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Overloaded Functions abs ✘ abs(z) = −z ⇐ x < 0 +z ⇐ x ≥ 0 tan ✔ tan(z) = e−iz−e iz e−iz +eiz asin ✔ arcsin(z) = −i log[iz + (1 − z2 ) 1 2] acos ✔ arccos(z) = −i log[z + (z2 − 1) 1 atan ✔ 2] arctan(z) = i 2 log 1−iz 1+iz sinh ✔ sinh(z) = ez−e −z cosh ✔ 2 cosh(z) = ez +e −z tanh ✔ 2 tanh(z) = ez−e −z ez +e−z dim ✘ z1 − z2 dim(z1, z2) = 0 ⇐ x1 > x2 ⇐ x1 ≤ x2 sign ✘ +|x1| sign(z1, z2) = −|x1| ⇐ x2 ≥ 0 ⇐ x2 < 0 max ✘ z1 max(z1, z2) = z2 ⇐ x1 ≥ x2 ⇐ x1 < x2 min ✘ min(z1, z2) = z1 z2 ⇐ x1 ≤ x2 ⇐ x1 > x2 – Joaquim Martins – January 12, 2000 – 9
Automatic Implementation • Cookbook procedure: – Substitute all real type variable declarations with complex declarations. – Define all functions and operators that are not defined for complex arguments. – A complex-step can then be added to the desired variable and the derivative can be estimated by f ′ ≈ Im[f(x + ih)]/h. • Fortran 77: write new subroutines, substitute some of the intrinsic function calls by the subroutine names, e.g. abs by c abs. But ... need to know variable types in original code. • Fortran 90: can overload intrinsic functions and operators, including comparison operators. Compiler knows variable types and chooses correct version of the function or operator. • Complexify.py: Python script processes code. complexify.f90: overloaded definitions. – Joaquim Martins – January 12, 2000 – 10
Page 1 and 2: AN AUTOMATED METHOD FOR SENSITIVITY
Page 3 and 4: Sensitivity Analysis Methods • Fi
Page 5 and 6: Cauchy-Riemann Equations • Extens
Page 7 and 8: Simple Numerical Example • Estima
Page 9: Complex Functions and Operators in
Page 13 and 14: Sensitivity Estimates vs. Step Size
Page 15 and 16: Aerodynamic Sensitivities: FLO82 RA
Page 17 and 18: Residual Convergence of Sensitivity
Page 19 and 20: Drag Coefficient Sensitivity 11 10
Page 21: Further Work • Continue to work o

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