Model fitting by least squares |

I suggest you skip over the remainder of this section and return after you have seen many examples and have developed some expertise, and have some technical problems.

The **conjugate-gradient method** was introduced
by **Hestenes** and **Stiefel** in 1952.
To read the standard literature and relate it to this book,
you should first realize that when I write fitting goals like

(112) | |||

(113) |

they are equivalent to minimizing the quadratic form:

The optimization theory (OT) literature starts from a minimization of

To relate equation (114) to (115) we expand the parentheses in (114) and abandon the constant term . Then gather the quadratic term in and the linear term in . There are two terms linear in that are transposes of each other. They are scalars so they are equal. Thus, to invoke ``standard methods,'' you take your problem-formulation operators , , and create two subroutines that apply:

(116) | |||

(117) |

The operators and operate on model space. Standard procedures do not require their adjoints because is its own adjoint and reduces model space to a scalar. You can see that computing and requires one temporary space the size of data space (whereas

When people have trouble with conjugate gradients or conjugate
directions, I always refer them to the **Paige and Saunders
algorithm** `LSQR`. Methods that form explicitly or
implicitly (including both the standard literature and the book3
method) square the condition number, that is, they are twice as
susceptible to rounding error as is `LSQR`. The Paige and
Saunders method is reviewed by Nolet in a geophysical context.

- It is possible to reject two dips with the operator:

(118)

(119) - Given and from the previous exercise, what are and ?
- Reduce
to the special case of one data point and two model points like this:

(120) - In 1695,
150 years before Lord Kelvin's absolute temperature scale,
120 years before Sadi Carnot's PhD. thesis,
40 years before Anders Celsius,
and 20 years before Gabriel Fahrenheit,
the French physicist Guillaume Amontons,
deaf since birth,
took a mercury manometer (pressure gauge) and sealed it inside a glass pipe (a constant volume of air).
He heated it to the boiling point of water at 100C.
As he lowered the temperature to freezing at 0C,
he observed the pressure dropped by 25% .
He could not drop the temperature any further, but he supposed that if he could drop it further by a factor of three,
the pressure would drop to zero (the lowest possible pressure), and the temperature would have been the lowest possible temperature.
Had he lived after Anders Celsius, he might have calculated this temperature to be C (Celsius).
Absolute zero is now known to be C.
It is your job to be Amontons' lab assistant. You make your

*i*-th measurement of temperature with Issac Newton's thermometer; and you measure pressure and volume in the metric system. Amontons needs you to fit his data with the regression and calculate the temperature shift that Newton should have made when he defined his temperature scale. Do not solve this problem! Instead, cast it in the form of equation (), identifying the data and the two column vectors and that are the fitting functions. Relate the model parameters and to the physical parameters and . Suppose you make ALL your measurements at room temperature, can you find ? Why or why not? - One way to remove a mean value from signal
is with the fitting goal
.
What operator matrix is involved?
- What linear operator subroutine from Chapter
can be used for finding the mean?
- How many CD iterations should be required to get the exact mean value?
- Write a mathematical expression for finding the mean by the CG method.

Model fitting by least squares |

2014-12-01