... a
multidimensionalminimizationstrategy and a one-dimensional minimizationroutine
results in some unnecessary copying of vectors hither and yon. That should not
420
Chapter 10. Minimization or Maximization of Functions
Sample ... equivalent information content, of order N
2
numbers.
In the direction set methods of §10.5, we collected the necessary information by
making on the o...
... a
one-dimensional minimization sub-algorithm such as Brent’s method (see
above). Storage is of order N
2
.
There are two major families of algorithms for multidimensional minimization
with calculation of first ... computation
of first derivatives.
• You must choose between methods that require storage of order N
2
and
those that require only of order N,whereNis the number of d...
... combined with (10 .6. 6) to solve for
λ. The result is exactly the expression (10 .6. 4). But with this value of λ, (10 .6. 6)
is the same as (10 .6. 2), q.e.d.
We have, then, the basis of an algorithm that ... knowledge of the
Hessian matrix A, nor even the storage necessary to store such a matrix. A sequence
of directions h
i
is constructed, using only line minimizations, evaluation...
... ncity);
jorder[nn++]=iorder[jj];
}
for (j=1;j<=m3;j++) { Finally, the segment from n [6] to n[3].
jj=1 + ((j+n [6] -2) % ncity);
jorder[nn++]=iorder[jj];
}
for (j=1;j<=ncity;j++) Copy jorder ... (j=1;j<=ncity;j++) Copy jorder back into iorder.
iorder[j]=jorder[j];
444
Chapter 10. Minimization or Maximization of Functions
Sample page from NUMERICAL RECIPES IN C: THE ART OF...
... books,diskettes, or CDROMs
visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to trade@cup.cam.ac.uk (outside North America).
estimate for the right-hand side of equation ... this way is that, for most functions,
the final square root is a number of order unity. Therefore, as a rule of thumb, it
is hopeless to ask for a bracketing interval...
... cycle. In the worst possible case, where the parabolic steps are acceptable but
404
Chapter 10. Minimization or Maximization of Functions
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC ... prohibited. To order Numerical Recipes books,diskettes, or CDROMs
visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to trade@cup.ca...
... consideration of multidimensional minimization,
that is, finding the minimum of a function of more than one independent variable.
This section stands apart from those which follow, however: All of the algorithms
after ... prohibited. To order Numerical Recipes books,diskettes, or CDROMs
visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email t...
... consideration of multidimensional minimization,
that is, finding the minimum of a function of more than one independent variable.
This section stands apart from those which follow, however: All of the algorithms
after ... tetrahedron. (The simplex method of linear programming,
412
Chapter 10. Minimization or Maximization of Functions
Sample page from NUMERICAL RECIPES IN...
... updates the information that is accumulated. Instead of requiring intermediate
storage on the order of N, the number of dimensions, it requires a matrix of size
N × N. Generally, for any moderate ... roundoff problem? The trick is not to store A but
rather a triangular decomposition of A, its Cholesky decomposition (cf. §2.9). The updating
formula used for the Cholesky decomposition...
... the form (10.8.3) or (10.8.4), but rather only equality constraints of the
form (10.8.5) and nonnegativity constraints of the form (10.8.2).
432
Chapter 10. Minimization or Maximization of Functions
Sample ... until the 1982 work
of Stephen Smale. (For a contemporary account, see
[3]
.)
Simplex Method for a Restricted Normal Form
A linear programming problem is said to be in...