## MB03MD

### Upper bound for L singular values of a bidiagonal matrix

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

Purpose

```  To compute an upper bound THETA using a bisection method such that
the bidiagonal matrix

|q(1) e(1)  0    ...   0   |
| 0   q(2) e(2)        .   |
J = | .                    .   |
| .                  e(N-1)|
| 0   ...        ...  q(N) |

has precisely L singular values less than or equal to THETA plus
a given tolerance TOL.

This routine is mainly intended to be called only by other SLICOT
routines.

```
Specification
```      SUBROUTINE MB03MD( N, L, THETA, Q, E, Q2, E2, PIVMIN, TOL, RELTOL,
\$                   IWARN, INFO )
C     .. Scalar Arguments ..
INTEGER           INFO, IWARN, L, N
DOUBLE PRECISION  PIVMIN, RELTOL, THETA, TOL
C     .. Array Arguments ..
DOUBLE PRECISION  E(*), E2(*), Q(*), Q2(*)

```
Arguments

Input/Output Parameters

```  N       (input) INTEGER
The order of the bidiagonal matrix J.  N >= 0.

L       (input/output) INTEGER
On entry, L must contain the number of singular values
of J which must be less than or equal to the upper bound
computed by the routine.  0 <= L <= N.
On exit, L may be increased if the L-th smallest singular
value of J has multiplicity greater than 1. In this case,
L is increased by the number of singular values of J which
are larger than its L-th smallest one and approach the
L-th smallest singular value of J within a distance less
than TOL.
If L has been increased, then the routine returns with
IWARN set to 1.

THETA   (input/output) DOUBLE PRECISION
On entry, THETA must contain an initial estimate for the
upper bound to be computed. If THETA < 0.0 on entry, then
one of the following default values is used.
If L = 0, THETA is set to 0.0 irrespective of the input
value of THETA; if L = 1, then THETA is taken as
MIN(ABS(Q(i))), for i = 1,2,...,N; otherwise, THETA is
taken as ABS(Q(N-L+1)).
On exit, THETA contains the computed upper bound such that
the bidiagonal matrix J has precisely L singular values
less than or equal to THETA + TOL.

Q       (input) DOUBLE PRECISION array, dimension (N)
This array must contain the diagonal elements q(1),
q(2),...,q(N) of the bidiagonal matrix J. That is,
Q(i) = J(i,i) for i = 1,2,...,N.

E       (input) DOUBLE PRECISION array, dimension (N-1)
This array must contain the superdiagonal elements
e(1),e(2),...,e(N-1) of the bidiagonal matrix J. That is,
E(k) = J(k,k+1) for k = 1,2,...,N-1.

Q2      (input) DOUBLE PRECISION array, dimension (N)
This array must contain the squares of the diagonal
elements q(1),q(2),...,q(N) of the bidiagonal matrix J.
That is, Q2(i) = J(i,i)**2 for i = 1,2,...,N.

E2      (input) DOUBLE PRECISION array, dimension (N-1)
This array must contain the squares of the superdiagonal
elements e(1),e(2),...,e(N-1) of the bidiagonal matrix J.
That is, E2(k) = J(k,k+1)**2 for k = 1,2,...,N-1.

PIVMIN  (input) DOUBLE PRECISION
The minimum absolute value of a "pivot" in the Sturm
sequence loop.
PIVMIN >= max( max( |q(i)|, |e(k)| )**2*sf_min, sf_min ),
where i = 1,2,...,N, k = 1,2,...,N-1, and sf_min is at
least the smallest number that can divide one without
overflow (see LAPACK Library routine DLAMCH).
Note that this condition is not checked by the routine.

```
Tolerances
```  TOL     DOUBLE PRECISION
This parameter defines the multiplicity of singular values
by considering all singular values within an interval of
length TOL as coinciding. TOL is used in checking how many
singular values are less than or equal to THETA. Also in
computing an appropriate upper bound THETA by a bisection
method, TOL is used as a stopping criterion defining the
minimum (absolute) subinterval width.  TOL >= 0.

RELTOL  DOUBLE PRECISION
This parameter specifies the minimum relative width of an
interval. When an interval is narrower than TOL, or than
RELTOL times the larger (in magnitude) endpoint, then it
is considered to be sufficiently small and bisection has
converged.
RELTOL >= BASE * EPS, where BASE is machine radix and EPS
is machine precision (see LAPACK Library routine DLAMCH).

```
Warning Indicator
```  IWARN   INTEGER
= 0:  no warnings;
= 1:  if the value of L has been increased as the L-th
smallest singular value of J coincides with the
(L+1)-th smallest one.

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

```
Method
```  Let s(i), i = 1,2,...,N, be the N non-negative singular values of
the bidiagonal matrix J arranged so that s(1) >= ... >= s(N) >= 0.
The routine then computes an upper bound T such that s(N-L) > T >=
s(N-L+1) as follows (see ).
First, if the initial estimate of THETA is not specified by the
user then the routine initialises THETA to be an estimate which
is close to the requested value of THETA if s(N-L) >> s(N-L+1).
Second, a bisection method (see [1, 8.5]) is used which generates
a sequence of shrinking intervals [Y,Z] such that either THETA in
[Y,Z] was found (so that J has L singular values less than or
equal to THETA), or

(number of s(i) <= Y) < L < (number of s(i) <= Z).

This bisection method is applied to an associated 2N-by-2N
symmetric tridiagonal matrix T" whose eigenvalues (see ) are
given by s(1),s(2),...,s(N),-s(1),-s(2),...,-s(N). One of the
starting values for the bisection method is the initial value of
THETA. If this value is an upper bound, then the initial lower
bound is set to zero, else the initial upper bound is computed
from the Gershgorin Circle Theorem [1, Theorem 7.2-1], applied to
T". The computation of the "number of s(i) <= Y (or Z)" is
achieved by calling SLICOT Library routine MB03ND, which applies
Sylvester's Law of Inertia or equivalently Sturm sequences
[1, 8.5] to the associated matrix T". If

Z - Y <= MAX( TOL, PIVMIN, RELTOL*MAX( ABS( Y ), ABS( Z ) ) )

at some stage of the bisection method, then at least two singular
values of J lie in the interval [Y,Z] within a distance less than
TOL from each other. In this case, s(N-L) and s(N-L+1) are assumed
to coincide, the upper bound T is set to the value of Z, the value
of L is increased and IWARN is set to 1.

```
References
```   Golub, G.H. and Van Loan, C.F.
Matrix Computations.
The Johns Hopkins University Press, Baltimore, Maryland, 1983.

 Van Huffel, S. and Vandewalle, J.
The Partial Total Least Squares Algorithm.
J. Comput. and Appl. Math., 21, pp. 333-341, 1988.

```
Numerical Aspects
```  None.

```
```  None
```
Example

Program Text

```*     MB03MD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER        ( ZERO = 0.0D0 )
INTEGER          NIN, NOUT
PARAMETER        ( NIN = 5, NOUT = 6 )
INTEGER          NMAX
PARAMETER        ( NMAX = 20 )
*     .. Local Scalars ..
DOUBLE PRECISION PIVMIN, RELTOL, SAFMIN, THETA, TOL
INTEGER          I, INFO, IWARN, L, N
*     .. Local Arrays ..
DOUBLE PRECISION E(NMAX-1), E2(NMAX-1), Q(NMAX), Q2(NMAX)
*     .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL         DLAMCH
*     .. External Subroutines ..
EXTERNAL         MB03MD
*     .. Intrinsic Functions ..
INTRINSIC        MAX
*     .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
*     Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, THETA, L, TOL, RELTOL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) N
ELSE IF ( L.LT.0 .OR. L.GT.N ) THEN
WRITE ( NOUT, FMT = 99990 ) L
ELSE
READ ( NIN, FMT = * ) ( Q(I), I = 1,N )
READ ( NIN, FMT = * ) ( E(I), I = 1,N-1 )
*        Print out the bidiagonal matrix J.
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N - 1
WRITE ( NOUT, FMT = 99996 ) I, I, Q(I), I, (I+1), E(I)
20    CONTINUE
WRITE ( NOUT, FMT = 99995 ) N, N, Q(N)
*        Compute Q**2, E**2, and PIVMIN.
Q2(N) = Q(N)**2
PIVMIN = Q2(N)
DO 40 I = 1, N - 1
Q2(I) = Q(I)**2
E2(I) = E(I)**2
PIVMIN = MAX( PIVMIN, Q2(I), E2(I) )
40    CONTINUE
SAFMIN = DLAMCH( 'Safe minimum' )
PIVMIN = MAX( PIVMIN*SAFMIN, SAFMIN )
TOL = MAX( TOL, ZERO )
IF ( RELTOL.LE.ZERO )
\$      RELTOL = DLAMCH( 'Base' )*DLAMCH( 'Epsilon' )
*        Compute an upper bound THETA such that J has 3 singular values
*        < =  THETA.
CALL MB03MD( N, L, THETA, Q, E, Q2, E2, PIVMIN, TOL, RELTOL,
\$                IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( IWARN.NE.0 ) WRITE ( NOUT, FMT = 99994 ) IWARN
WRITE ( NOUT, FMT = 99993 ) THETA
WRITE ( NOUT, FMT = 99992 ) L
END IF
END IF
STOP
*
99999 FORMAT (' MB03MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB03MD = ',I2)
99997 FORMAT (' The Bidiagonal Matrix J is',/)
99996 FORMAT (2(' (',I1,',',I1,') = ',F7.4,2X))
99995 FORMAT (' (',I1,',',I1,') = ',F7.4)
99994 FORMAT (' IWARN on exit from MB03MD = ',I2,/)
99993 FORMAT (/' The computed value of THETA is ',F7.4)
99992 FORMAT (/' J has ',I2,' singular values < =  THETA')
99991 FORMAT (/' N is out of range.',/' N = ',I5)
99990 FORMAT (/' L is out of range.',/' L = ',I5)
END
```
Program Data
``` MB03MD EXAMPLE PROGRAM DATA
5     -3.0     3     0.0     0.0
1.0  2.0  3.0  4.0  5.0
2.0  3.0  4.0  5.0
```
Program Results
``` MB03MD EXAMPLE PROGRAM RESULTS

The Bidiagonal Matrix J is

(1,1) =  1.0000   (1,2) =  2.0000
(2,2) =  2.0000   (2,3) =  3.0000
(3,3) =  3.0000   (3,4) =  4.0000
(4,4) =  4.0000   (4,5) =  5.0000
(5,5) =  5.0000

The computed value of THETA is  4.7500

J has  3 singular values < =  THETA
```