Description Details Report for linesearch methods Report for the pure method Report for the double dogleg global method Report for the single (Powell) dogleg global method Report for the hook step global method See Also

The format of the iteration report
provided by nleqslv when the `trace`

component of the `control`

argument
has been set to 1.

All iteration reports consist of a series of columns with a header summarising the contents. Common column headers are

`Iter`

Iteration counter

`Jac`

Jacobian type. The Jacobian type is indicated by

`N`

for a Newton Jacobian or`B`

for a Broyden updated matrix; optionally followed by the letter`s`

indicating a totally singular matrix or the letter`i`

indicating an ill-conditioned matrix. Unless the Jacobian is singular, the Jacobian type is followed by an estimate of the inverse condition number of the Jacobian in parentheses as computed by Lapack. This column will be blank when backtracking is active.`Fnorm`

square of the euclidean norm of function values / 2

`Largest |f|`

infinity norm of

*f(x)*at the current point

A sample iteration report for the linesearch global methods (`cline`

, `qline`

and `gline`

) is
(some intercolumn space has been removed to make the table fit)

1 2 3 4 5 6 7 8 9 10 11 | ```
Iter Jac Lambda Ftarg Fnorm Largest |f|
0 2.886812e+00 2.250000e+00
1 N(9.6e-03) 1.0000 2.886235e+00 5.787362e+05 1.070841e+03
1 0.1000 2.886754e+00 9.857947e+00 3.214799e+00
1 0.0100 2.886806e+00 2.866321e+00 2.237878e+00
2 B(2.2e-02) 1.0000 2.865748e+00 4.541965e+03 9.341610e+01
2 0.1000 2.866264e+00 3.253536e+00 2.242344e+00
2 0.0298 2.866304e+00 2.805872e+00 2.200544e+00
3 B(5.5e-02) 1.0000 2.805311e+00 2.919089e+01 7.073082e+00
3 0.1000 2.805816e+00 2.437606e+00 2.027297e+00
4 B(1.0e-01) 1.0000 2.437118e+00 9.839802e-01 1.142529e+00
``` |

The column headed by `Lambda`

shows the value of the line search parameter.
The column headed by `Ftarg`

follows from a sufficient decrease requirement and
is the value below which `Fnorm`

must drop if the current step is to be accepted.

The value of `Lambda`

may not be lower than a threshold determined
by the setting of control parameter `xtol`

to avoid reporting
false convergence. When no acceptable `Lambda`

is possible and the Broyden method
is being used, a new Jacobian will be computed.

The iteration report for the global method `none`

is almost the same as the above report,
except that the column labelled `Ftarg`

is omitted.
The column `Lambda`

gives the ratio of the maximum stepsize and the length of
the full Newton step. It is either exactly 1.0, indicating that the Newton step is smaller
than the maximum stepsize and therefore used unmodified, or smaller than 1.0,
indicating that the Newton step is larger than the maximum stepsize and therefore truncated.

A sample iteration report for the global method `dbldog`

is
(some intercolumn space has been removed to make the table fit)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | ```
Iter Jac Lambda Eta Dlt0 Dltn Fnorm Largest |f|
0 2.886812e+00 2.250000e+00
1 N(9.6e-03) C 0.9544 0.4671 0.9343* 1.699715e-01 5.421673e-01
1 W 0.0833 0.9544 0.9343 0.4671 1.699715e-01 5.421673e-01
2 B(1.1e-02) W 0.1154 0.4851 0.4671 0.4671 1.277667e-01 5.043571e-01
3 B(7.3e-02) W 0.7879 0.7289 0.4671 0.0759 5.067893e-01 7.973542e-01
3 C 0.7289 0.0759 0.1519 5.440250e-02 2.726084e-01
4 B(8.3e-02) W 0.5307 0.3271 0.1519 0.3037 3.576547e-02 2.657553e-01
5 B(1.8e-01) N 0.6674 0.2191 0.4383 6.566182e-03 8.555110e-02
6 B(1.8e-01) N 0.9801 0.0376 0.0752 4.921645e-04 3.094104e-02
7 B(1.9e-01) N 0.7981 0.0157 0.0313 4.960629e-06 2.826064e-03
8 B(1.6e-01) N 0.3942 0.0029 0.0058 1.545503e-08 1.757498e-04
9 B(1.5e-01) N 0.6536 0.0001 0.0003 2.968676e-11 5.983765e-06
10 B(1.5e-01) N 0.4730 0.0000 0.0000 4.741792e-14 2.198380e-07
11 B(1.5e-01) N 0.9787 0.0000 0.0000 6.451792e-19 8.118586e-10
``` |

After the column for the Jacobian the letters indicate the following

`C`

a fraction (

*<=1.0*) of the Cauchy or steepest descent step is taken where the fraction is the ratio of the trust region radius and the Cauchy steplength.`W`

a convex combination of the Cauchy and

`eta`

*(Newton step) is taken. The number in the column headed by`Lambda`

is the weight of the partial Newton step.`P`

a fraction (

*>=1.0*) of the partial Newton step, equal to`eta`

*(Newton step), is taken where the fraction is the ratio of the trust region radius and the partial Newton steplength.`N`

a normal full Newton step is taken.

The number in the column headed by `Eta`

is calculated from an upper limit on the ratio
of the length of the steepest descent direction and the length of the Newton step.
See Dennis and Schnabel (1996) pp.139ff for the details.
The column headed by `Dlt0`

gives the trust region size at the start of the current
iteration.
The column headed by `Dltn`

gives the trust region size when the current
step has been accepted by the dogleg algorithm.

The trust region size is decreased when the actual reduction of the function value norm
does not agree well with the predicted reduction from the linear approximation of the function
or does not exhibit sufficient decrease.
And increased when the actual and predicted reduction are sufficiently close.
The trust region size is not allowed to decrease beyond a threshold determined
by the setting of control parameter `xtol`

; when that happens the backtracking
is regarded as a failure and is terminated.
In that case a new Jacobian will be calculated if the Broyden method is being used.

The current trust region step is continued with a doubled trust region size
when the actual and predicted reduction agree extremely well. This is indicated
by `*`

immediately following the value in the column
headed by `Dltn`

. This could save gradient calculations.
However, a trial step is never taken if the current step is a Newton step.
If the trial step does not succeed then the previous trust region size is restored
by halving the trial size.
The exception is when a trial step takes a Newton step.
In that case the trust region size is
immediately set to the size of the Newton step which implies that a
halving of the new size leads to a smaller size for the trust region than before.

Normally the initial trust region radius is the same as the final trust region radius of the previous iteration but the trust region size is restricted by the size of the current Newton step. So when full Newton steps are being taken, the trust region radius at the start of an iteration may be less than the final value of the previous iteration. The double dogleg method and the trust region updating procedure are fully explained in sections 6.4.2 and 6.4.3 of Dennis and Schnabel (1996).

A sample iteration report for the global method `pwldog`

is
(some intercolumn space has been removed to make the table fit)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 | ```
Iter Jac Lambda Dlt0 Dltn Fnorm Largest |f|
0 2.886812e+00 2.250000e+00
1 N(9.6e-03) C 0.4671 0.9343* 1.699715e-01 5.421673e-01
1 W 0.0794 0.9343 0.4671 1.699715e-01 5.421673e-01
2 B(1.1e-02) W 0.0559 0.4671 0.4671 1.205661e-01 4.890487e-01
3 B(7.3e-02) W 0.5662 0.4671 0.0960 4.119560e-01 7.254441e-01
3 W 0.0237 0.0960 0.1921 4.426507e-02 2.139252e-01
4 B(8.8e-02) W 0.2306 0.1921 0.3842* 2.303135e-02 2.143943e-01
4 W 0.4769 0.3842 0.1921 2.303135e-02 2.143943e-01
5 B(1.9e-01) N 0.1375 0.2750 8.014508e-04 3.681498e-02
6 B(1.7e-01) N 0.0162 0.0325 1.355741e-05 5.084627e-03
7 B(1.3e-01) N 0.0070 0.0035 1.282776e-05 4.920962e-03
8 B(1.8e-01) N 0.0028 0.0056 3.678140e-08 2.643592e-04
9 B(1.9e-01) N 0.0001 0.0003 1.689182e-12 1.747622e-06
10 B(1.9e-01) N 0.0000 0.0000 9.568768e-16 4.288618e-08
11 B(1.9e-01) N 0.0000 0.0000 1.051357e-18 1.422036e-09
``` |

This is much simpler than the double dogleg report, since the single dogleg takes
either a steepest descent step, a convex combination of the steepest descent and Newton steps
or a full Newton step. The number in the column `Lambda`

is the weight of the Newton step.
The single dogleg method is a special case of the double dogleg method with `eta`

equal to 1.
It uses the same method of updating the trust region size as the double dogleg method.

A sample iteration report for the global method `hook`

is
(some intercolumn space has been removed to make the table fit)

1 2 3 4 5 6 7 8 9 10 11 12 | ```
Iter Jac mu dnorm Dlt0 Dltn Fnorm Largest |f|
0 2.886812e+00 2.250000e+00
1 N(9.6e-03) H 0.1968 0.4909 0.4671 0.9343* 1.806293e-01 5.749418e-01
1 H 0.0366 0.9381 0.9343 0.4671 1.806293e-01 5.749418e-01
2 B(2.5e-02) H 0.1101 0.4745 0.4671 0.2336 1.797759e-01 5.635028e-01
3 B(1.4e-01) H 0.0264 0.2341 0.2336 0.4671 3.768809e-02 2.063234e-01
4 B(1.6e-01) N 0.0819 0.0819 0.1637 3.002274e-03 7.736213e-02
5 B(1.8e-01) N 0.0513 0.0513 0.1025 5.355533e-05 1.018879e-02
6 B(1.5e-01) N 0.0090 0.0090 0.0179 1.357039e-06 1.224357e-03
7 B(1.5e-01) N 0.0004 0.0004 0.0008 1.846111e-09 6.070166e-05
8 B(1.4e-01) N 0.0000 0.0000 0.0001 3.292896e-12 2.555851e-06
9 B(1.5e-01) N 0.0000 0.0000 0.0000 7.281583e-18 3.800552e-09
``` |

The column headed by `mu`

shows the Levenberg-Marquardt parameter when the Newton step is
larger than the trust region radius. The column headed by `dnorm`

gives
the Euclidean norm of the step (adjustment of the current `x`

) taken by the algorithm.
The absolute value of the difference with `Dlt0`

is less than 0.1 times the trust region radius.

After the column for the Jacobian the letters indicate the following

`H`

a Levenberg-Marquardt restricted step is taken.

`N`

a normal full Newton step is taken.

The meaning of the columns headed by `Dlt0`

and `Dltn`

is identical to that of the same
columns for the double dogleg method.
The method of updating the trust region size is the same as in the double dogleg method.

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