NAG C Library Function Document

void	nag_lone_fit (Nag_OrderType order, Integer m, double a[], double b[], Integer nplus2, double toler, double x[], double resid, Integer rank, Integer iter, NagError fail)

3

Description

Given a matrix

A

with

m

rows and

n

columns

(m \geq n)

and a vector

b

with

m

elements, the function calculates an

l_{1}

solution to the over-determined system of equations

A x = b .

That is to say, it calculates a vector

x

, with

n

elements, which minimizes the

l_{1}

norm (the sum of the absolute values) of the residuals

r (x) = \sum_{i = 1}^{m} |r_{i}|,

where the residuals

r_{i}

are given by

r_{i} = b_{i} - \sum_{j = 1}^{n} a_{i j} x_{j}, i = 1, 2, \dots, m .

Here

a_{i j}

is the element in row

i

and column

j

A

b_{i}

is the

i

th element of

b

and

x_{j}

the

j

th element of

x

. The matrix

A

need not be of full rank.

Typically in applications to data fitting, data consisting of

m

points with coordinates

(t_{i}, y_{i})

are to be approximated in the

l_{1}

norm by a linear combination of known functions

ϕ_{j} (t)

α_{1} ϕ_{1} (t) + α_{2} ϕ_{2} (t) + \dots + α_{n} ϕ_{n} (t) .

This is equivalent to fitting an

l_{1}

solution to the over-determined system of equations

\sum_{j = 1}^{n} ϕ_{j} (t_{i}) α_{j} = y_{i}, i = 1, 2, \dots, m .

Thus if, for each value of

i

and

j

, the element

a_{i j}

of the matrix

A

in the previous paragraph is set equal to the value of

ϕ_{j} (t_{i})

and

b_{i}

is set equal to

y_{i}

, the solution vector

x

will contain the required values of the

α_{j}

. Note that the independent variable

t

above can, instead, be a vector of several independent variables (this includes the case where each

ϕ_{i}

is a function of a different variable, or set of variables).

The algorithm is a modification of the simplex method of linear programming applied to the primal formulation of the

l_{1}

problem (see Barrodale and Roberts (1973) and Barrodale and Roberts (1974)). The modification allows several neighbouring simplex vertices to be passed through in a single iteration, providing a substantial improvement in efficiency.

4

References

Barrodale I and Roberts F D K (1973) An improved algorithm for discrete

l_{1}

linear approximation SIAM J. Numer. Anal. 10 839–848

Barrodale I and Roberts F D K (1974) Solution of an overdetermined system of equations in the

l_{1}

-norm Comm. ACM 17(6) 319–320

5

Arguments

1: $order$ – Nag_OrderTypeInput

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $m$ – IntegerInput

On entry: the number of equations,

m

(the number of rows of the matrix

A

Constraint:

m \geq n \geq 1

3: $a [(m + 2) \times nplus2]$ – doubleInput/Output

Note: where

A (i, j)

appears in this document, it refers to the array element

$a [(j - 1) \times ((m + 2)) + i - 1]$ when $order = Nag_ColMajor$ ;
$a [(i - 1) \times nplus2 + j - 1]$ when $order = Nag_RowMajor$ .

On entry:

A (i, j)

must contain

a_{i j}

, the element in the

i

th row and

j

th column of the matrix

A

, for

i = 1, 2, \dots, m

and

j = 1, 2, \dots, n

. The remaining elements need not be set.

On exit: contains the last simplex tableau generated by the simplex method.

4: $b [m]$ – doubleInput/Output

On entry:

b [i - 1]

must contain

b_{i}

, the

i

th element of the vector

b

, for

i = 1, 2, \dots, m

On exit: the

i

th residual

r_{i}

corresponding to the solution vector

x

, for

i = 1, 2, \dots, m

5: $nplus2$ – IntegerInput

On entry:

n + 2

, where

n

is the number of unknowns (the number of columns of the matrix

A

Constraint:

3 \leq nplus2 \leq m + 2

6: $toler$ – doubleInput

On entry: a non-negative value. In general toler specifies a threshold below which numbers are regarded as zero. The recommended threshold value is

ε^{2 / 3}

where

ε

is the machine precision. The recommended value can be computed within the function by setting toler to zero. If premature termination occurs a larger value for toler may result in a valid solution.

Suggested value:

0.0

7: $x [nplus2]$ – doubleOutput

On exit:

x [j - 1]

contains the

j

th element of the solution vector

x

, for

j = 1, 2, \dots, n

. The elements

x [n]

and

x [n + 1]

are unused.

8: $resid$ – double *Output

On exit: the sum of the absolute values of the residuals for the solution vector

x

9: $rank$ – Integer *Output

On exit: the computed rank of the matrix

A

10: $iter$ – Integer *Output

On exit: the number of iterations taken by the simplex method.

11: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $nplus2 = 〈value〉$ .
Constraint: $nplus2 \geq 3$ .
NE_INT_2: On entry, $nplus2 = 〈value〉$ and $m = 〈value〉$ .
Constraint: $3 \leq nplus2 \leq m + 2$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_NON_UNIQUE: An optimal solution has been obtained, but may not be unique.
NE_TERMINATION_FAILURE: Premature termination due to rounding errors. Try using larger value of toler: $toler = 〈value〉$ .
NE_TOO_MANY_ITER: More than $1000 * n$ iterations were performed. nag_lone_fit (e02gac) has terminated without calculating a solution. The output data from the function is as computed on the last good iteration. Consider increasing the value of toler. Alternatively, $A$ may be ill conditioned—try scaling its columns.

7

Accuracy

Experience suggests that the computational accuracy of the solution

x

is comparable with the accuracy that could be obtained by applying Gaussian elimination with partial pivoting to the

n

equations satisfied by this algorithm (i.e., those equations with zero residuals). The accuracy therefore varies with the conditioning of the problem, but has been found generally very satisfactory in practice.

8

Parallelism and Performance

nag_lone_fit (e02gac) is not threaded in any implementation.

9

Further Comments

The effects of

m

and

n

on the time and on the number of iterations in the Simplex Method vary from problem to problem, but typically the number of iterations is a small multiple of

n

and the total time taken is approximately proportional to

m n^{2}

It is recommended that, before the function is entered, the columns of the matrix

A

are scaled so that the largest element in each column is of the order of unity. This should improve the conditioning of the matrix, and also enable the argument toler to perform its correct function. The solution

x

obtained will then, of course, relate to the scaled form of the matrix. Thus if the scaling is such that, for each

j = 1, 2, \dots, n

, the elements of the

j

th column are multiplied by the constant

k_{j}

, the element

x_{j}

of the solution vector

x

must be multiplied by

k_{j}

if it is desired to recover the solution corresponding to the original matrix

A

10

Example

Suppose we wish to approximate a set of data by a curve of the form

y = K e^{t} + L e^{- t} + M

where

K

L

and

M

are unknown. Given values

y_{i}

5

points

t_{i}

we may form the over-determined set of equations for

K

L

and

M

e^{x_{i}} K + e^{- x_{i}} L + M = y_{i}, i = 1, 2, \dots, 5 .

nag_lone_fit (e02gac) is used to solve these in the

l_{1}

sense.

NAG C Library Function Document

nag_lone_fit (e02gac)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results