## MB02QD

### Solution of a linear least squares problem corresponding to specified free elements

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

      To compute a solution, optionally corresponding to specified free
      elements, to a real linear least squares problem:

          minimize || A * X - B ||

      using a complete orthogonal factorization of the M-by-N matrix A,
      which may be rank-deficient.

      Several right hand side vectors b and solution vectors x can be
      handled in a single call; they are stored as the columns of the
      M-by-NRHS right hand side matrix B and the N-by-NRHS solution
      matrix X.

Specification

          SUBROUTINE MB02QD( JOB, INIPER, M, N, NRHS, RCOND, SVLMAX, A, LDA,
         $                   B, LDB, Y, JPVT, RANK, SVAL, DWORK, LDWORK,
         $                   INFO )
    C     .. Scalar Arguments ..
          CHARACTER          INIPER, JOB
          INTEGER            INFO, LDA, LDB, LDWORK, M, N, NRHS, RANK
          DOUBLE PRECISION   RCOND, SVLMAX
    C     .. Array Arguments ..
          INTEGER            JPVT( * )
          DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), DWORK( * ),
         $                   SVAL( 3 ), Y ( * )

Arguments

Mode Parameters

      JOB     CHARACTER*1
              Specifies whether or not a standard least squares solution
              must be computed, as follows:
              = 'L':  Compute a standard least squares solution (Y = 0);
              = 'F':  Compute a solution with specified free elements
                      (given in Y).

      INIPER  CHARACTER*1
              Specifies whether an initial column permutation, defined
              by JPVT, must be performed, as follows:
              = 'P':  Perform an initial column permutation;
              = 'N':  Do not perform an initial column permutation.

Input/Output Parameters

      M       (input) INTEGER
              The number of rows of the matrix A.  M >= 0.

      N       (input) INTEGER
              The number of columns of the matrix A.  N >= 0.

      NRHS    (input) INTEGER
              The number of right hand sides, i.e., the number of
              columns of the matrices B and X.  NRHS >= 0.

      RCOND   (input) DOUBLE PRECISION
              RCOND is used to determine the effective rank of A, which
              is defined as the order of the largest leading triangular
              submatrix R11 in the QR factorization with pivoting of A,
              whose estimated condition number is less than 1/RCOND.
              0 <= RCOND <= 1.

      SVLMAX  (input) DOUBLE PRECISION
              If A is a submatrix of another matrix C, and the rank
              decision should be related to that matrix, then SVLMAX
              should be an estimate of the largest singular value of C
              (for instance, the Frobenius norm of C).  If this is not
              the case, the input value SVLMAX = 0 should work.
              SVLMAX >= 0.

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading M-by-N part of this array must
              contain the given matrix A.
              On exit, the leading M-by-N part of this array contains
              details of its complete orthogonal factorization:
              the leading RANK-by-RANK upper triangular part contains
              the upper triangular factor T11 (see METHOD);
              the elements below the diagonal, with the entries 2 to
              min(M,N)+1 of the array DWORK, represent the orthogonal
              matrix Q as a product of min(M,N) elementary reflectors
              (see METHOD);
              the elements of the subarray A(1:RANK,RANK+1:N), with the
              next RANK entries of the array DWORK, represent the
              orthogonal matrix Z as a product of RANK elementary
              reflectors (see METHOD).

      LDA     INTEGER
              The leading dimension of the array A.  LDA >= max(1,M).

      B       (input/output) DOUBLE PRECISION array, dimension
              (LDB,NRHS)
              On entry, the leading M-by-NRHS part of this array must
              contain the right hand side matrix B.
              On exit, the leading N-by-NRHS part of this array contains
              the solution matrix X.
              If M >= N and RANK = N, the residual sum-of-squares for
              the solution in the i-th column is given by the sum of
              squares of elements N+1:M in that column.
              If NRHS = 0, this array is not referenced, and the routine
              returns the effective rank of A, and its QR factorization.

      LDB     INTEGER
              The leading dimension of the array B.  LDB >= max(1,M,N).

      Y       (input) DOUBLE PRECISION array, dimension ( N*NRHS )
              If JOB = 'F', the elements Y(1:(N-RANK)*NRHS) are used as
              free elements in computing the solution (see METHOD).
              The remaining elements are not referenced.
              If JOB = 'L', or NRHS = 0, this array is not referenced.

      JPVT    (input/output) INTEGER array, dimension (N)
              On entry with INIPER = 'P', if JPVT(i) <> 0, the i-th
              column of A is an initial column, otherwise it is a free
              column.  Before the QR factorization of A, all initial
              columns are permuted to the leading positions; only the
              remaining free columns are moved as a result of column
              pivoting during the factorization.
              If INIPER = 'N', JPVT need not be set on entry.
              On exit, if JPVT(i) = k, then the i-th column of A*P
              was the k-th column of A.

      RANK    (output) INTEGER
              The effective rank of A, i.e., the order of the submatrix
              R11.  This is the same as the order of the submatrix T11
              in the complete orthogonal factorization of A.

      SVAL    (output) DOUBLE PRECISION array, dimension ( 3 )
              The estimates of some of the singular values of the
              triangular factor R11:
              SVAL(1): largest singular value of  R(1:RANK,1:RANK);
              SVAL(2): smallest singular value of R(1:RANK,1:RANK);
              SVAL(3): smallest singular value of R(1:RANK+1,1:RANK+1),
                       if RANK < MIN( M, N ), or of R(1:RANK,1:RANK),
                       otherwise.
              If the triangular factorization is a rank-revealing one
              (which will be the case if the leading columns were well-
              conditioned), then SVAL(1) will also be an estimate for
              the largest singular value of A, and SVAL(2) and SVAL(3)
              will be estimates for the RANK-th and (RANK+1)-st singular
              values of A, respectively.
              By examining these values, one can confirm that the rank
              is well defined with respect to the chosen value of RCOND.
              The ratio SVAL(1)/SVAL(2) is an estimate of the condition
              number of R(1:RANK,1:RANK).

Workspace

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0, DWORK(1) returns the optimal value
              of LDWORK, and the entries 2 to min(M,N) + RANK + 1
              contain the scalar factors of the elementary reflectors
              used in the complete orthogonal factorization of A.
              Among the entries 2 to min(M,N) + 1, only the first RANK
              elements are useful, if INIPER = 'N'.

      LDWORK  INTEGER
              The length of the array DWORK.
              LDWORK >= max( min(M,N)+3*N+1, 2*min(M,N)+NRHS )
              For optimum performance LDWORK should be larger.

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value.

Method

      If INIPER = 'P', the routine first computes a QR factorization
      with column pivoting:
          A * P = Q * [ R11 R12 ]
                      [  0  R22 ]
      with R11 defined as the largest leading submatrix whose estimated
      condition number is less than 1/RCOND.  The order of R11, RANK,
      is the effective rank of A.
      If INIPER = 'N', the effective rank is estimated during a
      truncated QR factorization (with column pivoting) process, and
      the submatrix R22 is not upper triangular, but full and of small
      norm. (See SLICOT Library routines MB03OD or MB03OY, respectively,
      for further details.)

      Then, R22 is considered to be negligible, and R12 is annihilated
      by orthogonal transformations from the right, arriving at the
      complete orthogonal factorization:
         A * P = Q * [ T11 0 ] * Z
                     [  0  0 ]
      The solution is then
         X = P * Z' [ inv(T11)*Q1'*B ]
                    [        Y       ]
      where Q1 consists of the first RANK columns of Q, and Y contains
      free elements (if JOB = 'F'), or is zero (if JOB = 'L').

Numerical Aspects

      The algorithm is backward stable.

Further Comments

      Significant gain in efficiency is possible for small-rank problems
      using truncated QR factorization (option INIPER = 'N').

Example

Program Text

    *     MB02QD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX, MMAX, NRHSMX
          PARAMETER        ( NMAX = 20, MMAX = 20, NRHSMX = 20 )
          INTEGER          LDA, LDB
          PARAMETER        ( LDA = MMAX, LDB = MAX( MMAX, NMAX ) )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = MAX(   MIN( MMAX, NMAX) + 3*NMAX + 1,
         $                                 2*MIN( MMAX, NMAX) + NRHSMX ) )
    *     .. Local Scalars ..
          DOUBLE PRECISION RCOND, SVLMAX
          INTEGER          I, INFO, J, M, N, NRHS, RANK
          CHARACTER*1      INIPER, JOB
    *     .. Local Arrays ..
          INTEGER          JPVT(NMAX)
          DOUBLE PRECISION A(LDA,NMAX), B(LDB,NRHSMX), DWORK(LDWORK),
         $                 SVAL(3), Y(NMAX*NRHSMX)
    *     .. External Functions ..
          LOGICAL          LSAME
          EXTERNAL         LSAME
    *     .. External Subroutines ..
          EXTERNAL         MB02QD
    *     .. Intrinsic Functions ..
          INTRINSIC        MAX, MIN
    *     .. Executable Statements ..
    *
          WRITE ( NOUT, FMT = 99999 )
    *     Skip the heading in the data file and read the data.
          READ ( NIN, FMT = '()' )
          READ ( NIN, FMT = * ) M, N, NRHS, RCOND, SVLMAX, JOB, INIPER
          IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
             WRITE ( NOUT, FMT = 99994 ) M
          ELSE
             IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
                WRITE ( NOUT, FMT = 99993 ) N
             ELSE
                IF ( NRHS.LT.0 .OR. NRHS.GT.NRHSMX ) THEN
                   WRITE ( NOUT, FMT = 99992 ) NRHS
                ELSE
                   READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,M )
                   READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,NRHS ), I = 1,M )
                   IF ( LSAME( JOB, 'F' ) )
         $            READ ( NIN, FMT = * ) ( Y(I),  I = 1,N*NRHS )
                   IF ( LSAME( INIPER, 'P' ) )
         $            READ ( NIN, FMT = * ) ( JPVT(I),  I = 1,N )
    *              Find the least squares solution.
                   CALL MB02QD( JOB, INIPER, M, N, NRHS, RCOND, SVLMAX, A,
         $                      LDA, B, LDB, Y, JPVT, RANK, SVAL, DWORK,
         $                      LDWORK, INFO )
    *
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      WRITE ( NOUT, FMT = 99997 ) RANK, SVAL
                      WRITE ( NOUT, FMT = 99996 )
                      DO 10 I = 1, N
                         WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,NRHS )
       10             CONTINUE
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' MB02QD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from MB02QD =',I2)
    99997 FORMAT (' The effective rank of A =',I2,/
         $        ' Estimates of the singular values SVAL = '/3(1X,F8.4))
    99996 FORMAT (' The least squares solution is')
    99995 FORMAT (20(1X,F8.4))
    99994 FORMAT (/' M is out of range.',/' M = ',I5)
    99993 FORMAT (/' N is out of range.',/' N = ',I5)
    99992 FORMAT (/' NRHS is out of range.',/' NRHS = ',I5)
          END

Program Data

     MB02QD EXAMPLE PROGRAM DATA
       4   3   2 2.3D-16     0.0     L     N
       2.0  2.0 -3.0
       3.0  3.0 -1.0
       4.0  4.0 -5.0
      -1.0 -1.0 -2.0
       1.0  0.0
       0.0  0.0
       0.0  0.0
       0.0  1.0

Program Results

     MB02QD EXAMPLE PROGRAM RESULTS

     The effective rank of A = 2
     Estimates of the singular values SVAL =
       7.8659   2.6698   0.0000
     The least squares solution is
      -0.0034  -0.1054
      -0.0034  -0.1054
      -0.0816  -0.1973

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