Module test_functions

Functions

run()

func1(alpha, beta=2.0)
Test function 1.

dfunc1(alpha, beta=2.0)
Derivative of test function 1.

func2(alpha, beta=0.004)
Test function 2.

dfunc2(alpha, beta=0.004)
Derivative of test function 2.

func3(alpha, beta=0.01, l=39.0)
Test function 3.

dfunc3(alpha, beta=0.01, l=39.0)
Derivative of test function 3.

func456(alpha, beta1, beta2)
Test functions 4, 5, and 6.

dfunc456(alpha, beta1, beta2)
Test functions 4, 5, and 6.

Imports: cos, pi, sin, sqrt, Float64, array, dot, more_thuente

Function Details

func1(alpha, beta=2.0)

Test function 1.

From More, J. J., and Thuente, D. J. 1994, Line search algorithms with guaranteed sufficient decrease.
ACM Trans. Math. Softw. 20, 286-307.

The function is:
                              alpha
        phi(alpha)  =  - ---------------
                         alpha**2 + beta

dfunc1(alpha, beta=2.0)

Derivative of test function 1.

From More, J. J., and Thuente, D. J. 1994, Line search algorithms with guaranteed sufficient decrease.
ACM Trans. Math. Softw. 20, 286-307.

The gradient is:
                             2*alpha**2                 1
        phi'(alpha)  =  --------------------  -  ---------------
                        (alpha**2 + beta)**2     alpha**2 + beta

func2(alpha, beta=0.004)

Test function 2.

From More, J. J., and Thuente, D. J. 1994, Line search algorithms with guaranteed sufficient decrease.
ACM Trans. Math. Softw. 20, 286-307.

The function is:

        phi(alpha)  =  (alpha + beta)**5 - 2(alpha + beta)**4

dfunc2(alpha, beta=0.004)

Derivative of test function 2.

From More, J. J., and Thuente, D. J. 1994, Line search algorithms with guaranteed sufficient decrease.
ACM Trans. Math. Softw. 20, 286-307.

The gradient is:

        phi'(alpha)  =  5(alpha + beta)**4 - 8(alpha + beta)**3

func3(alpha, beta=0.01, l=39.0)

Test function 3.

From More, J. J., and Thuente, D. J. 1994, Line search algorithms with guaranteed sufficient decrease.
ACM Trans. Math. Softw. 20, 286-307.

The function is:

                                     2(1 - beta)       / l*pi         \ 
        phi(alpha)  =  phi0(alpha) + ----------- . sin | ---- . alpha |
                                        l*pi           \  2           /

        where:
                                /  1 - alpha,                           if alpha <= 1 - beta,
                                |
                                |  alpha - 1,                           if alpha >= 1 + beta,
                phi0(alpha) =  <
                                |   1                    1
                                | ------(alpha - 1)**2 + - beta,        if alpha in [1 - beta, 1 + beta].
                                \ 2*beta                 2

dfunc3(alpha, beta=0.01, l=39.0)

Derivative of test function 3.

From More, J. J., and Thuente, D. J. 1994, Line search algorithms with guaranteed sufficient decrease.
ACM Trans. Math. Softw. 20, 286-307.

The gradient is:
                                                       / l*pi         \ 
        phi(alpha)  =  phi0'(alpha) + (1 - beta) . cos | ---- . alpha |
                                                       \  2           /

        where:
                                 /  -1,                 if alpha <= 1 - beta,
                                 |
                                 |  1,                  if alpha >= 1 + beta,
                phi0'(alpha) =  <
                                 | alpha - 1
                                 | ---------,           if alpha in [1 - beta, 1 + beta].
                                 \   beta

func456(alpha, beta1, beta2)

Test functions 4, 5, and 6.

From More, J. J., and Thuente, D. J. 1994, Line search algorithms with guaranteed sufficient decrease.
ACM Trans. Math. Softw. 20, 286-307.

The function is:

        phi(alpha)  =  gamma(beta1) * sqrt((1 - alpha)**2 + beta2**2) + gamma(beta2) * sqrt(alpha**2 + beta1**2)

        where:
                gamma(beta) = sqrt(1 + beta**2) - beta

dfunc456(alpha, beta1, beta2)

Test functions 4, 5, and 6.

From More, J. J., and Thuente, D. J. 1994, Line search algorithms with guaranteed sufficient decrease.
ACM Trans. Math. Softw. 20, 286-307.

The function is:

                                                  (1 - alpha)                                         a
        phi'(alpha)  =  - gamma(beta1) * -------------------------------  +  gamma(beta2) * -------------------------
                                         sqrt((1 - alpha)**2 + beta2**2)                    sqrt(alpha**2 + beta1**2)

        where:
                gamma(beta) = sqrt(1 + beta**2) - beta