improved back_substitute, cleaned up a few other functions, removed

some non-breaking space characters.

math.scad

@@ -773,26 +773,26 @@ function _qr_factor(A,Q, column, m, n) =
 //   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 [].  
-function back_substitute(R, b, x=[],transpose = false) =
+function back_substitute(R, b, transpose = false) =
     assert(is_matrix(R, square=true))
     let(n=len(R))
     assert(is_vector(b,n) || is_matrix(b,n),str("R and b are not compatible in back_substitute ",n, len(b)))
-    !is_vector(b) ? transpose([for(i=[0:len(b[0])-1]) back_substitute(R,subindex(b,i),transpose=transpose)]) :
-    transpose?
-        reverse(back_substitute(
-            [for(i=[0:n-1]) [for(j=[0:n-1]) R[n-1-j][n-1-i]]],
-            reverse(b), x, false
-        )) :
-    len(x) == n ? x :
-    let(
-        ind = n - len(x) - 1
-    )
-    R[ind][ind] == 0 ? [] :
-    let(
-        newvalue =
-            len(x)==0? b[ind]/R[ind][ind] : 
-            (b[ind]-select(R[ind],ind+1,-1) * x)/R[ind][ind]
-    ) back_substitute(R, b, concat([newvalue],x));
+    transpose
+      ? reverse(_back_substitute([for(i=[0:n-1]) [for(j=[0:n-1]) R[n-1-j][n-1-i]]],
+                                 reverse(b)))  
+      : _back_substitute(R,b);
+
+function _back_substitute(R, b, x=[]) =
+      let(n=len(R))
+      len(x) == n ? x
+    : let(ind = n - len(x) - 1)
+      R[ind][ind] == 0 ? []
+    : let(
+          newvalue = len(x)==0
+            ? b[ind]/R[ind][ind]
+            : (b[ind]-select(R[ind],ind+1,-1) * x)/R[ind][ind]
+      )
+      _back_substitute(R, b, concat([newvalue],x));
 
 
 // Function: det2()
@@ -1120,19 +1120,21 @@ function _deriv_nonuniform(data, h, closed) =
 //   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? [
+    assert( L>=3, "Input list has less than 3 elements.") 
+    closed
+    ? [
         for(i=[0:1:L-1])
         (data[(i+1)%L]-2*data[i]+data[(L+i-1)%L])/h/h
-    ] :
+      ]
+    :
     let(
-        first = L<3? undef : 
+        first = 
             L==3? data[0] - 2*data[1] + data[2] :
             L==4? 2*data[0] - 5*data[1] + 4*data[2] - data[3] :
             (35*data[0] - 104*data[1] + 114*data[2] - 56*data[3] + 11*data[4])/12, 
-        last = L<3? undef :
+        last = 
             L==3? data[L-1] - 2*data[L-2] + data[L-3] :
             L==4? -2*data[L-1] + 5*data[L-2] - 4*data[L-3] + data[L-4] :
             (35*data[L-1] - 104*data[L-2] + 114*data[L-3] - 56*data[L-4] + 11*data[L-5])/12
@@ -1212,34 +1214,13 @@ function C_div(z1,z2) =
 //   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) =
-    is_undef(_k)  
-    ?   assert( is_vector(p), "Input polynomial coefficients must be a vector." )
-        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]);
+    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:
@@ -1266,18 +1247,18 @@ 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")
+  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]
-                               ])
-                   ]);
+         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()