cosh(x) =
+    assert(is_finite(x), "The input must be a finite number.")
     (exp(x)+exp(-x))/2;
 
 
 // Function: tanh()
 // Description: Takes a value `x`, and returns the hyperbolic tangent of it.
 function tanh(x) =
+    assert(is_finite(x), "The input must be a finite number.")
     sinh(x)/cosh(x);
 
 
 // Function: asinh()
 // Description: Takes a value `x`, and returns the inverse hyperbolic sine of it.
 function asinh(x) =
+    assert(is_finite(x), "The input must be a finite number.")
     ln(x+sqrt(x*x+1));
 
 
 // Function: acosh()
 // Description: Takes a value `x`, and returns the inverse hyperbolic cosine of it.
 function acosh(x) =
+    assert(is_finite(x), "The input must be a finite number.")
     ln(x+sqrt(x*x-1));
 
 
 // Function: atanh()
 // Description: Takes a value `x`, and returns the inverse hyperbolic tangent of it.
 function atanh(x) =
+    assert(is_finite(x), "The input must be a finite number.")
     ln((1+x)/(1-x))/2;
 
 
-
 // Section: Quantization
 
 // Function: quant()
@@ -185,7 +240,10 @@ function atanh(x) =
 //   quant([9,10,10.4,10.5,11,12],3);      // Returns: [9,9,9,12,12,12]
 //   quant([[9,10,10.4],[10.5,11,12]],3);  // Returns: [[9,9,9],[12,12,12]]
 function quant(x,y) =
-    is_list(x)? [for (v=x) quant(v,y)] :
+    assert(is_finite(y) && !approx(y,0,eps=1e-24), "The multiple must be a non zero integer.")
+    is_list(x)
+    ?   [for (v=x) quant(v,y)]
+    :   assert( is_finite(x), "The input to quantize must be a number or a list of numbers.")
         floor(x/y+0.5)*y;
 
 
@@ -214,7 +272,10 @@ function quant(x,y) =
 //   quantdn([9,10,10.4,10.5,11,12],3);      // Returns: [9,9,9,9,9,12]
 //   quantdn([[9,10,10.4],[10.5,11,12]],3);  // Returns: [[9,9,9],[9,9,12]]
 function quantdn(x,y) =
-    is_list(x)? [for (v=x) quantdn(v,y)] :
+    assert(is_finite(y) && !approx(y,0,eps=1e-24), "The multiple must be a non zero integer.")
+    is_list(x)
+    ?    [for (v=x) quantdn(v,y)]
+    :    assert( is_finite(x), "The input to quantize must be a number or a list of numbers.")
         floor(x/y)*y;
 
 
@@ -243,7 +304,10 @@ function quantdn(x,y) =
 //   quantup([9,10,10.4,10.5,11,12],3);      // Returns: [9,12,12,12,12,12]
 //   quantup([[9,10,10.4],[10.5,11,12]],3);  // Returns: [[9,12,12],[12,12,12]]
 function quantup(x,y) =
-    is_list(x)? [for (v=x) quantup(v,y)] :
+    assert(is_finite(y) && !approx(y,0,eps=1e-24), "The multiple must be a non zero integer.")
+    is_list(x)
+    ?    [for (v=x) quantup(v,y)]
+    :    assert( is_finite(x), "The input to quantize must be a number or a list of numbers.")
         ceil(x/y)*y;
 
 
@@ -264,7 +328,9 @@ function quantup(x,y) =
 //   constrain(0.3, -1, 1);  // Returns: 0.3
 //   constrain(9.1, 0, 9);   // Returns: 9
 //   constrain(-0.1, 0, 9);  // Returns: 0
-function constrain(v, minval, maxval) = min(maxval, max(minval, v));
+function constrain(v, minval, maxval) = 
+    assert( is_finite(v+minval+maxval), "Input must be finite number(s).")
+    min(maxval, max(minval, v));
 
 
 // Function: posmod()
@@ -283,7 +349,9 @@ function constrain(v, minval, maxval) = min(maxval, max(minval, v));
 //   posmod(270,360);   // Returns: 270
 //   posmod(700,360);   // Returns: 340
 //   posmod(3,2.5);     // Returns: 0.5
-function posmod(x,m) = (x%m+m)%m;
+function posmod(x,m) = 
+    assert( is_finite(x) && is_finite(m) && !approx(m,0) , "Input must be finite numbers. The divisor cannot be zero.")
+    (x%m+m)%m;
 
 
 // Function: modang(x)
@@ -299,6 +367,7 @@ function posmod(x,m) = (x%m+m)%m;
 //   modang(270,360);   // Returns: -90
 //   modang(700,360);   // Returns: -20
 function modang(x) =
+    assert( is_finite(x), "Input must be a finite number.")
     let(xx = posmod(x,360)) xx<180? xx : xx-360;
 
 
@@ -306,7 +375,7 @@ function modang(x) =
 // Usage:
 //   modrange(x, y, m, [step])
 // Description:
-//   Returns a normalized list of values from `x` to `y`, by `step`, modulo `m`.  Wraps if `x` > `y`.
+//   Returns a normalized list of numbers from `x` to `y`, by `step`, modulo `m`.  Wraps if `x` > `y`.
 // Arguments:
 //   x = The start value to constrain.
 //   y = The end value to constrain.
@@ -318,6 +387,7 @@ function modang(x) =
 //   modrange(90,270,360, step=-45);  // Returns: [90,45,0,315,270]
 //   modrange(270,90,360, step=-45);  // Returns: [270,225,180,135,90]
 function modrange(x, y, m, step=1) =
+    assert( is_finite(x+y+step+m) && !approx(m,0), "Input must be finite numbers. The module value cannot be zero.")
     let(
         a = posmod(x, m),
         b = posmod(y, m),
@@ -330,20 +400,21 @@ function modrange(x, y, m, step=1) =
 
 // Function: rand_int()
 // Usage:
-//   rand_int(min,max,N,[seed]);
+//   rand_int(minval,maxval,N,[seed]);
 // Description:
-//   Return a list of random integers in the range of min to max, inclusive.
+//   Return a list of random integers in the range of minval to maxval, inclusive.
 // Arguments:
-//   min = Minimum integer value to return.
+//   minval = Minimum integer value to return.
-//   max = Maximum integer value to return.
+//   maxval = Maximum integer value to return.
 //   N = Number of random integers to return.
 //   seed = If given, sets the random number seed.
 // Example:
 //   ints = rand_int(0,100,3);
 //   int = rand_int(-10,10,1)[0];
-function rand_int(min, max, N, seed=undef) =
+function rand_int(minval, maxval, N, seed=undef) =
-    assert(max >= min, "Max value cannot be smaller than min")
+    assert( is_finite(minval+maxval+N) && (is_undef(seed) || is_finite(seed) ), "Input must be finite numbers.")
-    let (rvect = is_def(seed) ? rands(min,max+1,N,seed) : rands(min,max+1,N))
+    assert(maxval >= minval, "Max value cannot be smaller than minval")
+    let (rvect = is_def(seed) ? rands(minval,maxval+1,N,seed) : rands(minval,maxval+1,N))
     [for(entry = rvect) floor(entry)];
 
 
@@ -358,6 +429,7 @@ function rand_int(min, max, N, seed=undef) =
 //   N = Number of random numbers to return.  Default: 1
 //   seed = If given, sets the random number seed.
 function gaussian_rands(mean, stddev, N=1, seed=undef) =
+    assert( is_finite(mean+stddev+N) && (is_undef(seed) || is_finite(seed) ), "Input must be finite numbers.")
     let(nums = is_undef(seed)? rands(0,1,N*2) : rands(0,1,N*2,seed))
     [for (i = list_range(N)) mean + stddev*sqrt(-2*ln(nums[i*2]))*cos(360*nums[i*2+1])];
 
@@ -374,6 +446,10 @@ function gaussian_rands(mean, stddev, N=1, seed=undef) =
 //   N = Number of random numbers to return.  Default: 1
 //   seed = If given, sets the random number seed.
 function log_rands(minval, maxval, factor, N=1, seed=undef) =
+    assert( is_finite(minval+maxval+N) 
+            && (is_undef(seed) || is_finite(seed) )
+            && factor>0, 
+            "Input must be finite numbers. `factor` should be greater than zero.")
     assert(maxval >= minval, "maxval cannot be smaller than minval")
     let(
         minv = 1-1/pow(factor,minval),
@@ -395,18 +471,18 @@ function gcd(a,b) =
     b==0 ? abs(a) : gcd(b,a % b);
 
 
-// Computes lcm for two scalars
+// Computes lcm for two integers
 function _lcm(a,b) =
-    assert(is_int(a), "Invalid non-integer parameters to lcm")
+    assert(is_int(a) && is_int(b), "Invalid non-integer parameters to lcm")
-    assert(is_int(b), "Invalid non-integer parameters to lcm")
     assert(a!=0 && b!=0, "Arguments to lcm must be non zero")
     abs(a*b) / gcd(a,b);
 
 
 // Computes lcm for a list of values
 function _lcmlist(a) =
-    len(a)==1 ? a[0] :
+    len(a)==1 
-    _lcmlist(concat(slice(a,0,len(a)-2),[lcm(a[len(a)-2],a[len(a)-1])]));
+    ?   a[0] 
+    :   _lcmlist(concat(slice(a,0,len(a)-2),[lcm(a[len(a)-2],a[len(a)-1])]));
 
 
 // Function: lcm()
@@ -418,11 +494,10 @@ function _lcmlist(a) =
 //   be non-zero integers.  The output is always a positive integer.  It is an error to pass zero
 //   as an argument.  
 function lcm(a,b=[]) =
-    !is_list(a) && !is_list(b) ? _lcm(a,b) : 
+    !is_list(a) && !is_list(b) 
-    let(
+    ?   _lcm(a,b) 
-        arglist = concat(force_list(a),force_list(b))
+    :   let( arglist = concat(force_list(a),force_list(b)) )
-    )
+        assert(len(arglist)>0, "Invalid call to lcm with empty list(s)")
-    assert(len(arglist)>0,"invalid call to lcm with empty list(s)")
         _lcmlist(arglist);
 
 
@@ -431,8 +506,9 @@ function lcm(a,b=[]) =
 
 // Function: sum()
 // Description:
-//   Returns the sum of all entries in the given list.
+//   Returns the sum of all entries in the given consistent list.
-//   If passed an array of vectors, returns a vector of sums of each part.
+//   If passed an array of vectors, returns the sum the vectors.
+//   If passed an array of matrices, returns the sum of the matrices.
 //   If passed an empty list, the value of `dflt` will be returned.
 // Arguments:
 //   v = The list to get the sum of.
@@ -441,10 +517,9 @@ function lcm(a,b=[]) =
 //   sum([1,2,3]);  // returns 6.
 //   sum([[1,2,3], [3,4,5], [5,6,7]]);  // returns [9, 12, 15]
 function sum(v, dflt=0) =
-    is_vector(v) ? [for(i=v) 1]*v :
+    is_list(v) && len(v) == 0 ? dflt :
+    is_vector(v) || is_matrix(v)? [for(i=v) 1]*v :
     assert(is_consistent(v), "Input to sum is non-numeric or inconsistent")
-    is_vector(v[0]) ? [for(i=v) 1]*v :
-    len(v) == 0 ? dflt :
     _sum(v,v[0]*0);
 
 function _sum(v,_total,_i=0) = _i>=len(v) ? _total : _sum(v,_total+v[_i], _i+1);
@@ -495,10 +570,11 @@ function sum_of_squares(v) = sum(vmul(v,v));
 // Examples:
 //   v = sum_of_sines(30, [[10,3,0], [5,5.5,60]]);
 function sum_of_sines(a, sines) =
-    sum([
+    assert( is_finite(a) && is_matrix(sines,undef,3), "Invalid input.")
-        for (s = sines) let(
+    sum([ for (s = sines) 
+            let(
               ss=point3d(s),
-            v=ss.x*sin(a*ss.y+ss.z)
+              v=ss[0]*sin(a*ss[1]+ss[2])
             ) v
         ]);
 
@@ -506,26 +582,39 @@ function sum_of_sines(a, sines) =
 // Function: deltas()
 // Description:
 //   Returns a list with the deltas of adjacent entries in the given list.
+//   The list should be a consistent list of numeric components (numbers, vectors, matrix, etc).
 //   Given [a,b,c,d], returns [b-a,c-b,d-c].
 // Arguments:
 //   v = The list to get the deltas of.
 // Example:
 //   deltas([2,5,9,17]);  // returns [3,4,8].
 //   deltas([[1,2,3], [3,6,8], [4,8,11]]);  // returns [[2,4,5], [1,2,3]]
-function deltas(v) = [for (p=pair(v)) p.y-p.x];
+function deltas(v) = 
+    assert( is_consistent(v) && len(v)>1 , "Inconsistent list or with length<=1.")
+    [for (p=pair(v)) p[1]-p[0]] ;
 
 
 // Function: product()
 // Description:
 //   Returns the product of all entries in the given list.
-//   If passed an array of vectors, returns a vector of products of each part.
+//   If passed a list of vectors of same dimension, returns a vector of products of each part.
-//   If passed an array of matrices, returns a the resulting product matrix.
+//   If passed a list of square matrices, returns a the resulting product matrix.
 // Arguments:
 //   v = The list to get the product of.
 // Example:
 //   product([2,3,4]);  // returns 24.
 //   product([[1,2,3], [3,4,5], [5,6,7]]);  // returns [15, 48, 105]
-function product(v, i=0, tot=undef) = i>=len(v)? tot : product(v, i+1, ((tot==undef)? v[i] : is_vector(v[i])? vmul(tot,v[i]) : tot*v[i]));
+function product(v) = 
+    assert( is_vector(v) || is_matrix(v) || ( is_matrix(v[0],square=true) && is_consistent(v)), 
+            "Invalid input.")
+    _product(v, 1, v[0]);
+
+function _product(v, i=0, _tot) = 
+    i>=len(v) ? _tot :
+    _product( v, 
+              i+1, 
+              ( is_vector(v[i])? vmul(_tot,v[i]) : _tot*v[i] ) );
+               
 
 
 // Function: outer_product()
@@ -534,21 +623,22 @@ function product(v, i=0, tot=undef) = i>=len(v)? tot : product(v, i+1, ((tot==un
 // Usage:
 //   M = outer_product(u,v);
 function outer_product(u,v) =
-  assert(is_vector(u) && is_vector(v))
+  assert(is_vector(u) && is_vector(v), "The inputs must be vectors.")
-  assert(len(u)==len(v))
+  [for(ui=u) ui*v];
-  [for(i=[0:len(u)-1]) [for(j=[0:len(u)-1]) u[i]*v[j]]];
 
 
 // Function: mean()
 // Description:
-//   Returns the arithmatic mean/average of all entries in the given array.
+//   Returns the arithmetic mean/average of all entries in the given array.
 //   If passed a list of vectors, returns a vector of the mean of each part.
 // Arguments:
 //   v = The list of values to get the mean of.
 // Example:
 //   mean([2,3,4]);  // returns 3.
 //   mean([[1,2,3], [3,4,5], [5,6,7]]);  // returns [3, 4, 5]
-function mean(v) = sum(v)/len(v);
+function mean(v) = 
+    assert(is_list(v) && len(v)>0, "Invalid list.")
+    sum(v)/len(v);
 
 
 // Function: median()
@@ -556,18 +646,33 @@ function mean(v) = sum(v)/len(v);
 //   x = median(v);
 // Description:
 //   Given a list of numbers or vectors, finds the median value or midpoint.
-//   If passed a list of vectors, returns the vector of the median of each part.
+//   If passed a list of vectors, returns the vector of the median of each component.
 function median(v) =
-    assert(is_list(v))
+    is_vector(v) ? (min(v)+max(v))/2 :
-    assert(len(v)>0)
+    is_matrix(v) ? [for(ti=transpose(v))  (min(ti)+max(ti))/2 ]
-    is_vector(v[0])? (
+    :   assert(false , "Invalid input.");
-        assert(is_consistent(v))
+
-        [
+// Function: convolve()
-            for (i=idx(v[0]))
+// Usage:
-            let(vals = subindex(v,i))
+//   x = convolve(p,q);
-            (min(vals)+max(vals))/2
+// Description:
-        ]
+//   Given two vectors, finds the convolution of them.
-    ) : (min(v)+max(v))/2;
+//   The length of the returned vector is len(p)+len(q)-1 .
+// Arguments:
+//   p = The first vector.
+//   q = The second vector.
+// Example:
+//   a = convolve([1,1],[1,2,1]); // Returns: [1,3,3,1]
+//   b = convolve([1,2,3],[1,2,1])); // Returns: [1,4,8,8,3]
+function convolve(p,q) =
+    p==[] || q==[] ? [] :
+    assert( is_vector(p) && is_vector(q), "The inputs should be vectors.")
+    let( n = len(p),
+         m = len(q))
+    [for(i=[0:n+m-2], k1 = max(0,i-n+1), k2 = min(i,m-1) )
+       [for(j=[k1:k2]) p[i-j] ] * [for(j=[k1:k2]) q[j] ] 
+    ];
+
 
 
 // Section: Matrix math
@@ -582,7 +687,7 @@ function median(v) =
 //   want to solve Ax=b1 and Ax=b2 that you need to form the matrix transpose([b1,b2]) for the right hand side and then
 //   transpose the returned value.  
 function linear_solve(A,b) =
-    assert(is_matrix(A))
+    assert(is_matrix(A), "Input should be a matrix.")
     let(
         m = len(A),
         n = len(A[0])
@@ -619,7 +724,11 @@ function matrix_inverse(A) =
 // Description:
 //   Returns a submatrix with the specified index ranges or index sets.  
 function submatrix(M,ind1,ind2) =
-    [for(i=ind1) [for(j=ind2) M[i][j] ] ];
+    assert( is_matrix(M), "Input must be a matrix." )
+    [for(i=ind1) 
+        [for(j=ind2) 
+            assert( ! is_undef(M[i][j]), "Invalid indexing." )
+            M[i][j] ] ];
     
 
 // Function: qr_factor()
@@ -628,7 +737,7 @@ function submatrix(M,ind1,ind2) =
 //   Calculates the QR factorization of the input matrix A and returns it as the list [Q,R].  This factorization can be
 //   used to solve linear systems of equations.  
 function qr_factor(A) =
-    assert(is_matrix(A))
+    assert(is_matrix(A), "Input must be a matrix." )
     let(
       m = len(A),
       n = len(A[0])
@@ -659,8 +768,8 @@ function _qr_factor(A,Q, column, m, n) =
 // Function: back_substitute()
 // Usage: back_substitute(R, b, [transpose])
 // Description:
-//   Solves the problem Rx=b where R is an upper triangular square matrix.  No check is made that the lower triangular entries
+//   Solves the problem Rx=b where R is an upper triangular square matrix.  The lower triangular entries of R are
-//   are actually zero.  If transpose==true then instead solve transpose(R)*x=b.
+//   ignored.  If transpose==true then instead solve transpose(R)*x=b.
 //   You can supply a compatible matrix b and it will produce the solution for every column of b.  Note that if you want to
 //   solve Rx=b1 and Rx=b2 you must set b to transpose([b1,b2]) and then take the transpose of the result.  If the matrix
 //   is singular (e.g. has a zero on the diagonal) then it returns [].  
@@ -694,7 +803,9 @@ function back_substitute(R, b, x=[],transpose = false) =
 // Example:
 //   M = [ [6,-2], [1,8] ];
 //   det = det2(M);  // Returns: 50
-function det2(M) = M[0][0] * M[1][1] - M[0][1]*M[1][0];
+function det2(M) = 
+    assert( is_matrix(M,2,2), "Matrix should be 2x2." )
+    M[0][0] * M[1][1] - M[0][1]*M[1][0];
 
 
 // Function: det3()
@@ -706,6 +817,7 @@ function det2(M) = M[0][0] * M[1][1] - M[0][1]*M[1][0];
 //   M = [ [6,4,-2], [1,-2,8], [1,5,7] ];
 //   det = det3(M);  // Returns: -334
 function det3(M) =
+    assert( is_matrix(M,3,3), "Matrix should be 3x3." )
     M[0][0] * (M[1][1]*M[2][2]-M[2][1]*M[1][2]) -
     M[1][0] * (M[0][1]*M[2][2]-M[2][1]*M[0][2]) +
     M[2][0] * (M[0][1]*M[1][2]-M[1][1]*M[0][2]);
@@ -720,7 +832,7 @@ function det3(M) =
 //   M = [ [6,4,-2,9], [1,-2,8,3], [1,5,7,6], [4,2,5,1] ];
 //   det = determinant(M);  // Returns: 2267
 function determinant(M) =
-    assert(len(M)==len(M[0]))
+    assert(is_matrix(M,square=true), "Input should be a square matrix." )
     len(M)==1? M[0][0] :
     len(M)==2? det2(M) :
     len(M)==3? det3(M) :
@@ -753,8 +865,11 @@ function determinant(M) =
 //   n = optional width of matrix
 //   square = set to true to require a square matrix.  Default: false        
 function is_matrix(A,m,n,square=false) =
-    is_vector(A[0],n) && is_vector(A*(0*A[0]),m) &&
+    is_list(A[0]) 
-    (!square || len(A)==len(A[0]));
+    && ( let(v = A*A[0]) is_num(0*(v*v)) ) // a matrix of finite numbers 
+    && (is_undef(n) || len(A[0])==n )
+    && (is_undef(m) || len(A)==m )
+    && ( !square || len(A)==len(A[0]));
 
 
 // Section: Comparisons and Logic
@@ -777,8 +892,10 @@ function is_matrix(A,m,n,square=false) =
 function approx(a,b,eps=EPSILON) = 
     a==b? true :
     a*0!=b*0? false :
-    is_list(a)? ([for (i=idx(a)) if(!approx(a[i],b[i],eps=eps)) 1] == []) :
+    is_list(a)
-    (abs(a-b) <= eps);
+    ? ([for (i=idx(a)) if( !approx(a[i],b[i],eps=eps)) 1] == [])
+    : is_num(a) && is_num(b) && (abs(a-b) <= eps);
+    
 
 
 function _type_num(x) =
@@ -796,7 +913,7 @@ function _type_num(x) =
 // Description:
 //   Compares two values.  Lists are compared recursively.
 //   Returns <0 if a<b.  Returns >0 if a>b.  Returns 0 if a==b.
-//   If types are not the same, then undef < bool < num < str < list < range.
+//   If types are not the same, then undef < bool < nan < num < str < list < range.
 // Arguments:
 //   a = First value to compare.
 //   b = Second value to compare.
@@ -820,13 +937,14 @@ function compare_vals(a, b) =
 //   a = First list to compare.
 //   b = Second list to compare.
 function compare_lists(a, b) =
-    a==b? 0 : let(
+    a==b? 0 
-        cmps = [
+    :   let(
-            for(i=[0:1:min(len(a),len(b))-1]) let(
+          cmps = [ for(i=[0:1:min(len(a),len(b))-1]) 
-                cmp = compare_vals(a[i],b[i])
+                      let( cmp = compare_vals(a[i],b[i]) )
-            ) if(cmp!=0) cmp
+                      if(cmp!=0) cmp 
                  ]
-    ) cmps==[]? (len(a)-len(b)) : cmps[0];
+           ) 
+        cmps==[]? (len(a)-len(b)) : cmps[0];
 
 
 // Function: any()
@@ -843,14 +961,13 @@ function compare_lists(a, b) =
 //   any([[0,0], [1,0]]);   // Returns true.
 function any(l, i=0, succ=false) =
     (i>=len(l) || succ)? succ :
-    any(
+    any( l, 
-        l, i=i+1, succ=(
+         i+1, 
-            is_list(l[i])? any(l[i]) :
+         succ = is_list(l[i]) ? any(l[i]) : !(!l[i])
-            !(!l[i])
-        )
         );
 
 
+
 // Function: all()
 // Description:
 //   Returns true if all items in list `l` evaluate as true.
@@ -865,15 +982,14 @@ function any(l, i=0, succ=false) =
 //   all([[0,0], [1,0]]);   // Returns false.
 //   all([[1,1], [1,1]]);   // Returns true.
 function all(l, i=0, fail=false) =
-    (i>=len(l) || fail)? (!fail) :
+    (i>=len(l) || fail)? !fail :
-    all(
+    all( l, 
-        l, i=i+1, fail=(
+         i+1,
-            is_list(l[i])? !all(l[i]) :
+         fail = is_list(l[i]) ? !all(l[i]) : !l[i]
-            !l[i]
-        )
         ) ;
 
 
+
 // Function: count_true()
 // Usage:
 //   count_true(l)
@@ -904,6 +1020,21 @@ function count_true(l, nmax=undef, i=0, cnt=0) =
     );
 
 
+function count_true(l, nmax) = 
+    !is_list(l) ? !(!l) ? 1: 0 :
+    let( c = [for( i = 0,
+                   n = !is_list(l[i]) ? !(!l[i]) ? 1: 0 : undef,
+                   c = !is_undef(n)? n : count_true(l[i], nmax),
+                   s = c;
+                 i<len(l) && (is_undef(nmax) || s<nmax);
+                   i = i+1,
+                   n = !is_list(l[i]) ? !(!l[i]) ? 1: 0 : undef,
+                   c = !is_undef(n) || (i==len(l))? n : count_true(l[i], nmax-s),
+                   s = s+c
+                 )  s ] )
+    len(c)<len(l)? nmax: c[len(c)-1];
+
+
 
 // Section: Calculus
 
@@ -921,21 +1052,30 @@ function count_true(l, nmax=undef, i=0, cnt=0) =
 //   between data[i+1] and data[i], and the data values will be linearly resampled at each corner
 //   to produce a uniform spacing for the derivative estimate.  At the endpoints a single point method
 //   is used: f'(t) = (f(t+h)-f(t))/h.  
+// Arguments:
+//   data = the list of the elements to compute the derivative of.
+//   h = the parametric sampling of the data.
+//   closed = boolean to indicate if the data set should be wrapped around from the end to the start.
 function deriv(data, h=1, closed=false) =
+    assert( is_consistent(data) , "Input list is not consistent or not numerical.") 
+    assert( len(data)>=2, "Input `data` should have at least 2 elements.") 
+    assert( is_finite(h) || is_vector(h), "The sampling `h` must be a number or a list of numbers." )
+    assert( is_num(h) || len(h) == len(data)-(closed?0:1),
+            str("Vector valued `h` must have length ",len(data)-(closed?0:1)))
     is_vector(h) ? _deriv_nonuniform(data, h, closed=closed) :
     let( L = len(data) )
-    closed? [
+    closed
+    ? [
         for(i=[0:1:L-1])
         (data[(i+1)%L]-data[(L+i-1)%L])/2/h
-    ] :
+      ]
-    let(
+    : let(
-        first =
+        first = L<3 ? data[1]-data[0] : 
-            L<3? data[1]-data[0] : 
                 3*(data[1]-data[0]) - (data[2]-data[1]),
-        last =
+        last = L<3 ? data[L-1]-data[L-2]:
-            L<3? data[L-1]-data[L-2]:
                (data[L-3]-data[L-2])-3*(data[L-2]-data[L-1])
-    ) [
+         ) 
+      [
         first/2/h,
         for(i=[1:1:L-2]) (data[i+1]-data[i-1])/2/h,
         last/2/h
@@ -947,15 +1087,13 @@ function _dnu_calc(f1,fc,f2,h1,h2) =
         f1 = h2<h1 ? lerp(fc,f1,h2/h1) : f1 , 
         f2 = h1<h2 ? lerp(fc,f2,h1/h2) : f2
        )
-    (f2-f1) / 2 / min([h1,h2]);
+    (f2-f1) / 2 / min(h1,h2);
 
 
 function _deriv_nonuniform(data, h, closed) =
-    assert(len(h) == len(data)-(closed?0:1),str("Vector valued h must be length ",len(data)-(closed?0:1)))
+    let( L = len(data) )
-    let(
+    closed
-      L = len(data)
+    ? [for(i=[0:1:L-1])
-    )
-    closed? [for(i=[0:1:L-1])
           _dnu_calc(data[(L+i-1)%L], data[i], data[(i+1)%L], select(h,i-1), h[i]) ]
     : [
         (data[1]-data[0])/h[0],
@@ -967,15 +1105,23 @@ function _deriv_nonuniform(data, h, closed) =
 // Function: deriv2()
 // Usage: deriv2(data, [h], [closed])
 // Description:
-//   Computes a numerical esimate of the second derivative of the data, which may be scalar or vector valued.
+//   Computes a numerical estimate of the second derivative of the data, which may be scalar or vector valued.
 //   The `h` parameter gives the step size of your sampling so the derivative can be scaled correctly. 
 //   If the `closed` parameter is true the data is assumed to be defined on a loop with data[0] adjacent to
 //   data[len(data)-1].  For internal points this function uses the approximation 
-//   f''(t) = (f(t-h)-2*f(t)+f(t+h))/h^2.  For the endpoints (when closed=false) the algorithm
+//   f''(t) = (f(t-h)-2*f(t)+f(t+h))/h^2.  For the endpoints (when closed=false),
-//   when sufficient points are available the method is either the four point expression
+//   when sufficient points are available, the method is either the four point expression
-//   f''(t) = (2*f(t) - 5*f(t+h) + 4*f(t+2*h) - f(t+3*h))/h^2 or if five points are available
+//   f''(t) = (2*f(t) - 5*f(t+h) + 4*f(t+2*h) - f(t+3*h))/h^2 or 
 //   f''(t) = (35*f(t) - 104*f(t+h) + 114*f(t+2*h) - 56*f(t+3*h) + 11*f(t+4*h)) / 12h^2
+//   if five points are available.
+// Arguments:
+//   data = the list of the elements to compute the derivative of.
+//   h = the constant parametric sampling of the data.
+//   closed = boolean to indicate if the data set should be wrapped around from the end to the start.
 function deriv2(data, h=1, closed=false) =
+    assert( is_consistent(data) , "Input list is not consistent or not numerical.") 
+    assert( len(data)>=3, "Input list has less than 3 elements.") 
+    assert( is_finite(h), "The sampling `h` must be a number." )
     let( L = len(data) )
     closed? [
         for(i=[0:1:L-1])
@@ -1003,16 +1149,19 @@ function deriv2(data, h=1, closed=false) =
 //   Computes a numerical third derivative estimate of the data, which may be scalar or vector valued.
 //   The `h` parameter gives the step size of your sampling so the derivative can be scaled correctly. 
 //   If the `closed` parameter is true the data is assumed to be defined on a loop with data[0] adjacent to
-//   data[len(data)-1].  This function uses a five point derivative estimate, so the input must include five points:
+//   data[len(data)-1].  This function uses a five point derivative estimate, so the input data must include 
+//   at least five points:
 //   f'''(t) = (-f(t-2*h)+2*f(t-h)-2*f(t+h)+f(t+2*h)) / 2h^3.  At the first and second points from the end
 //   the estimates are f'''(t) = (-5*f(t)+18*f(t+h)-24*f(t+2*h)+14*f(t+3*h)-3*f(t+4*h)) / 2h^3 and
 //   f'''(t) = (-3*f(t-h)+10*f(t)-12*f(t+h)+6*f(t+2*h)-f(t+3*h)) / 2h^3.
 function deriv3(data, h=1, closed=false) =
+    assert( is_consistent(data) , "Input list is not consistent or not numerical.") 
+    assert( len(data)>=5, "Input list has less than 5 elements.") 
+    assert( is_finite(h), "The sampling `h` must be a number." )
     let(
         L = len(data),
         h3 = h*h*h
     )
-    assert(L>=5, "Need five points for 3rd derivative estimate")
     closed? [
         for(i=[0:1:L-1])
         (-data[(L+i-2)%L]+2*data[(L+i-1)%L]-2*data[(i+1)%L]+data[(i+2)%L])/2/h3
@@ -1036,16 +1185,22 @@ function deriv3(data, h=1, closed=false) =
 // Function: C_times()
 // Usage: C_times(z1,z2)
 // Description:
-//   Multiplies two complex numbers.  
+//   Multiplies two complex numbers represented by 2D vectors.  
-function C_times(z1,z2) = [z1.x*z2.x-z1.y*z2.y,z1.x*z2.y+z1.y*z2.x];
+function C_times(z1,z2) = 
+    assert( is_vector(z1+z2,2), "Complex numbers should be represented by 2D vectors." )
+    [ z1.x*z2.x - z1.y*z2.y, z1.x*z2.y + z1.y*z2.x ];
 
 // Function: C_div()
 // Usage: C_div(z1,z2)
 // Description:
-//   Divides z1 by z2.  
+//   Divides two complex numbers represented by 2D vectors.  
-function C_div(z1,z2) = let(den = z2.x*z2.x + z2.y*z2.y)
+function C_div(z1,z2) = 
+    assert( is_vector(z1,2) && is_vector(z2), "Complex numbers should be represented by 2D vectors." )
+    assert( !approx(z2,0), "The divisor `z2` cannot be zero." ) 
+    let(den = z2.x*z2.x + z2.y*z2.y)
     [(z1.x*z2.x + z1.y*z2.y)/den, (z1.y*z2.x - z1.x*z2.y)/den];
 
+// For the sake of consistence with Q_mul and vmul, C_times should be called C_mul
 
 // Section: Polynomials
 
@@ -1056,14 +1211,35 @@ function C_div(z1,z2) = let(den = z2.x*z2.x + z2.y*z2.y)
 //   Evaluates specified real polynomial, p, at the complex or real input value, z.
 //   The polynomial is specified as p=[a_n, a_{n-1},...,a_1,a_0]
 //   where a_n is the z^n coefficient.  Polynomial coefficients are real.
+//   The result is a number if `z` is a number and a complex number otherwise.
 
 // Note: this should probably be recoded to use division by [1,-z], which is more accurate
 // and avoids overflow with large coefficients, but requires poly_div to support complex coefficients.  
-function polynomial(p, z, k, zk, total) =
+function polynomial(p, z, _k, _zk, _total) =
-   is_undef(k) ? polynomial(p, z, len(p)-1, is_num(z)? 1 : [1,0], is_num(z) ? 0 : [0,0]) :
+    is_undef(_k)  
-   k==-1 ? total :
+    ?   assert( is_vector(p), "Input polynomial coefficients must be a vector." )
-   polynomial(p, z, k-1, is_num(z) ? zk*z : C_times(zk,z), total+zk*p[k]);
+        let(p = _poly_trim(p))
+        assert( is_finite(z) || is_vector(z,2), "The value of `z` must be a real or a complex number." )
+        polynomial( p, 
+                    z, 
+                    len(p)-1, 
+                    is_num(z)? 1 : [1,0], 
+                    is_num(z) ? 0 : [0,0]) 
+    :   _k==0 
+        ? _total + +_zk*p[0]
+        : polynomial( p, 
+                      z, 
+                      _k-1, 
+                      is_num(z) ? _zk*z : C_times(_zk,z), 
+                      _total+_zk*p[_k]);
 
+function polynomial(p,z,k,total) =
+     is_undef(k)
+   ?    assert( is_vector(p) , "Input polynomial coefficients must be a vector." )
+        assert( is_finite(z) || is_vector(z,2), "The value of `z` must be a real or a complex number." )
+        polynomial( _poly_trim(p), z, 0, is_num(z) ? 0 : [0,0])
+   : k==len(p) ? total
+   : polynomial(p,z,k+1, is_num(z) ? total*z+p[k] : C_times(total,z)+[p[k],0]);
 
 // Function: poly_mult()
 // Usage:
@@ -1074,7 +1250,9 @@ function polynomial(p, z, k, zk, total) =
 //   computes the coefficient list of the product polynomial.  
 function poly_mult(p,q) = 
     is_undef(q) ?
-     assert(is_list(p) && (is_vector(p[0]) || p[0]==[]), "Invalid arguments to poly_mult")
+       assert( is_list(p) 
+               && []==[for(pi=p) if( !is_vector(pi) && pi!=[]) 0], 
+               "Invalid arguments to poly_mult")
        len(p)==2 ? poly_mult(p[0],p[1]) 
                  : poly_mult(p[0], poly_mult(select(p,1,-1)))
     :
@@ -1087,6 +1265,20 @@ function poly_mult(p,q) =
                  ])
      ]);        
      
+function poly_mult(p,q) = 
+    is_undef(q) ?
+       len(p)==2 ? poly_mult(p[0],p[1]) 
+                 : poly_mult(p[0], poly_mult(select(p,1,-1)))
+    :
+    assert( is_vector(p) && is_vector(q),"Invalid arguments to poly_mult")
+    _poly_trim( [
+                  for(n = [len(p)+len(q)-2:-1:0])
+                      sum( [for(i=[0:1:len(p)-1])
+                           let(j = len(p)+len(q)- 2 - n - i)
+                           if (j>=0 && j<len(q)) p[i]*q[j]
+                               ])
+                   ]);
+
     
 // Function: poly_div()
 // Usage:
@@ -1096,9 +1288,13 @@ function poly_mult(p,q) =
 //    a list of two polynomials, [quotient, remainder].  If the division has no remainder then
 //    the zero polynomial [] is returned for the remainder.  Similarly if the quotient is zero
 //    the returned quotient will be [].  
-function poly_div(n,d,q=[]) =
+function poly_div(n,d,q) =
-    assert(len(d)>0 && d[0]!=0 , "Denominator is zero or has leading zero coefficient")
+    is_undef(q) 
-    len(n)<len(d) ? [q,_poly_trim(n)] : 
+    ?   assert( is_vector(n) && is_vector(d) , "Invalid polynomials." )
+        let( d = _poly_trim(d) )
+        assert( d!=[0] , "Denominator cannot be a zero polynomial." )
+        poly_div(n,d,q=[])
+    :   len(n)<len(d) ? [q,_poly_trim(n)] : 
         let(
           t = n[0] / d[0], 
           newq = concat(q,[t]),
@@ -1115,7 +1311,7 @@ function poly_div(n,d,q=[]) =
 //    or give epsilon for approximate zeros.  
 function _poly_trim(p,eps=0) =
     let( nz = [for(i=[0:1:len(p)-1]) if ( !approx(p[i],0,eps)) i])
-  len(nz)==0 ? [] : select(p,nz[0],-1);
+    len(nz)==0 ? [0] : select(p,nz[0],-1);
 
 
 // Function: poly_add()
@@ -1124,6 +1320,7 @@ function _poly_trim(p,eps=0) =
 // Description:
 //    Computes the sum of two polynomials.  
 function poly_add(p,q) = 
+    assert( is_vector(p) && is_vector(q), "Invalid input polynomial(s)." )
     let(  plen = len(p),
           qlen = len(q),
           long = plen>qlen ? p : q,
@@ -1150,11 +1347,10 @@ function poly_add(p,q) =
 //
 // Dario Bini. "Numerical computation of polynomial zeros by means of Aberth's Method", Numerical Algorithms, Feb 1996.
 // https://www.researchgate.net/publication/225654837_Numerical_computation_of_polynomial_zeros_by_means_of_Aberth's_method
-
 function poly_roots(p,tol=1e-14,error_bound=false) =
-  assert(p!=[], "Input polynomial must have a nonzero coefficient")
+    assert( is_vector(p), "Invalid polynomial." )
-  assert(is_vector(p), "Input must be a vector")
+    let( p = _poly_trim(p,eps=0) )
-  p[0] == 0 ? poly_roots(slice(p,1,-1),tol=tol,error_bound=error_bound) :    // Strip leading zero coefficients
+    assert( p!=[0], "Input polynomial cannot be zero." )
     p[len(p)-1] == 0 ?                                       // Strip trailing zero coefficients
         let( solutions = poly_roots(select(p,0,-2),tol=tol, error_bound=error_bound))
         (error_bound ? [ [[0,0], each solutions[0]], [0, each solutions[1]]]
@@ -1182,6 +1378,7 @@ function poly_roots(p,tol=1e-14,error_bound=false) =
       )
       error_bound ? [roots, error] : roots;
 
+// Internal function
 // p = polynomial
 // pderiv = derivative polynomial of p
 // z = current guess for the roots
@@ -1222,12 +1419,16 @@ function _poly_roots(p, pderiv, s, z, tol, i=0) =
 //   tol = tolerance for the complex polynomial root finder
 
 function real_roots(p,eps=undef,tol=1e-14) =
+    assert( is_vector(p), "Invalid polynomial." )
+    let( p = _poly_trim(p,eps=0) )
+    assert( p!=[0], "Input polynomial cannot be zero." )
     let( 
        roots_err = poly_roots(p,error_bound=true),
        roots = roots_err[0],
        err = roots_err[1]
     )
-   is_def(eps) ? [for(z=roots) if (abs(z.y)/(1+norm(z))<eps) z.x]
+    is_def(eps) 
+    ? [for(z=roots) if (abs(z.y)/(1+norm(z))<eps) z.x]
     : [for(i=idx(roots)) if (abs(roots[i].y)<=err[i]) roots[i].x];
 
 // vim: expandtab tabstop=4 shiftwidth=4 softtabstop=4 nowrap

tests/polyhedra.scad

@@ -2,10 +2,10 @@ include<../std.scad>
 include<../polyhedra.scad>
 
 
-$fn=96;
-
 if (true) {
 
+   $fn=96;
+
   // Display of all solids with insphere, midsphere and circumsphere
     
   for(i=[0:len(_polyhedra_)-1]) {

tests/test_arrays.scad

@@ -1,37 +1,14 @@
 include <../std.scad>
 
-// List/Array Ops
 
-module test_repeat() {
+// Section: List Query Operations
-    assert(repeat(1, 4) == [1,1,1,1]);
+
-    assert(repeat(8, [2,3]) == [[8,8,8], [8,8,8]]);
+module test_is_simple_list() {
-    assert(repeat(0, [2,2,3]) == [[[0,0,0],[0,0,0]], [[0,0,0],[0,0,0]]]);
+		assert(is_simple_list([1,2,3,4]));
-    assert(repeat([1,2,3],3) == [[1,2,3], [1,2,3], [1,2,3]]);
+		assert(is_simple_list([]));
+		assert(!is_simple_list([1,2,[3,4]]));
 } 
-test_repeat();
+test_is_simple_list();
-
-
-module test_in_list() {
-    assert(in_list("bar", ["foo", "bar", "baz"]));
-    assert(!in_list("bee", ["foo", "bar", "baz"]));
-    assert(in_list("bar", [[2,"foo"], [4,"bar"], [3,"baz"]], idx=1));
-    assert(!in_list(undef, [3,4,5]));
-    assert(in_list(undef,[3,4,undef,5]));
-    assert(!in_list(3,[]));
-    assert(!in_list(3,[4,5,[3]]));
-    
-}
-test_in_list();
-
-
-module test_slice() {
-    assert(slice([3,4,5,6,7,8,9], 3, 5) == [6,7]);
-    assert(slice([3,4,5,6,7,8,9], 2, -1) == [5,6,7,8,9]);
-    assert(slice([3,4,5,6,7,8,9], 1, 1) == []);
-    assert(slice([3,4,5,6,7,8,9], 6, -1) == [9]);
-    assert(slice([3,4,5,6,7,8,9], 2, -2) == [5,6,7,8]);
-}
-test_slice();
 
 
 module test_select() {
@@ -49,6 +26,74 @@ module test_select() {
 test_select();
 
 
+module test_slice() {
+    assert(slice([3,4,5,6,7,8,9], 3, 5) == [6,7]);
+    assert(slice([3,4,5,6,7,8,9], 2, -1) == [5,6,7,8,9]);
+    assert(slice([3,4,5,6,7,8,9], 1, 1) == []);
+    assert(slice([3,4,5,6,7,8,9], 6, -1) == [9]);
+    assert(slice([3,4,5,6,7,8,9], 2, -2) == [5,6,7,8]);
+    assert(slice([], 2, -2) == []);
+}
+test_slice();
+
+
+module test_in_list() {
+    assert(in_list("bar", ["foo", "bar", "baz"]));
+    assert(!in_list("bee", ["foo", "bar", "baz"]));
+    assert(in_list("bar", [[2,"foo"], [4,"bar"], [3,"baz"]], idx=1));
+    assert(!in_list("bee", ["foo", "bar", ["bee"]]));
+    assert(in_list(NAN, [NAN])==false);
+    assert(!in_list(undef, [3,4,5]));
+    assert(in_list(undef,[3,4,undef,5]));
+    assert(!in_list(3,[]));
+    assert(!in_list(3,[4,5,[3]]));
+}
+test_in_list();
+
+
+module test_min_index() {
+    assert(min_index([5,3,9,6,2,7,8,2,1])==8);
+    assert(min_index([5,3,9,6,2,7,8,2,7],all=true)==[4,7]);
+//    assert(min_index([],all=true)==[]);
+}
+test_min_index();
+
+
+module test_max_index() {
+    assert(max_index([5,3,9,6,2,7,8,9,1])==2);
+    assert(max_index([5,3,9,6,2,7,8,9,7],all=true)==[2,7]);
+//    assert(max_index([],all=true)==[]);
+}
+test_max_index();
+
+
+module test_list_increasing() {
+    assert(list_increasing([1,2,3,4]) == true);
+    assert(list_increasing([1,3,2,4]) == false);
+    assert(list_increasing([4,3,2,1]) == false);
+}
+test_list_increasing();
+
+
+module test_list_decreasing() {
+    assert(list_decreasing([1,2,3,4]) == false);
+    assert(list_decreasing([4,2,3,1]) == false);
+    assert(list_decreasing([4,3,2,1]) == true);
+}
+test_list_decreasing();
+
+// Section: Basic List Generation
+
+module test_repeat() {
+    assert(repeat(1, 4) == [1,1,1,1]);
+    assert(repeat(8, [2,3]) == [[8,8,8], [8,8,8]]);
+    assert(repeat(0, [2,2,3]) == [[[0,0,0],[0,0,0]], [[0,0,0],[0,0,0]]]);
+    assert(repeat([1,2,3],3) == [[1,2,3], [1,2,3], [1,2,3]]);
+    assert(repeat(4, [2,-1]) == [[], []]);
+}
+test_repeat();
+
+
 module test_list_range() {
     assert(list_range(4) == [0,1,2,3]);
     assert(list_range(n=4, step=2) == [0,2,4,6]);
@@ -66,6 +111,8 @@ test_list_range();
 
 module test_reverse() {
     assert(reverse([3,4,5,6]) == [6,5,4,3]);
+    assert(reverse("abcd") == ["d","c","b","a"]);
+    assert(reverse([]) == []);
 }
 test_reverse();
 
@@ -90,6 +137,8 @@ module test_deduplicate() {
     assert(deduplicate(closed=true, [8,3,4,4,4,8,2,3,3,8,8]) == [8,3,4,8,2,3]);
     assert(deduplicate("Hello") == ["H","e","l","o"]);
     assert(deduplicate([[3,4],[7,1.99],[7,2],[1,4]],eps=0.1) == [[3,4],[7,2],[1,4]]);
+    assert(deduplicate([], closed=true) == []);
+    assert(deduplicate([[1,[1,[undef]]],[1,[1,[undef]]],[1,[2]],[1,[2,[0]]]])==[[1, [1,[undef]]],[1,[2]],[1,[2,[0]]]]);
 }
 test_deduplicate();
 
@@ -148,22 +197,6 @@ module test_list_bset() {
 test_list_bset();
 
 
-module test_list_increasing() {
-    assert(list_increasing([1,2,3,4]) == true);
-    assert(list_increasing([1,3,2,4]) == false);
-    assert(list_increasing([4,3,2,1]) == false);
-}
-test_list_increasing();
-
-
-module test_list_decreasing() {
-    assert(list_decreasing([1,2,3,4]) == false);
-    assert(list_decreasing([4,2,3,1]) == false);
-    assert(list_decreasing([4,3,2,1]) == true);
-}
-test_list_decreasing();
-
-
 module test_list_shortest() {
     assert(list_shortest(["foobar", "bazquxx", "abcd"]) == 4);
 }
@@ -315,6 +348,13 @@ test_set_intersection();
 // Arrays
 
 
+module test_add_scalar() {
+    assert(add_scalar([1,2,3],3) == [4,5,6]);
+    assert(add_scalar([[1,2,3],[3,4,5]],3) == [[4,5,6],[6,7,8]]);
+}
+test_add_scalar();
+
+
 module test_subindex() {
     v = [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]];
     assert(subindex(v,2) == [3, 7, 11, 15]);
@@ -402,10 +442,18 @@ test_array_group();
 
 module test_flatten() {
     assert(flatten([[1,2,3], [4,5,[6,7,8]]]) == [1,2,3,4,5,[6,7,8]]);
+    assert(flatten([]) == []);
 }
 test_flatten();
 
 
+module test_full_flatten() {
+    assert(full_flatten([[1,2,3], [4,5,[6,[7],8]]]) == [1,2,3,4,5,6,7,8]);
+    assert(full_flatten([]) == []);
+}
+test_full_flatten();
+
+
 module test_array_dim() {
     assert(array_dim([[[1,2,3],[4,5,6]],[[7,8,9],[10,11,12]]]) == [2,2,3]);
     assert(array_dim([[[1,2,3],[4,5,6]],[[7,8,9],[10,11,12]]], 0) == 2);

tests/test_common.scad

@@ -18,6 +18,10 @@ module test_typeof() {
     assert(typeof([0:1:5]) == "range");
     assert(typeof([-3:2:5]) == "range");
     assert(typeof([10:-2:-10]) == "range");
+    assert(typeof([0:NAN:INF]) == "invalid");
+    assert(typeof([0:"a":INF]) == "undef"); 
+    assert(typeof([0:[]:INF]) == "undef"); 
+    assert(typeof([true:1:INF]) == "undef"); 
 }
 test_typeof();
 
@@ -102,6 +106,8 @@ module test_is_int() {
     assert(!is_int(-99.1));
     assert(!is_int(99.1));
     assert(!is_int(undef));
+    assert(!is_int(INF));
+    assert(!is_int(NAN));
     assert(!is_int(false));
     assert(!is_int(true));
     assert(!is_int("foo"));
@@ -124,6 +130,8 @@ module test_is_integer() {
     assert(!is_integer(-99.1));
     assert(!is_integer(99.1));
     assert(!is_integer(undef));
+    assert(!is_integer(INF));
+    assert(!is_integer(NAN));
     assert(!is_integer(false));
     assert(!is_integer(true));
     assert(!is_integer("foo"));
@@ -161,16 +169,33 @@ module test_is_range() {
     assert(!is_range(5));
     assert(!is_range(INF));
     assert(!is_range(-INF));
-    assert(!is_nan(NAN));
     assert(!is_range(""));
     assert(!is_range("foo"));
     assert(!is_range([]));
     assert(!is_range([3,4,5]));
+    assert(!is_range([INF:4:5]));
+    assert(!is_range([3:NAN:5]));
+    assert(!is_range([3:4:"a"]));
     assert(is_range([3:1:5]));
 }
-test_is_nan();
+test_is_range();
 
 
+module test_valid_range() {
+    assert(valid_range([0:0]));
+    assert(valid_range([0:1:0]));
+    assert(valid_range([0:1:10]));
+    assert(valid_range([0.1:1.1:2.1]));
+    assert(valid_range([0:-1:0]));
+    assert(valid_range([10:-1:0]));
+    assert(valid_range([2.1:-1.1:0.1]));
+    assert(!valid_range([10:1:0]));
+    assert(!valid_range([2.1:1.1:0.1]));
+    assert(!valid_range([0:-1:10]));
+    assert(!valid_range([0.1:-1.1:2.1]));
+}
+test_valid_range();
+
 module test_is_list_of() {
     assert(is_list_of([3,4,5], 0));
     assert(!is_list_of([3,4,undef], 0));
@@ -181,10 +206,14 @@ module test_is_list_of() {
 }
 test_is_list_of();
 
-
 module test_is_consistent() {
+    assert(is_consistent([]));
+    assert(is_consistent([[],[]]));
     assert(is_consistent([3,4,5]));
     assert(is_consistent([[3,4],[4,5],[6,7]]));
+    assert(is_consistent([[[3],4],[[4],5]]));
+    assert(!is_consistent(5));
+    assert(!is_consistent(undef));
     assert(!is_consistent([[3,4,5],[3,4]]));
     assert(is_consistent([[3,[3,4,[5]]], [5,[2,9,[9]]]]));
     assert(!is_consistent([[3,[3,4,[5]]], [5,[2,9,9]]]));
@@ -331,11 +360,25 @@ module test_scalar_vec3() {
     assert(scalar_vec3([3]) == [3,0,0]);
     assert(scalar_vec3([3,4]) == [3,4,0]);
     assert(scalar_vec3([3,4],dflt=1) == [3,4,1]);
+    assert(scalar_vec3([3,"a"],dflt=1) == [3,"a",1]);
+    assert(scalar_vec3([3,[2]],dflt=1) == [3,[2],1]);
     assert(scalar_vec3([3],dflt=1) == [3,1,1]);
     assert(scalar_vec3([3,4,5]) == [3,4,5]);
     assert(scalar_vec3([3,4,5,6]) == [3,4,5]);
+    assert(scalar_vec3([3,4,5,6]) == [3,4,5]);
 }
 test_scalar_vec3();
 
 
+module test_segs() {
+    assert_equal(segs(50,$fn=8), 8);
+    assert_equal(segs(50,$fa=2,$fs=2), 158);
+    assert(segs(1)==5);
+    assert(segs(11)==30);
+  //  assert(segs(1/0)==5);
+  //  assert(segs(0/0)==5);
+  //  assert(segs(undef)==5);
+}
+test_segs();
+
 // vim: expandtab tabstop=4 shiftwidth=4 softtabstop=4 nowrap

tests/test_math.scad

@@ -110,6 +110,8 @@ module test_approx() {
     assert_equal(approx(1/3, 0.3333333333), true);
     assert_equal(approx(-1/3, -0.3333333333), true);
     assert_equal(approx(10*[cos(30),sin(30)], 10*[sqrt(3)/2, 1/2]), true);
+    assert_equal(approx([1,[1,undef]], [1+1e-12,[1,true]]), false);
+    assert_equal(approx([1,[1,undef]], [1+1e-12,[1,undef]]), true);
 }
 test_approx();
 
@@ -389,7 +391,6 @@ module test_mean() {
 }
 test_mean();
 
-
 module test_median() {
     assert_equal(median([2,3,7]), 4.5);
     assert_equal(median([[1,2,3], [3,4,5], [8,9,10]]), [4.5,5.5,6.5]);
@@ -397,6 +398,16 @@ module test_median() {
 test_median();
 
 
+module test_convolve() {
+    assert_equal(convolve([],[1,2,1]), []);
+    assert_equal(convolve([1,1],[]), []);
+    assert_equal(convolve([1,1],[1,2,1]), [1,3,3,1]);
+    assert_equal(convolve([1,2,3],[1,2,1]), [1,4,8,8,3]);
+}
+test_convolve();
+
+
+
 module test_matrix_inverse() {
     assert_approx(matrix_inverse(rot([20,30,40])), [[0.663413948169,0.556670399226,-0.5,0],[-0.47302145844,0.829769465589,0.296198132726,0],[0.579769465589,0.0400087565481,0.813797681349,0],[0,0,0,1]]);
 }
@@ -583,6 +594,24 @@ module test_factorial() {
 }
 test_factorial();
 
+module test_binomial() {
+    assert_equal(binomial(1), [1,1]);
+    assert_equal(binomial(2), [1,2,1]);
+    assert_equal(binomial(3), [1,3,3,1]);
+    assert_equal(binomial(5), [1,5,10,10,5,1]);
+}
+test_binomial();
+
+module test_binomial_coefficient() {
+    assert_equal(binomial_coefficient(2,1), 2);
+    assert_equal(binomial_coefficient(3,2), 3);
+    assert_equal(binomial_coefficient(4,2), 6);
+    assert_equal(binomial_coefficient(10,7), 120);
+    assert_equal(binomial_coefficient(10,7), binomial(10)[7]);
+    assert_equal(binomial_coefficient(15,4), binomial(15)[4]);
+}
+test_binomial_coefficient();
+
 
 module test_gcd() {
     assert_equal(gcd(15,25), 5);
@@ -682,6 +711,7 @@ test_linear_solve();
 
 module test_outer_product(){
   assert_equal(outer_product([1,2,3],[4,5,6]), [[4,5,6],[8,10,12],[12,15,18]]);
+  assert_equal(outer_product([1,2],[4,5,6]), [[4,5,6],[8,10,12]]);
   assert_equal(outer_product([9],[7]), [[63]]);
 }
 test_outer_product();
@@ -782,8 +812,10 @@ test_deriv3();
 
 
 module test_polynomial(){
-  assert_equal(polynomial([],12),0);
+  assert_equal(polynomial([0],12),0);
-  assert_equal(polynomial([],[12,4]),[0,0]);
+  assert_equal(polynomial([0],[12,4]),[0,0]);
+//  assert_equal(polynomial([],12),0);
+//  assert_equal(polynomial([],[12,4]),[0,0]);
   assert_equal(polynomial([1,2,3,4],3),58);
   assert_equal(polynomial([1,2,3,4],[3,-1]),[47,-41]);
   assert_equal(polynomial([0,0,2],4),2);
@@ -879,16 +911,20 @@ test_qr_factor();
 
 module test_poly_mult(){
   assert_equal(poly_mult([3,2,1],[4,5,6,7]),[12,23,32,38,20,7]);
-  assert_equal(poly_mult([3,2,1],[]),[]);
+  assert_equal(poly_mult([3,2,1],[0]),[0]);
+//  assert_equal(poly_mult([3,2,1],[]),[]);
   assert_equal(poly_mult([[1,2],[3,4],[5,6]]), [15,68,100,48]);
-  assert_equal(poly_mult([[1,2],[],[5,6]]), []);
+  assert_equal(poly_mult([[1,2],[0],[5,6]]), [0]);
-  assert_equal(poly_mult([[3,4,5],[0,0,0]]),[]);
+//  assert_equal(poly_mult([[1,2],[],[5,6]]), []);
+  assert_equal(poly_mult([[3,4,5],[0,0,0]]),[0]);
+//  assert_equal(poly_mult([[3,4,5],[0,0,0]]),[]);
 }
 test_poly_mult();
 
 
 module test_poly_div(){
-  assert_equal(poly_div(poly_mult([4,3,3,2],[2,1,3]), [2,1,3]),[[4,3,3,2],[]]);
+  assert_equal(poly_div(poly_mult([4,3,3,2],[2,1,3]), [2,1,3]),[[4,3,3,2],[0]]);
+//  assert_equal(poly_div(poly_mult([4,3,3,2],[2,1,3]), [2,1,3]),[[4,3,3,2],[]]);
   assert_equal(poly_div([1,2,3,4],[1,2,3,4,5]), [[], [1,2,3,4]]);
   assert_equal(poly_div(poly_add(poly_mult([1,2,3,4],[2,0,2]), [1,1,2]), [1,2,3,4]), [[2,0,2],[1,1,2]]);
   assert_equal(poly_div([1,2,3,4], [1,-3]), [[1,5,18],[58]]);
@@ -899,7 +935,8 @@ test_poly_div();
 module test_poly_add(){
   assert_equal(poly_add([2,3,4],[3,4,5,6]),[3,6,8,10]);
   assert_equal(poly_add([1,2,3,4],[-1,-2,3,4]), [6,8]);
-  assert_equal(poly_add([1,2,3],-[1,2,3]),[]);
+  assert_equal(poly_add([1,2,3],-[1,2,3]),[0]);
+//  assert_equal(poly_add([1,2,3],-[1,2,3]),[]);
 }
 test_poly_add();
 

tests/test_vectors.scad

@@ -9,6 +9,7 @@ module test_is_vector() {
     assert(is_vector(1) == false);
     assert(is_vector("foo") == false);
     assert(is_vector(true) == false);
+
     assert(is_vector([0,0,0],zero=true) == true);
     assert(is_vector([0,0,0],zero=false) == false);
     assert(is_vector([0,1,0],zero=true) == false);
@@ -17,13 +18,6 @@ module test_is_vector() {
 test_is_vector();
 
 
-module test_add_scalar() {
-    assert(add_scalar([1,2,3],3) == [4,5,6]);
-    assert(add_scalar([[1,2,3],[3,4,5]],3) == [[4,5,6],[6,7,8]]);
-}
-test_add_scalar();
-
-
 module test_vfloor() {
     assert_equal(vfloor([2.0, 3.14, 18.9, 7]), [2,3,18,7]);
     assert_equal(vfloor([-2.0, -3.14, -18.9, -7]), [-2,-4,-19,-7]);

vectors.scad

@@ -19,8 +19,6 @@
 // Arguments:
 //   v = The value to test to see if it is a vector.
 //   length = If given, make sure the vector is `length` items long.
-//   zero = If false, require that the length of the vector is not approximately zero.  If true, require the length of the vector to be approx zero-length.  Default: `undef` (don't check vector length.)
-//   eps = The minimum vector length that is considered non-zero.  Default: `EPSILON` (`1e-9`)
 // Example:
 //   is_vector(4);                          // Returns false
 //   is_vector([4,true,false]);             // Returns false
@@ -30,31 +28,16 @@
 //   is_vector([3,4,5],3);                  // Returns true
 //   is_vector([3,4,5],4);                  // Returns true
 //   is_vector([]);                         // Returns false
-//   is_vector([0,0,0],zero=true);   // Returns true
+//   is_vector([0,4,0],3,zero=false);     // Returns true
 //   is_vector([0,0,0],zero=false);       // Returns false
-//   is_vector([0,1,0],zero=true);   // Returns false
+//   is_vector([0,0,1e-12],zero=false);   // Returns false
-//   is_vector([0,0,1],zero=false);  // Returns true
+//   is_vector([],zero=false);            // Returns false
 function is_vector(v,length,zero,eps=EPSILON) =
     is_list(v) && is_num(0*(v*v))
     && (is_undef(length) || len(v)==length)
     && (is_undef(zero) || ((norm(v) >= eps) == !zero));
 
 
-// Function: add_scalar()
-// Usage:
-//   add_scalar(v,s);
-// Description:
-//   Given a vector and a scalar, returns the vector with the scalar added to each item in it.
-//   If given a list of vectors, recursively adds the scalar to the each vector.
-// Arguments:
-//   v = The initial list of values.
-//   s = A scalar value to add to every item in the vector.
-// Example:
-//   add_scalar([1,2,3],3);            // Returns: [4,5,6]
-//   add_scalar([[1,2,3],[3,4,5]],3);  // Returns: [[4,5,6],[6,7,8]]
-function add_scalar(v,s) = [for (x=v) is_list(x)? add_scalar(x,s) : x+s];
-
-
 // Function: vang()
 // Usage:
 //   theta = vang([X,Y]);
@@ -63,6 +46,7 @@ function add_scalar(v,s) = [for (x=v) is_list(x)? add_scalar(x,s) : x+s];
 //   Given a 2D vector, returns the angle in degrees counter-clockwise from X+ on the XY plane.
 //   Given a 3D vector, returns [THETA,PHI] where THETA is the number of degrees counter-clockwise from X+ on the XY plane, and PHI is the number of degrees up from the X+ axis along the XZ plane.
 function vang(v) =
+    assert( is_vector(v,2) || is_vector(v,3) , "Invalid vector")
     len(v)==2? atan2(v.y,v.x) :
     let(res=xyz_to_spherical(v)) [res[1], 90-res[2]];
 
@@ -70,13 +54,19 @@ function vang(v) =
 // Function: vmul()
 // Description:
 //   Element-wise vector multiplication.  Multiplies each element of vector `v1` by
-//   the corresponding element of vector `v2`.  Returns a vector of the products.
+//   the corresponding element of vector `v2`. The vectors should have the same dimension.
+//   Returns a vector of the products.
 // Arguments:
 //   v1 = The first vector.
 //   v2 = The second vector.
 // Example:
 //   vmul([3,4,5], [8,7,6]);  // Returns [24, 28, 30]
-function vmul(v1, v2) = [for (i = [0:1:len(v1)-1]) v1[i]*v2[i]];
+function vmul(v1, v2) = 
+// this thighter check can be done yet because it would break other codes in the library
+//    assert( is_vector(v1) && is_vector(v2,len(v1)), "Incompatible vectors")
+    assert( is_vector(v1) && is_vector(v2), "Invalid vector(s)")
+    [for (i = [0:1:len(v1)-1]) v1[i]*v2[i]];
+    
 
 
 // Function: vdiv()
@@ -88,7 +78,9 @@ function vmul(v1, v2) = [for (i = [0:1:len(v1)-1]) v1[i]*v2[i]];
 //   v2 = The second vector.
 // Example:
 //   vdiv([24,28,30], [8,7,6]);  // Returns [3, 4, 5]
-function vdiv(v1, v2) = [for (i = [0:1:len(v1)-1]) v1[i]/v2[i]];
+function vdiv(v1, v2) = 
+    assert( is_vector(v1) && is_vector(v2,len(v1)), "Incompatible vectors")
+    [for (i = [0:1:len(v1)-1]) v1[i]/v2[i]];
 
 
 // Function: vabs()
@@ -97,19 +89,25 @@ function vdiv(v1, v2) = [for (i = [0:1:len(v1)-1]) v1[i]/v2[i]];
 //   v = The vector to get the absolute values of.
 // Example:
 //   vabs([-1,3,-9]);  // Returns: [1,3,9]
-function vabs(v) = [for (x=v) abs(x)];
+function vabs(v) =
+    assert( is_vector(v), "Invalid vector" ) 
+    [for (x=v) abs(x)];
 
 
 // Function: vfloor()
 // Description:
 //   Returns the given vector after performing a `floor()` on all items.
-function vfloor(v) = [for (x=v) floor(x)];
+function vfloor(v) =
+    assert( is_vector(v), "Invalid vector" ) 
+    [for (x=v) floor(x)];
 
 
 // Function: vceil()
 // Description:
 //   Returns the given vector after performing a `ceil()` on all items.
-function vceil(v) = [for (x=v) ceil(x)];
+function vceil(v) =
+    assert( is_vector(v), "Invalid vector" ) 
+    [for (x=v) ceil(x)];
 
 
 // Function: unit()
@@ -137,6 +135,7 @@ function unit(v, error=[[["ASSERT"]]]) =
 // Function: vector_angle()
 // Usage:
 //   vector_angle(v1,v2);
+//   vector_angle([v1,v2]);
 //   vector_angle(PT1,PT2,PT3);
 //   vector_angle([PT1,PT2,PT3]);
 // Description:
@@ -156,20 +155,21 @@ function unit(v, error=[[["ASSERT"]]]) =
 //   vector_angle([10,0,10], [0,0,0], [-10,10,0]);  // Returns: 120
 //   vector_angle([[10,0,10], [0,0,0], [-10,10,0]]);  // Returns: 120
 function vector_angle(v1,v2,v3) =
-    let(
+    assert( ( is_undef(v3) && ( is_undef(v2) || same_shape(v1,v2) ) )
-        vecs = !is_undef(v3)? [v1-v2,v3-v2] :
+            || is_consistent([v1,v2,v3]) ,
+            "Bad arguments.")
+    assert( is_vector(v1) || is_consistent(v1), "Bad arguments.") 
+    let( vecs = ! is_undef(v3) ? [v1-v2,v3-v2] :
                 ! is_undef(v2) ? [v1,v2] :
-            len(v1) == 3? [v1[0]-v1[1],v1[2]-v1[1]] :
+                len(v1) == 3   ? [v1[0]-v1[1], v1[2]-v1[1]] 
-            len(v1) == 2? v1 :
+                               : v1
-            assert(false, "Bad arguments to vector_angle()"),
-        is_valid = is_vector(vecs[0]) && is_vector(vecs[1]) && vecs[0]*0 == vecs[1]*0
     )
-    assert(is_valid, "Bad arguments to vector_angle()")
+    assert(is_vector(vecs[0],2) || is_vector(vecs[0],3), "Bad arguments.")
     let(
         norm0 = norm(vecs[0]),
         norm1 = norm(vecs[1])
     )
-    assert(norm0>0 && norm1>0,"Zero length vector given to vector_angle()")
+    assert(norm0>0 && norm1>0, "Zero length vector.")
     // NOTE: constrain() corrects crazy FP rounding errors that exceed acos()'s domain.
     acos(constrain((vecs[0]*vecs[1])/(norm0*norm1), -1, 1));
     
@@ -177,13 +177,14 @@ function vector_angle(v1,v2,v3) =
 // Function: vector_axis()
 // Usage:
 //   vector_axis(v1,v2);
+//   vector_axis([v1,v2]);
 //   vector_axis(PT1,PT2,PT3);
 //   vector_axis([PT1,PT2,PT3]);
 // Description:
 //   If given a single list of two vectors, like `vector_axis([V1,V2])`, returns the vector perpendicular the two vectors V1 and V2.
-//   If given a single list of three points, like `vector_axis([A,B,C])`, returns the vector perpendicular the line segments AB and BC.
+//   If given a single list of three points, like `vector_axis([A,B,C])`, returns the vector perpendicular to the plane through a, B and C.
-//   If given two vectors, like `vector_axis(V1,V1)`, returns the vector perpendicular the two vectors V1 and V2.
+//   If given two vectors, like `vector_axis(V1,V2)`, returns the vector perpendicular to the two vectors V1 and V2.
-//   If given three points, like `vector_axis(A,B,C)`, returns the vector perpendicular the line segments AB and BC.
+//   If given three points, like `vector_axis(A,B,C)`, returns the vector perpendicular to the plane through a, B and C.
 // Arguments:
 //   v1 = First vector or point.
 //   v2 = Second vector or point.
@@ -199,20 +200,14 @@ function vector_axis(v1,v2=undef,v3=undef) =
     is_vector(v3)
     ?   assert(is_consistent([v3,v2,v1]), "Bad arguments.")
         vector_axis(v1-v2, v3-v2)
-    :
+    :   assert( is_undef(v3), "Bad arguments.")
-    assert( is_undef(v3), "Bad arguments.")
         is_undef(v2)
         ?   assert( is_list(v1), "Bad arguments.")
             len(v1) == 2 
             ?   vector_axis(v1[0],v1[1]) 
             :   vector_axis(v1[0],v1[1],v1[2])
-    :
+        :   assert( is_vector(v1,zero=false) && is_vector(v2,zero=false) && is_consistent([v1,v2])
-    assert(
+                    , "Bad arguments.")  
-        is_vector(v1,zero=false) &&
-            is_vector(v2,zero=false) &&
-            is_consistent([v1,v2]),
-        "Bad arguments."
-    )
             let(
               eps = 1e-6,
               w1 = point3d(v1/norm(v1)),
@@ -223,4 +218,5 @@ function vector_axis(v1,v2=undef,v3=undef) =
             ) unit(cross(w1,w3));
 
 
+
 // vim: expandtab tabstop=4 shiftwidth=4 softtabstop=4 nowrap