SECTION 8.

MUKAI

This is a conjugate gradient method. For a description see
Mukai,H., Readily Implementable Conjugate Gradient Methods, *Mathematical
Programming*, 17 (1979), 298-319.

STORAGE REQUIREMENTS

NQ = i2 = 6*NP + 4

TERMINATION FLAGS

IER = Greater than 0 if optimum is achieved

IER = 3 Function value is accurate to relative ACC

IER = 2 Norm of gradient is less than ACC

IER = 1 Attempted stepsize is less than ACC (relatively)

IER = -1 Iteration limit exceeded

IER = -2 Function error on first call of FP

IER = -3 Input error

IER = -4 Not enough scratch storage

IER = -5 Continual straying into forbidden area

IER = -9 Function error in first FUNC call

IER =-16 Small gradient without meeting termination criteria

IER =-24 Gradient inaccurate---does not point uphill

SECTION 8.1 OPTIONS

Besides the general options, the following are available.

COMMON/BOPT8/IV8,IV3

IV8 = 1 Use formula (14) in paper (Dixon method)

IV8 = 2 Use formula (15) in paper (Fletcher and Reeves) (Default
= 2)

IV8 = 3 Use formula (22) in paper (Polak,Ribiere,Polyak)

IV8 = 4 Use formula (23) in paper (Mukai)

IV3 = 0 Use Step 3a in paper (Default = 0)

IV3 = 1 Use Step 3b in paper

COMMON/BOPT7/ALPHA,BETA,OMEGA,EPSTIL,RH,DELTIL,SIGMAS,BIGL,AKAPPA

ALPHA = Type of Lipschitz constant for stepsize (Default=0.1)

BETA = Used in searching for suitable step-length (Default=0.6)
OMEGA = Used to insure that we have an ascent direction
(Default=0.3)

EPSTIL = Upper bound for epsilon in Step 1 of algorithm
(Default=1.D-3)

RH = Default step-length in Step 3. Function dependent.
(Default=0.3)

DELTIL = First guess at smallest eigenvalue of quadratic
approximation

(Default= 1.D-3)

SIGMA = Shrinkage factor for deltahat in Step 3b. (Default=0.5)

BIGL = Affects direction choice (Step 10). Function dependent.

(Default=1000.)

AKAPPA = Relates to estimate of deltahat (Step 2). (Default=30.)

Note: Compared to other algorithms, MUKAI employs a large number of parameters. Experimentation with alternative values is likely to be profitable. The algorithm tends to do relatively well far from the optimum and relatively less well close to it if the function has narrow valleys or ridges.To refine the location of the optimum, the user may wish to use GRADX or some other algorithm after MUKAI terminates.