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1-D minimization using golden section search

`[a,gX,perf,retcode,delta,tol] = srchgol(net,X,Pd,Tl,Ai,Q,TS,dX,gX,perf,dperf,delta,tol,ch_perf)`

`srchgol`

is a linear search routine. It searches
in a given direction to locate the minimum of the performance function
in that direction. It uses a technique called the golden section search.

`[a,gX,perf,retcode,delta,tol] = srchgol(net,X,Pd,Tl,Ai,Q,TS,dX,gX,perf,dperf,delta,tol,ch_perf)`

takes
these inputs,

`net` | Neural network |

`X` | Vector containing current values of weights and biases |

`Pd` | Delayed input vectors |

`Tl` | Layer target vectors |

`Ai` | Initial input delay conditions |

`Q` | Batch size |

`TS` | Time steps |

`dX` | Search direction vector |

`gX` | Gradient vector |

`perf` | Performance value at current |

`dperf` | Slope of performance value at current |

`delta` | Initial step size |

`tol` | Tolerance on search |

`ch_perf` | Change in performance on previous step |

and returns

`a` | Step size that minimizes performance |

`gX` | Gradient at new minimum point |

`perf` | Performance value at new minimum point |

`retcode` | Return code that has three elements. The first two elements correspond to the number of function evaluations in the two stages of the search. The third element is a return code. These have different meanings for different search algorithms. Some might not be used in this function. |

`0` Normal | |

`1` Minimum
step taken | |

`2` Maximum
step taken | |

`3` Beta
condition not met | |

`delta` | New initial step size, based on the current step size |

`tol` | New tolerance on search |

Parameters used for the golden section algorithm are

`alpha` | Scale factor that determines sufficient reduction in |

`bmax` | Largest step size |

`scale_tol` | Parameter that relates the tolerance |

The defaults for these parameters are set in the training function
that calls them. See `traincgf`

, `traincgb`

, `traincgp`

, `trainbfg`

, and `trainoss`

.

Dimensions for these variables are

`Pd` |
| Each element |

`Tl` |
| Each element |

`Ai` |
| Each element |

where

`Ni ` | `=` | `net.numInputs` |

`Nl ` | `=` | `net.numLayers` |

`LD ` | `=` | `net.numLayerDelays` |

`Ri ` | `=` | `net.inputs{i}.size` |

`Si` | `=` | `net.layers{i}.size` |

`Vi` | `=` | `net.targets{i}.size` |

`Dij` | `=` | `Ri * length(net.inputWeights{i,j}.delays)` |

Here is a problem consisting of inputs `p`

and
targets `t`

to be solved with a network.

p = [0 1 2 3 4 5]; t = [0 0 0 1 1 1];

A two-layer feed-forward network is created. The network's
input ranges from `[0 to 10]`

. The first layer has
two `tansig`

neurons, and the second layer has one `logsig`

neuron.
The `traincgf`

network training function and the `srchgol`

search
function are to be used.

net = newff([0 5],[2 1],{'tansig','logsig'},'traincgf'); a = sim(net,p)

net.trainParam.searchFcn = 'srchgol'; net.trainParam.epochs = 50; net.trainParam.show = 10; net.trainParam.goal = 0.1; net = train(net,p,t); a = sim(net,p)

You can create a standard network that uses `srchgol`

with `newff`

, `newcf`

,
or `newelm`

.

To prepare a custom network to be trained with `traincgf`

,
using the line search function `srchgol`

,

Set

`net.trainFcn`

to`'traincgf'`

. This sets`net.trainParam`

to`traincgf`

's default parameters.Set

`net.trainParam.searchFcn`

to`'srchgol'`

.

The `srchgol`

function can be used with any
of the following training functions: `traincgf`

, `traincgb`

, `traincgp`

, `trainbfg`

, `trainoss`

.

`srchgol`

locates the minimum of the performance
function in the search direction `dX`

, using the
golden section search. It is based on the algorithm as described on
page 33 of Scales (see reference below).

Scales, L.E., *Introduction to Non-Linear Optimization*,
New York, Springer-Verlag, 1985

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