3.8 Analysis of Computational Cost

The subroutines H2LC, H3LC and H3ALC were written with the aim of minimising the computational cost of evaluating the discrete form when the point p does not lie on the panel. When p does lie on the panel then each of the subroutines will generally be more costly, this is particularly true for H3ALC. However, in integral equation methods the cost of evaluating the discrete form when p does not lie on the panel is the key to estimating the overall computational cost.


Table 3.A: Required Operations
Op.	L_k	M_k	M_k^t	N_k
A	×	×	×	×
B		×		×
C				×
D	×	×	×	×
E	×	×	×	×
F				°
G		×		×
H			×	×
I				×
J				×
K	×	×	×	×
L				×
M	×	×	×	×
N	×
O	×
P		×	×	×
Q		×	×
R		×	×
S			×
T				×
U				×
× : Operation required in 2D and 3D
° : Operation required in 3D only


Table 3.B: Cost of Operations A to J
Op.	2D	3D	3Daxi
A	2C_f	6C_f	2C_f
B	0	0	2C_f
C	0	0	2C_f
D	0	0	0
E	C_s+2C_f	C_s+3C_f	C_s+3C_f
F	-	C_f	C_f
G	3C_f	4C_f	4C_f
H	3C_f	4C_f	3C_f
I	C_f	C_f	C_f
J	C_f	C_f	C_f
C_f: Cost of floating point operation
C_s: Cost of square root evaluation

Table 3.C: Cost of Operations K to U
k is zero k is real k is imaginary k is complex
Op. 2D 3D 2D 3D 2D 3D 2D 3D
K - - C_f C_f C_f C_f 2C_f 2C_f
L - - C_f C_f C_f C_f 2C_f 2C_f
M - - C_h C_e C_h C_e C_h C_e
N C_l+C_f C_f 2C_f 2C_f C_f C_f 2C_f 2C_f
O C_f C_f 2C_f 2C_f C_f C_f 2C_f 2C_f
P C_f C_f 2C_f 4C_f C_f C_f 4C_f 6C_f
Q C_f C_f 2C_f 2C_f C_f C_f 2C_f 2C_f
R C_f C_f C_f C_f C_f C_f 2C_f 2C_f
S C_f C_f 2C_f 2C_f C_f C_f 2C_f 2C_f
T C_f C_f 4C_f 4C_f 2C_f 2C_f 8C_f 6C_f
U 3C_f 3C_f 6C_f 6C_f 3C_f 3C_f 6C_f 6C_f
C_f: Cost of floating point operation
C_s: Cost of square root evaluation
C_l: Cost of logarithm evaluation
C_h: Cost of Hankel function evaluation
C_e: Cost of complex exponential evaluation

The list of operations necessary for computing all the integral operators when p does not lie on the panel is given in Section 3.4. The computations required for the calculation of each of the integral operators are given Table 3.A.

Operations A to J are independent of k whereas operations K to U do depend on the wavenumber. Savings are made in the subroutine when k is zero, purely real or purely imaginary over when k is complex. The computational cost of the set of operations is listed in Tables 3.B and 3.C for each of these cases.

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Table 3.C: Cost of Operations K to U
	k is zero		k is real		k is imaginary		k is complex
Op.	2D	3D	2D	3D	2D	3D	2D	3D
K	-	-	C_f	C_f	C_f	C_f	2C_f	2C_f
L	-	-	C_f	C_f	C_f	C_f	2C_f	2C_f
M	-	-	C_h	C_e	C_h	C_e	C_h	C_e
N	C_l+C_f	C_f	2C_f	2C_f	C_f	C_f	2C_f	2C_f
O	C_f	C_f	2C_f	2C_f	C_f	C_f	2C_f	2C_f
P	C_f	C_f	2C_f	4C_f	C_f	C_f	4C_f	6C_f
Q	C_f	C_f	2C_f	2C_f	C_f	C_f	2C_f	2C_f
R	C_f	C_f	C_f	C_f	C_f	C_f	2C_f	2C_f
S	C_f	C_f	2C_f	2C_f	C_f	C_f	2C_f	2C_f
T	C_f	C_f	4C_f	4C_f	2C_f	2C_f	8C_f	6C_f
U	3C_f	3C_f	6C_f	6C_f	3C_f	3C_f	6C_f	6C_f
C_f: Cost of floating point operation
C_s: Cost of square root evaluation
C_l: Cost of logarithm evaluation
C_h: Cost of Hankel function evaluation
C_e: Cost of complex exponential evaluation