slahqr.c

来自「提供矩阵类的函数库」· C语言 代码 · 共 597 行 · 第 1/2 页

#include "blaswrap.h"
/*  -- translated by f2c (version 19990503).
   You must link the resulting object file with the libraries:
	-lf2c -lm   (in that order)
*/

#include "f2c.h"

/* Common Block Declarations */

struct {
    real ops, itcnt;
} latime_;

#define latime_1 latime_

/* Table of constant values */

static integer c__1 = 1;

/* Subroutine */ int slahqr_(logical *wantt, logical *wantz, integer *n, 
	integer *ilo, integer *ihi, real *h__, integer *ldh, real *wr, real *
	wi, integer *iloz, integer *ihiz, real *z__, integer *ldz, integer *
	info)
{
    /* System generated locals */
    integer h_dim1, h_offset, z_dim1, z_offset, i__1, i__2, i__3, i__4;
    real r__1, r__2;

    /* Builtin functions */
    double sqrt(doublereal), r_sign(real *, real *);

    /* Local variables */
    static real h43h34, disc, unfl, ovfl, work[1], opst;
    extern /* Subroutine */ int srot_(integer *, real *, integer *, real *, 
	    integer *, real *, real *);
    static integer i__, j, k, l, m;
    static real s, v[3];
    static integer i1, i2;
    extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *, 
	    integer *);
    static real t1, t2, t3, v1, v2, v3;
    extern /* Subroutine */ int slanv2_(real *, real *, real *, real *, real *
	    , real *, real *, real *, real *, real *);
    static real h00, h10, h11, h12, h21, h22, h33, h44;
    static integer nh;
    static real cs;
    extern /* Subroutine */ int slabad_(real *, real *);
    static integer nr;
    static real sn;
    extern doublereal slamch_(char *);
    static integer nz;
    extern /* Subroutine */ int slarfg_(integer *, real *, real *, integer *, 
	    real *);
    extern doublereal slanhs_(char *, integer *, real *, integer *, real *);
    static real smlnum, ave, h33s, h44s;
    static integer itn, its;
    static real ulp, sum, tst1;


#define h___ref(a_1,a_2) h__[(a_2)*h_dim1 + a_1]
#define z___ref(a_1,a_2) z__[(a_2)*z_dim1 + a_1]


/*  -- LAPACK auxiliary routine (instrum. to count ops. version 3.0) --   
       Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,   
       Courant Institute, Argonne National Lab, and Rice University   
       June 30, 1999   

       Common block to return operation count.   

    Purpose   
    =======   

    SLAHQR is an auxiliary routine called by SHSEQR to update the   
    eigenvalues and Schur decomposition already computed by SHSEQR, by   
    dealing with the Hessenberg submatrix in rows and columns ILO to IHI.   

    Arguments   
    =========   

    WANTT   (input) LOGICAL   
            = .TRUE. : the full Schur form T is required;   
            = .FALSE.: only eigenvalues are required.   

    WANTZ   (input) LOGICAL   
            = .TRUE. : the matrix of Schur vectors Z is required;   
            = .FALSE.: Schur vectors are not required.   

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

    ILO     (input) INTEGER   
    IHI     (input) INTEGER   
            It is assumed that H is already upper quasi-triangular in   
            rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless   
            ILO = 1). SLAHQR works primarily with the Hessenberg   
            submatrix in rows and columns ILO to IHI, but applies   
            transformations to all of H if WANTT is .TRUE..   
            1 <= ILO <= max(1,IHI); IHI <= N.   

    H       (input/output) REAL array, dimension (LDH,N)   
            On entry, the upper Hessenberg matrix H.   
            On exit, if WANTT is .TRUE., H is upper quasi-triangular in   
            rows and columns ILO:IHI, with any 2-by-2 diagonal blocks in   
            standard form. If WANTT is .FALSE., the contents of H are   
            unspecified on exit.   

    LDH     (input) INTEGER   
            The leading dimension of the array H. LDH >= max(1,N).   

    WR      (output) REAL array, dimension (N)   
    WI      (output) REAL array, dimension (N)   
            The real and imaginary parts, respectively, of the computed   
            eigenvalues ILO to IHI are stored in the corresponding   
            elements of WR and WI. If two eigenvalues are computed as a   
            complex conjugate pair, they are stored in consecutive   
            elements of WR and WI, say the i-th and (i+1)th, with   
            WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., the   
            eigenvalues are stored in the same order as on the diagonal   
            of the Schur form returned in H, with WR(i) = H(i,i), and, if   
            H(i:i+1,i:i+1) is a 2-by-2 diagonal block,   
            WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).   

    ILOZ    (input) INTEGER   
    IHIZ    (input) INTEGER   
            Specify the rows of Z to which transformations must be   
            applied if WANTZ is .TRUE..   
            1 <= ILOZ <= ILO; IHI <= IHIZ <= N.   

    Z       (input/output) REAL array, dimension (LDZ,N)   
            If WANTZ is .TRUE., on entry Z must contain the current   
            matrix Z of transformations accumulated by SHSEQR, and on   
            exit Z has been updated; transformations are applied only to   
            the submatrix Z(ILOZ:IHIZ,ILO:IHI).   
            If WANTZ is .FALSE., Z is not referenced.   

    LDZ     (input) INTEGER   
            The leading dimension of the array Z. LDZ >= max(1,N).   

    INFO    (output) INTEGER   
            = 0: successful exit   
            > 0: SLAHQR failed to compute all the eigenvalues ILO to IHI   
                 in a total of 30*(IHI-ILO+1) iterations; if INFO = i,   
                 elements i+1:ihi of WR and WI contain those eigenvalues   
                 which have been successfully computed.   

    Further Details   
    ===============   

    2-96 Based on modifications by   
       David Day, Sandia National Laboratory, USA   

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


       Parameter adjustments */
    h_dim1 = *ldh;
    h_offset = 1 + h_dim1 * 1;
    h__ -= h_offset;
    --wr;
    --wi;
    z_dim1 = *ldz;
    z_offset = 1 + z_dim1 * 1;
    z__ -= z_offset;

    /* Function Body */
    *info = 0;
/* **   
       Initialize */
    opst = 0.f;
/* **   

       Quick return if possible */

    if (*n == 0) {
	return 0;
    }
    if (*ilo == *ihi) {
	wr[*ilo] = h___ref(*ilo, *ilo);
	wi[*ilo] = 0.f;
	return 0;
    }

    nh = *ihi - *ilo + 1;
    nz = *ihiz - *iloz + 1;

/*     Set machine-dependent constants for the stopping criterion.   
       If norm(H) <= sqrt(OVFL), overflow should not occur. */

    unfl = slamch_("Safe minimum");
    ovfl = 1.f / unfl;
    slabad_(&unfl, &ovfl);
    ulp = slamch_("Precision");
    smlnum = unfl * (nh / ulp);

/*     I1 and I2 are the indices of the first row and last column of H   
       to which transformations must be applied. If eigenvalues only are   
       being computed, I1 and I2 are set inside the main loop. */

    if (*wantt) {
	i1 = 1;
	i2 = *n;
    }

/*     ITN is the total number of QR iterations allowed. */

    itn = nh * 30;

/*     The main loop begins here. I is the loop index and decreases from   
       IHI to ILO in steps of 1 or 2. Each iteration of the loop works   
       with the active submatrix in rows and columns L to I.   
       Eigenvalues I+1 to IHI have already converged. Either L = ILO or   
       H(L,L-1) is negligible so that the matrix splits. */

    i__ = *ihi;
L10:
    l = *ilo;
    if (i__ < *ilo) {
	goto L150;
    }

/*     Perform QR iterations on rows and columns ILO to I until a   
       submatrix of order 1 or 2 splits off at the bottom because a   
       subdiagonal element has become negligible. */

    i__1 = itn;
    for (its = 0; its <= i__1; ++its) {

/*        Look for a single small subdiagonal element. */

	i__2 = l + 1;
	for (k = i__; k >= i__2; --k) {
	    tst1 = (r__1 = h___ref(k - 1, k - 1), dabs(r__1)) + (r__2 = 
		    h___ref(k, k), dabs(r__2));
	    if (tst1 == 0.f) {
		i__3 = i__ - l + 1;
		tst1 = slanhs_("1", &i__3, &h___ref(l, l), ldh, work);
/* **   
                Increment op count */
		latime_1.ops += (i__ - l + 1) * (i__ - l + 2) / 2;
/* ** */
	    }
/* Computing MAX */
	    r__2 = ulp * tst1;
	    if ((r__1 = h___ref(k, k - 1), dabs(r__1)) <= dmax(r__2,smlnum)) {
		goto L30;
	    }
/* L20: */
	}
L30:
	l = k;
/* **   
          Increment op count */
	opst += (i__ - l + 1) * 3;
/* ** */
	if (l > *ilo) {

/*           H(L,L-1) is negligible */

	    h___ref(l, l - 1) = 0.f;
	}

/*        Exit from loop if a submatrix of order 1 or 2 has split off. */

	if (l >= i__ - 1) {
	    goto L140;
	}

/*        Now the active submatrix is in rows and columns L to I. If   
          eigenvalues only are being computed, only the active submatrix   
          need be transformed. */

	if (! (*wantt)) {
	    i1 = l;
	    i2 = i__;
	}

	if (its == 10 || its == 20) {

/*           Exceptional shift. */

	    s = (r__1 = h___ref(i__, i__ - 1), dabs(r__1)) + (r__2 = h___ref(
		    i__ - 1, i__ - 2), dabs(r__2));
	    h44 = s * .75f + h___ref(i__, i__);
	    h33 = h44;
	    h43h34 = s * -.4375f * s;
/* **   
             Increment op count */
	    opst += 5;
/* ** */
	} else {

/*           Prepare to use Francis' double shift   
             (i.e. 2nd degree generalized Rayleigh quotient) */

	    h44 = h___ref(i__, i__);
	    h33 = h___ref(i__ - 1, i__ - 1);
	    h43h34 = h___ref(i__, i__ - 1) * h___ref(i__ - 1, i__);