Simplified bernoulli polynomials

(cherry picked from commit 3b557d56)

src/maths_functions.f90

@@ -253,44 +253,55 @@ CONTAINS
   END FUNCTION trilog
 
   FUNCTION BERNOULLI_POLYNOMIAL(n, x) result(res)
+    integer, parameter :: maxn = 15
     integer n
-    complex(kind=prec) :: x, res
-    select case(n)
-      case(1)
-        res = -1/2. + x
-      case(2)
-        res = 1/6. - x + x**2
-      case(3)
-        res = x/2. - 3*x**2/2. + x**3
-      case(4)
-        res = -1/30. + x**2 - 2*x**3 + x**4
-      case(5)
-        res = -x/6. + 5*x**3/3 - 5*x**4/2 + x**5
-      case(6)
-        res = 1/42. - x**2/2 + 5*x**4/2 - 3*x**5 + x**6
-      case(7)
-        res = x/6. - 7*x**3/6 + 7*x**5/2 - 7*x**6/2 + x**7
-      case(8)
-        res = -1/30. + 2*x**2/3 - 7*x**4/3 + 14*x**6/3 - 4*x**7 + x**8
-      case(9)
-        res = -3*x/10 + 2*x**3 - 21*x**5/5 + 6*x**7 - 9*x**8/2 + x**9
-      case(10)
-        res = 5/66. - 3*x**2/2 + 5*x**4 - 7*x**6 + 15*x**8/2 - 5*x**9 + x**10
-      case(11)
-        res = 5*x/6 - 11*x**3/2 + 11*x**5 - 11*x**7 + 55*x**9/6 - 11*x**10/2 + x**11
-      case(12)
-        res = -691/2730. + 5*x**2 - 33*x**4/2 + 22*x**6 - 33*x**8/2 + 11*x**10 - 6*x**11 + x**12
-      case(13)
-        res = -691*x/210 + 65*x**3/3 - 429*x**5/10 + 286*x**7/7 - 143*x**9/6 + 13*x**11 - 13*x**12/2 + x**13
-      case(14)
-        res = 7/6. - 691*x**2/30 + 455*x**4/6 - 1001*x**6/10 + 143*x**8/2 - 1001*x**10/30 + 91*x**12/6 - 7*x**13 + x**14
-      case(15)
-        res = 35*x/2 - 691*x**3/6 + 455*x**5/2 - 429*x**7/2 + 715*x**9/6 - 91*x**11/2 + 35*x**13/2 - 15*x**14/2 + x**15
-      case default
-        print*,"Bernoulli beyond 15 is not implemented"
-        stop
-    end select
+    restype :: x, res
+    restype :: xpow(maxn+1)
+    integer, parameter :: coeffN(maxn+1, maxn) = reshape((/ &
+        -   1, +   1,     0,     0,     0,     0,     0,     0,     0,     0,     0,     0,     0,     0,     0,     0, &
+        +   1, -   1, +   1,     0,     0,     0,     0,     0,     0,     0,     0,     0,     0,     0,     0,     0, &
+            0, +   1, -   3, +   1,     0,     0,     0,     0,     0,     0,     0,     0,     0,     0,     0,     0, &
+        -   1,     0, +   1, -   2, +   1,     0,     0,     0,     0,     0,     0,     0,     0,     0,     0,     0, &
+            0, -   1,     0, +   5, -   5, +   1,     0,     0,     0,     0,     0,     0,     0,     0,     0,     0, &
+        +   1,     0, -   1,     0, +   5, -   3, +   1,     0,     0,     0,     0,     0,     0,     0,     0,     0, &
+            0, +   1,     0, -   7,     0, +   7, -   7, +   1,     0,     0,     0,     0,     0,     0,     0,     0, &
+        -   1,     0, +   2,     0, -   7,     0, +  14, -   4, +   1,     0,     0,     0,     0,     0,     0,     0, &
+            0, -   3,     0, +   2,     0, -  21,     0, +   6, -   9, +   1,     0,     0,     0,     0,     0,     0, &
+        +   5,     0, -   3,     0, +   5,     0, -   7,     0, +  15, -   5, +   1,     0,     0,     0,     0,     0, &
+            0, +   5,     0, -  11,     0, +  11,     0, -  11,     0, +  55, -  11, +   1,     0,     0,     0,     0, &
+        - 691,     0, +   5,     0, -  33,     0, +  22,     0, -  33,     0, +  11, -   6, +   1,     0,     0,     0, &
+            0, - 691,     0, +  65,     0, - 429,     0, + 286,     0, - 143,     0, +  13, -  13, +   1,     0,     0, &
+        +   7,     0, - 691,     0, + 455,     0, -1001,     0, + 143,     0, -1001,     0, +  91, -   7, +   1,     0, &
+            0, +  35,     0, - 691,     0, + 455,     0, - 429,     0, + 715,     0, -  91,     0, +  35, -  15, +   1 /), &
+            (/maxn+1, maxn/))
+    integer, parameter :: coeffD(maxn+1, maxn) = reshape((/ &
+        +   2, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, &
+        +   6, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, &
+        +   1, +   2, +   2, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, &
+        +  30, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, &
+        +   1, +   6, +   1, +   3, +   2, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, &
+        +  42, +   1, +   2, +   1, +   2, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, &
+        +   1, +   6, +   1, +   6, +   1, +   2, +   2, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, &
+        +  30, +   1, +   3, +   1, +   3, +   1, +   3, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, +   1, &
+        +   1, +  10, +   1, +   1, +   1, +   5, +   1, +   1, +   2, +   1, +   1, +   1, +   1, +   1, +   1, +   1, &
+        +  66, +   1, +   2, +   1, +   1, +   1, +   1, +   1, +   2, +   1, +   1, +   1, +   1, +   1, +   1, +   1, &
+        +   1, +   6, +   1, +   2, +   1, +   1, +   1, +   1, +   1, +   6, +   2, +   1, +   1, +   1, +   1, +   1, &
+        +2730, +   1, +   1, +   1, +   2, +   1, +   1, +   1, +   2, +   1, +   1, +   1, +   1, +   1, +   1, +   1, &
+        +   1, + 210, +   1, +   3, +   1, +  10, +   1, +   7, +   1, +   6, +   1, +   1, +   2, +   1, +   1, +   1, &
+        +   6, +   1, +  30, +   1, +   6, +   1, +  10, +   1, +   2, +   1, +  30, +   1, +   6, +   1, +   1, +   1, &
+        +   1, +   2, +   1, +   6, +   1, +   2, +   1, +   2, +   1, +   6, +   1, +   2, +   1, +   2, +   2, +   1 /), &
+            (/maxn+1, maxn/))
+
+    real(kind=prec), parameter :: coeff(maxn+1,maxn) = coeffN/real(coeffD)
+    integer i
+
+    if (n>maxn) then
+      print*,"Bernoulli beyond 15 is not implemented"
+      stop
+    endif
 
+    xpow(1:n+1) = (/ ( x**i, i = 0, n ) /)
+    res = sum( xpow(1:n+1) * coeff(1:n+1,n) )
 
   END FUNCTION