How To Solve A Routine
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1 NAG Fortran Library Routine Document Note: before using this routine, please read the Users Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details. 1 Purpose calculates an l 1 solution to an over-determined system of linear equations. 2 Specification SUBROUTINE (M, N, MDIM, NDIM, A, B, TOL, RELERR, X, RESMAX, IRANK, 1 ITER, IFAIL) INTEGER M, N, MDIM, NDIM, IRANK, ITER, IFAIL real A(NDIM,MDIM), B(M), TOL, RELERR, X(N), RESMAX 3 Description Given a matrix A with m rows and n columns ðm nþ and a vector b with m elements, the routine calculates an l 1 solution to the over-determined system of equations Ax ¼ b: That is to say, it calculates a vector x, with n elements, which minimizes the l 1 norm of the residuals (the absolutely largest residual) rðxþ ¼ max jr ij 1im where the residuals r i are given by r i ¼ b i Xn a ij x j; i ¼ 1; 2;...;m: j¼1 Here a ij is the element in row i and column j of A, b i is the ith element of b and x j the jth element of x. The matrix A need not be of full rank. The solution is not unique in this case, and may not be unique even if A is of full rank. Alternatively, in applications where a complete minimization of the l 1 norm is not necessary, the user may obtain an approximate solution, usually in shorter time, by giving an appropriate value to the parameter RELERR. Typically in applications to data fitting, data consisting of m points with co-ordinates ðt i ;y i Þ is to be approximated in the l 1 norm by a linear combination of known functions j ðtþ, 1 1 ðtþþ 2 2 ðtþþþ n n ðtþ: This is equivalent to finding an l 1 solution to the over-determined system of equations X n j¼1 j ðt i Þ j ¼ y i; i ¼ 1; 2;...;m: Thus if, for each value of i and j the element a ij of the matrix A above is set equal to the value of j ðt i Þ and b i is set equal to y i, the solution vector x will contain the required values of the j. Note that the independent variable t above can, instead, be a vector of several independent variables (this includes the case where each 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 dual formation of the l 1 problem (see Barrodale and Phillips (1974) and Barrodale and Phillips (1975)). The.1
2 NAG Fortran Library Manual modifications are designed to improve the efficiency and stability of the simplex method for this particular application. 4 References Barrodale I and Phillips C (1974) An improved algorithm for discrete Chebyshev linear approximation Proc. 4th Manitoba Conf. Numerical Mathematics University of Manitoba, Canada Barrodale I and Phillips C (1975) Solution of an overdetermined system of linear equations in the Chebyshev norm [F4] (Algorithm 495) ACM Trans. Math. Software 1 (3) Parameters 1: M INTEGER Input On entry: the number of equations, m (the number of rows of the matrix A). Constraint: M N. 2: N INTEGER Input On entry: the number of unknowns, n (the number of columns of the matrix A). Constraint: N 1. 3: MDIM INTEGER Input On entry: the second dimension of the array A as declared in the (sub)program from which is called. Constraint: MDIM M þ 1. 4: NDIM INTEGER Input On entry: the first dimension of the array A as declared in the (sub)program from which is called. Constraint: NDIM N þ 3. 5: A(NDIM,MDIM) real array Input/Output On entry: Aðj; iþ must contain a ij, the element in the ith row and jth column of the matrix A for, i ¼ 1; 2;...;m; j ¼ 1; 2;...;n (that is, the transpose of the matrix). The remaining elements need not be set. Preferably, the columns of the matrix A (rows of the parameter A) should be scaled before entry: see Section 7. On exit: A contains the last simplex tableau. 6: B(M) real array Input/Output On entry: b i, the ith element of the vector b, for i ¼ 1; 2;...;m. On exit: the ith residual r i corresponding to the solution vector x, for i ¼ 1; 2;...;m. Note however that these residuals may contain few significant figures, especially when RESMAX is within one or two orders of magnitude of TOL. Indeed if RESMAX TOL, the elements BðiÞ may all be set to zero. It is therefore often advisable to compute the residuals directly. 7: TOL real Input On entry: a threshold below which numbers are regarded as zero. The recommended threshold value is 10:0, where is the machine precision. If TOL 0:0 on entry, the recommended value is used within the routine. If premature termination occurs, a larger value for TOL may result in a valid solution. Suggested value:
3 8: RELERR real Input/Output On entry: RELERR must be set to a bound on the relative error acceptable in the maximum residual at the solution. If RELERR 0:0, then the l 1 solution is computed, and RELERR is set to 0.0 on exit. If RELERR > 0:0, then the routine obtains instead an approximate solution for which the largest residual is less than 1:0 þ RELERR times that of the l 1 solution; on exit, RELERR contains a smaller value such that the above bound still applies. (The usual result of this option, say with RELERR ¼ 0:1, is a saving in the number of simplex iterations). On exit: RELERR is altered as described above. 9: X(N) real array Output On exit: if IFAIL ¼ 0 or 1, XðjÞ contains the jth element of the solution vector x, for j ¼ 1; 2;...;n. Whether this is an l 1 solution or an approximation to one, depends on the value of RELERR on entry. 10: RESMAX real Output On exit: if IFAIL ¼ 0 or 1, RESMAX contains the absolute value of the largest residual(s) for the solution vector x. (See B above.) 11: IRANK INTEGER Output On exit: if IFAIL ¼ 0 or 1, IRANK contains the computed rank of the matrix A. 12: ITER INTEGER Output On exit: if IFAIL ¼ 0 or 1, ITER contains the number of iterations taken by the simplex method. 13: IFAIL INTEGER Input/Output On entry: IFAIL must be set to 0, 1 or 1. Users who are unfamiliar with this parameter should refer to Chapter P01 for details. On exit: IFAIL ¼ 0 unless the routine detects an error (see Section 6). 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, because for this routine the values of the output parameters may be useful even if IFAIL 6¼ 0 on exit, the recommended value is 1. When the value 1 or 1 is used it is essential to test the value of IFAIL on exit. 6 Error Indicators and Warnings If on entry IFAIL ¼ 0or 1, explanatory error messages are output on the current error message unit (as defined by X04AAF). Errors or warnings detected by the routine: IFAIL ¼ 1 An optimal solution has been obtained but this may not be unique (perhaps simply because the matrix A is not of full rank, i.e., IRANK < N). IFAIL ¼ 2 The calculations have terminated prematurely due to rounding errors. Experiment with larger values of TOL or try rescaling the columns of the matrix (see Section 8)..3
4 NAG Fortran Library Manual IFAIL ¼ 3 On entry, NDIM < N þ 3, or MDIM < M þ 1, or M < N, or N < 1. 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 þ 1 equations which have residuals of largest absolute value. 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 is approximately proportional to mn 2. 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 TOL 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 ¼ 1; 2;...;n, the elements of the jth 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. 9 Example Suppose we wish to approximate a set of data by a curve of the form y ¼ Ke t þ Le t þ M where K, L and M are unknown. Given values y i at 5 points t i we may form the over-determined set of equations for K, L and M e t i K þ e t i L þ M ¼ y i ; i ¼ 1; 2;...; 5: is used to solve these in the l 1 sense. 9.1 Program Text Note: the listing of the example program presented below uses bold italicised terms to denote precision-dependent details. Please read the Users Note for your implementation to check the interpretation of these terms. As explained in the Essential Introduction to this manual, the results produced may not be identical for all implementations. Example Program Text Mark 14 Revised. NAG Copyright Parameters.. INTEGER N, MMAX, NDIM, MDIM PARAMETER (N=3,MMAX=5,NDIM=N+3,MDIM=MMAX+1) INTEGER NIN, NOUT PARAMETER (NIN=5,NOUT=6).. Local Scalars.. real RELERR, RESMAX, T, TOL INTEGER I, IFAIL, IRANK, ITER, M.. Local Arrays.. real A(NDIM,MDIM), B(MMAX), X(N).. External Subroutines.. EXTERNAL.. Intrinsic Functions.. INTRINSIC EXP.. Executable Statements...4
5 WRITE (NOUT,) Example Program Results Skip heading in data file READ (NIN,) READ (NIN,) M IF (M.GT.0.AND. M.LE.MMAX) THEN DO 20 I = 1, M READ (NIN,) T, B(I) A(1,I) = EXP(T) A(2,I) = EXP(-T) A(3,I) = 1.0e0 20 CONTINUE TOL = 0.0e0 RELERR = 0.0e0 IFAIL = 1 CALL (M,N,MDIM,NDIM,A,B,TOL,RELERR,X,RESMAX,IRANK,ITER, + IFAIL) WRITE (NOUT,) IF (IFAIL.LE.1) THEN WRITE (NOUT,99999) RESMAX =, RESMAX, Rank =, IRANK, + Iterations =, ITER, IFAIL =, IFAIL WRITE (NOUT,) WRITE (NOUT,) Solution WRITE (NOUT,99998) (X(I),I=1,N) ELSE WRITE (NOUT,99997) fails with error, IFAIL END IF END IF STOP FORMAT (1X,A,e10.2,A,I5,A,I5,A,I5) FORMAT (1X,6F10.4) FORMAT (1X,A,I2) END 9.2 Program Data Example Program Data Program Results Example Program Results RESMAX = 0.10E-02 Rank = 3 Iterations = 4 IFAIL = 0 Solution (last)
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