NAG Library Routine Document
E02GAF
1 Purpose
E02GAF calculates an l1 solution to an over-determined system of linear equations.
2 Specification
SUBROUTINE E02GAF ( |
M, A, LDA, B, NPLUS2, TOLER, X, RESID, IRANK, ITER, IWORK, IFAIL) |
INTEGER |
M, LDA, NPLUS2, IRANK, ITER, IWORK(M), IFAIL |
REAL (KIND=nag_wp) |
A(LDA,NPLUS2), B(M), TOLER, X(NPLUS2), RESID |
|
3 Description
Given a matrix
A with
m rows and
n columns
m≥n and a vector
b with
m elements, the routine calculates an
l1 solution to the over-determined system of equations
That is to say, it calculates a vector
x, with
n elements, which minimizes the
l1 norm (the sum of the absolute values) of the residuals
where the residuals
ri are given by
Here
aij is the element in row
i and column
j of
A,
bi is the
ith element of
b and
xj the
jth 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
ti,yi are to be approximated in the
l1 norm by a linear combination of known functions
ϕjt,
This is equivalent to fitting an
l1 solution to the over-determined system of equations
Thus if, for each value of
i and
j, the element
aij of the matrix
A in the previous paragraph is set equal to the value of
ϕjti and
bi is set equal to
yi, the solution vector
x will contain the required values of the
αj. Note that the independent variable
t above can, instead, be a vector of several independent variables (this includes the case where each
ϕi 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
l1 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
l1 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
l1-norm
Comm. ACM 17(6) 319–320
5 Parameters
- 1: M – INTEGERInput
On entry: the number of equations, m (the number of rows of the matrix A).
Constraint:
M≥n≥1.
- 2: A(LDA,NPLUS2) – REAL (KIND=nag_wp) arrayInput/Output
On entry: Aij must contain aij, the element in the ith row and jth column of the matrix A, for i=1,2,…,m and j=1,2,…,n. The remaining elements need not be set.
On exit: contains the last simplex tableau generated by the simplex method.
- 3: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which E02GAF is called.
Constraint:
LDA≥M+2.
- 4: B(M) – REAL (KIND=nag_wp) arrayInput/Output
On entry: Bi must contain bi, the ith element of the vector b, for i=1,2,…,m.
On exit: the
ith residual ri corresponding to the solution vector x, for i=1,2,…,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≤NPLUS2≤M+2.
- 6: TOLER – REAL (KIND=nag_wp)Input
On entry: a non-negative value. In general
TOLER specifies a threshold below which numbers are regarded as zero. The recommended threshold value is
ε2/3 where
ε is the
machine precision. The recommended value can be computed within the routine 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) – REAL (KIND=nag_wp) arrayOutput
On exit: Xj contains the jth element of the solution vector x, for j=1,2,…,n. The elements Xn+1 and Xn+2 are unused.
- 8: RESID – REAL (KIND=nag_wp)Output
On exit: the sum of the absolute values of the residuals for the solution vector x.
- 9: IRANK – INTEGEROutput
On exit: the computed rank of the matrix A.
- 10: ITER – INTEGEROutput
On exit: the number of iterations taken by the simplex method.
- 11: IWORK(M) – INTEGER arrayWorkspace
- 12: IFAIL – INTEGERInput/Output
-
On entry:
IFAIL must be set to
0,
-1 or 1. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
-1 or 1 is recommended. If the output of error messages is undesirable, then the value
1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
0.
When the value -1 or 1 is used it is essential to test the value of IFAIL on exit.
On exit:
IFAIL=0 unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
- IFAIL=1
An optimal solution has been obtained but this may not be unique.
- IFAIL=2
The calculations have terminated prematurely due to rounding errors. Experiment with larger values of
TOLER or try scaling the columns of the matrix (see
Section 8).
- IFAIL=3
On entry, | NPLUS2<3, |
or | NPLUS2>M+2, |
or | LDA<M+2. |
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 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 mn2.
It is recommended that, before the routine 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 parameter
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,…,n, the elements of the
jth column are multiplied by the constant
kj, the element
xj of the solution vector
x must be multiplied by
kj if it is desired to recover the solution corresponding to the original matrix
A.
9 Example
Suppose we wish to approximate a set of data by a curve of the form
where
K,
L and
M are unknown. Given values
yi at
5 points
ti we may form the over-determined set of equations for
K,
L and
M
E02GAF is used to solve these in the
l1 sense.
9.1 Program Text
Program Text (e02gafe.f90)
9.2 Program Data
Program Data (e02gafe.d)
9.3 Program Results
Program Results (e02gafe.r)