sstebz.c

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/*
* ======================================================================
* NIST Guide to Available Math Software.
* Fullsource for module SSYEVX.C from package CLAPACK.
* Retrieved from NETLIB on Fri Mar 10 14:23:44 2000.
* ======================================================================
*/
#include <f2c.h>

/* Subroutine */ int sstebz_(char *range, char *order, integer *n, real *vl, 
	real *vu, integer *il, integer *iu, real *abstol, real *d, real *e, 
	integer *m, integer *nsplit, real *w, integer *iblock, integer *
	isplit, real *work, integer *iwork, integer *info)
{
/*  -- LAPACK routine (version 2.0) --   
       Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,   
       Courant Institute, Argonne National Lab, and Rice University   
       September 30, 1994   


    Purpose   
    =======   

    SSTEBZ computes the eigenvalues of a symmetric tridiagonal   
    matrix T.  The user may ask for all eigenvalues, all eigenvalues   
    in the half-open interval (VL, VU], or the IL-th through IU-th   
    eigenvalues.   

    To avoid overflow, the matrix must be scaled so that its   
    largest element is no greater than overflow**(1/2) *   
    underflow**(1/4) in absolute value, and for greatest   
    accuracy, it should not be much smaller than that.   

    See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal   
    Matrix", Report CS41, Computer Science Dept., Stanford   
    University, July 21, 1966.   

    Arguments   
    =========   

    RANGE   (input) CHARACTER   
            = 'A': ("All")   all eigenvalues will be found.   
            = 'V': ("Value") all eigenvalues in the half-open interval   
                             (VL, VU] will be found.   
            = 'I': ("Index") the IL-th through IU-th eigenvalues (of the 
  
                             entire matrix) will be found.   

    ORDER   (input) CHARACTER   
            = 'B': ("By Block") the eigenvalues will be grouped by   
                                split-off block (see IBLOCK, ISPLIT) and 
  
                                ordered from smallest to largest within   
                                the block.   
            = 'E': ("Entire matrix")   
                                the eigenvalues for the entire matrix   
                                will be ordered from smallest to   
                                largest.   

    N       (input) INTEGER   
            The order of the tridiagonal matrix T.  N >= 0.   

    VL      (input) REAL   
    VU      (input) REAL   
            If RANGE='V', the lower and upper bounds of the interval to   
            be searched for eigenvalues.  Eigenvalues less than or equal 
  
            to VL, or greater than VU, will not be returned.  VL < VU.   
            Not referenced if RANGE = 'A' or 'I'.   

    IL      (input) INTEGER   
    IU      (input) INTEGER   
            If RANGE='I', the indices (in ascending order) of the   
            smallest and largest eigenvalues to be returned.   
            1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.   
            Not referenced if RANGE = 'A' or 'V'.   

    ABSTOL  (input) REAL   
            The absolute tolerance for the eigenvalues.  An eigenvalue   
            (or cluster) is considered to be located if it has been   
            determined to lie in an interval whose width is ABSTOL or   
            less.  If ABSTOL is less than or equal to zero, then ULP*|T| 
  
            will be used, where |T| means the 1-norm of T.   

            Eigenvalues will be computed most accurately when ABSTOL is   
            set to twice the underflow threshold 2*SLAMCH('S'), not zero. 
  

    D       (input) REAL array, dimension (N)   
            The n diagonal elements of the tridiagonal matrix T.   

    E       (input) REAL array, dimension (N-1)   
            The (n-1) off-diagonal elements of the tridiagonal matrix T. 
  

    M       (output) INTEGER   
            The actual number of eigenvalues found. 0 <= M <= N.   
            (See also the description of INFO=2,3.)   

    NSPLIT  (output) INTEGER   
            The number of diagonal blocks in the matrix T.   
            1 <= NSPLIT <= N.   

    W       (output) REAL array, dimension (N)   
            On exit, the first M elements of W will contain the   
            eigenvalues.  (SSTEBZ may use the remaining N-M elements as   
            workspace.)   

    IBLOCK  (output) INTEGER array, dimension (N)   
            At each row/column j where E(j) is zero or small, the   
            matrix T is considered to split into a block diagonal   
            matrix.  On exit, if INFO = 0, IBLOCK(i) specifies to which   
            block (from 1 to the number of blocks) the eigenvalue W(i)   
            belongs.  (SSTEBZ may use the remaining N-M elements as   
            workspace.)   

    ISPLIT  (output) INTEGER array, dimension (N)   
            The splitting points, at which T breaks up into submatrices. 
  
            The first submatrix consists of rows/columns 1 to ISPLIT(1), 
  
            the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),   
            etc., and the NSPLIT-th consists of rows/columns   
            ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.   
            (Only the first NSPLIT elements will actually be used, but   
            since the user cannot know a priori what value NSPLIT will   
            have, N words must be reserved for ISPLIT.)   

    WORK    (workspace) REAL array, dimension (4*N)   

    IWORK   (workspace) INTEGER array, dimension (3*N)   

    INFO    (output) INTEGER   
            = 0:  successful exit   
            < 0:  if INFO = -i, the i-th argument had an illegal value   
            > 0:  some or all of the eigenvalues failed to converge or   
                  were not computed:   
                  =1 or 3: Bisection failed to converge for some   
                          eigenvalues; these eigenvalues are flagged by a 
  
                          negative block number.  The effect is that the 
  
                          eigenvalues may not be as accurate as the   
                          absolute and relative tolerances.  This is   
                          generally caused by unexpectedly inaccurate   
                          arithmetic.   
                  =2 or 3: RANGE='I' only: Not all of the eigenvalues   
                          IL:IU were found.   
                          Effect: M < IU+1-IL   
                          Cause:  non-monotonic arithmetic, causing the   
                                  Sturm sequence to be non-monotonic.   
                          Cure:   recalculate, using RANGE='A', and pick 
  
                                  out eigenvalues IL:IU.  In some cases, 
  
                                  increasing the PARAMETER "FUDGE" may   
                                  make things work.   
                  = 4:    RANGE='I', and the Gershgorin interval   
                          initially used was too small.  No eigenvalues   
                          were computed.   
                          Probable cause: your machine has sloppy   
                                          floating-point arithmetic.   
                          Cure: Increase the PARAMETER "FUDGE",   
                                recompile, and try again.   

    Internal Parameters   
    ===================   

    RELFAC  REAL, default = 2.0e0   
            The relative tolerance.  An interval (a,b] lies within   
            "relative tolerance" if  b-a < RELFAC*ulp*max(|a|,|b|),   
            where "ulp" is the machine precision (distance from 1 to   
            the next larger floating point number.)   

    FUDGE   REAL, default = 2   
            A "fudge factor" to widen the Gershgorin intervals.  Ideally, 
  
            a value of 1 should work, but on machines with sloppy   
            arithmetic, this needs to be larger.  The default for   
            publicly released versions should be large enough to handle   
            the worst machine around.  Note that this has no effect   
            on accuracy of the solution.   

    ===================================================================== 
  


    
   Parameter adjustments   
       Function Body */
    /* Table of constant values */
    static integer c__1 = 1;
    static integer c_n1 = -1;
    static integer c__3 = 3;
    static integer c__2 = 2;
    static integer c__0 = 0;
    
    /* System generated locals */
    integer i__1, i__2, i__3;
    real r__1, r__2, r__3, r__4, r__5;
    /* Builtin functions */
    double sqrt(doublereal), log(doublereal);
    /* Local variables */
    static integer iend, ioff, iout, itmp1, j, jdisc;
    extern logical lsame_(char *, char *);
    static integer iinfo;
    static real atoli;
    static integer iwoff;
    static real bnorm;
    static integer itmax;
    static real wkill, rtoli, tnorm;
    static integer ib, jb, ie, je, nb;
    static real gl;
    static integer im, in, ibegin;
    static real gu;
    static integer iw;
    static real wl;
    static integer irange, idiscl;
    extern doublereal slamch_(char *);
    static real safemn, wu;
    static integer idumma[1];
    extern integer ilaenv_(integer *, char *, char *, integer *, integer *, 
	    integer *, integer *, ftnlen, ftnlen);
    extern /* Subroutine */ int xerbla_(char *, integer *);
    static integer idiscu;
    extern /* Subroutine */ int slaebz_(integer *, integer *, integer *, 
	    integer *, integer *, integer *, real *, real *, real *, real *, 
	    real *, real *, integer *, real *, real *, integer *, integer *, 
	    real *, integer *, integer *);
    static integer iorder;
    static logical ncnvrg;
    static real pivmin;
    static logical toofew;
    static integer nwl;
    static real ulp, wlu, wul;
    static integer nwu;
    static real tmp1, tmp2;



#define IDUMMA(I) idumma[(I)]
#define IWORK(I) iwork[(I)-1]
#define WORK(I) work[(I)-1]
#define ISPLIT(I) isplit[(I)-1]
#define IBLOCK(I) iblock[(I)-1]
#define W(I) w[(I)-1]
#define E(I) e[(I)-1]
#define D(I) d[(I)-1]


    *info = 0;

/*     Decode RANGE */

    if (lsame_(range, "A")) {
	irange = 1;
    } else if (lsame_(range, "V")) {
	irange = 2;
    } else if (lsame_(range, "I")) {
	irange = 3;
    } else {
	irange = 0;
    }

/*     Decode ORDER */

    if (lsame_(order, "B")) {
	iorder = 2;
    } else if (lsame_(order, "E")) {
	iorder = 1;
    } else {
	iorder = 0;
    }

/*     Check for Errors */

    if (irange <= 0) {
	*info = -1;
    } else if (iorder <= 0) {
	*info = -2;
    } else if (*n < 0) {
	*info = -3;
    } else if (irange == 2 && *vl >= *vu) {
	*info = -5;
    } else if (irange == 3 && (*il < 1 || *il > max(1,*n))) {
	*info = -6;
    } else if (irange == 3 && (*iu < min(*n,*il) || *iu > *n)) {
	*info = -7;
    }

    if (*info != 0) {
	i__1 = -(*info);
	xerbla_("SSTEBZ", &i__1);
	return 0;
    }

/*     Initialize error flags */

    *info = 0;
    ncnvrg = FALSE_;
    toofew = FALSE_;

/*     Quick return if possible */

    *m = 0;
    if (*n == 0) {
	return 0;
    }

/*     Simplifications: */

    if (irange == 3 && *il == 1 && *iu == *n) {
	irange = 1;
    }

/*     Get machine constants   
       NB is the minimum vector length for vector bisection, or 0   
       if only scalar is to be done. */

    safemn = slamch_("S");
    ulp = slamch_("P");
    rtoli = ulp * 2.f;
    nb = ilaenv_(&c__1, "SSTEBZ", " ", n, &c_n1, &c_n1, &c_n1, 6L, 1L);
    if (nb <= 1) {
	nb = 0;
    }

/*     Special Case when N=1 */

    if (*n == 1) {
	*nsplit = 1;
	ISPLIT(1) = 1;
	if (irange == 2 && (*vl >= D(1) || *vu < D(1))) {
	    *m = 0;
	} else {
	    W(1) = D(1);
	    IBLOCK(1) = 1;
	    *m = 1;
	}
	return 0;
    }

/*     Compute Splitting Points */

    *nsplit = 1;
    WORK(*n) = 0.f;
    pivmin = 1.f;

    i__1 = *n;
    for (j = 2; j <= *n; ++j) {
/* Computing 2nd power */
	r__1 = E(j - 1);
	tmp1 = r__1 * r__1;
/* Computing 2nd power */
	r__2 = ulp;
	if ((r__1 = D(j) * D(j - 1), dabs(r__1)) * (r__2 * r__2) + safemn > 
		tmp1) {
	    ISPLIT(*nsplit) = j - 1;
	    ++(*nsplit);
	    WORK(j - 1) = 0.f;
	} else {
	    WORK(j - 1) = tmp1;
	    pivmin = dmax(pivmin,tmp1);
	}
/* L10: */
    }
    ISPLIT(*nsplit) = *n;
    pivmin *= safemn;

/*     Compute Interval and ATOLI */

    if (irange == 3) {

/*        RANGE='I': Compute the interval containing eigenvalues   
                     IL through IU.   

          Compute Gershgorin interval for entire (split) matrix   
          and use it as the initial interval */

	gu = D(1);
	gl = D(1);
	tmp1 = 0.f;