Actual source code: filtlan.c
slepc-3.10.1 2018-10-23
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      1: /*
      2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
      3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
      4:    Copyright (c) 2002-2018, Universitat Politecnica de Valencia, Spain

      6:    This file is part of SLEPc.
      7:    SLEPc is distributed under a 2-clause BSD license (see LICENSE).
      8:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
      9: */
     10: /*
     11:    This file is an adaptation of several subroutines from FILTLAN, the
     12:    Filtered Lanczos Package, authored by Haw-ren Fang and Yousef Saad.

     14:    More information at:
     15:    http://www-users.cs.umn.edu/~saad/software/filtlan

     17:    References:

     19:        [1] H. Fang and Y. Saad, "A filtered Lanczos procedure for extreme and interior
     20:            eigenvalue problems", SIAM J. Sci. Comput. 34(4):A2220-A2246, 2012.
     21: */

     23: #include <slepc/private/stimpl.h>
     24: #include "./filter.h"

     26: static PetscErrorCode FILTLAN_FilteredConjugateResidualPolynomial(PetscReal*,PetscReal*,PetscInt,PetscReal*,PetscInt,PetscReal*,PetscInt);
     27: static PetscReal FILTLAN_PiecewisePolynomialEvaluationInChebyshevBasis(PetscReal*,PetscInt,PetscReal*,PetscInt,PetscReal);
     28: static PetscErrorCode FILTLAN_ExpandNewtonPolynomialInChebyshevBasis(PetscInt,PetscReal,PetscReal,PetscReal*,PetscReal*,PetscReal*,PetscReal*);

     30: /* ////////////////////////////////////////////////////////////////////////////
     31:    //    Newton - Hermite Polynomial Interpolation
     32:    //////////////////////////////////////////////////////////////////////////// */

     34: /*
     35:    FILTLAN function NewtonPolynomial

     37:    build P(z) by Newton's divided differences in the form
     38:        P(z) = a(1) + a(2)*(z-x(1)) + a(3)*(z-x(1))*(z-x(2)) + ... + a(n)*(z-x(1))*...*(z-x(n-1)),
     39:    such that P(x(i)) = y(i) for i=1,...,n, where
     40:        x,y are input vectors of length n, and a is the output vector of length n
     41:    if x(i)==x(j) for some i!=j, then it is assumed that the derivative of P(z) is to be zero at x(i),
     42:        and the Hermite polynomial interpolation is applied
     43:    in general, if there are k x(i)'s with the same value x0, then
     44:        the j-th order derivative of P(z) is zero at z=x0 for j=1,...,k-1
     45: */
     46: PETSC_STATIC_INLINE PetscErrorCode FILTLAN_NewtonPolynomial(PetscInt n,PetscReal *x,PetscReal *y,PetscReal *sa,PetscReal *sf)
     47: {
     49:   PetscReal      d,*sx=x,*sy=y;
     50:   PetscInt       j,k;

     53:   PetscMemcpy(sf,sy,n*sizeof(PetscReal));

     55:   /* apply Newton's finite difference method */
     56:   sa[0] = sf[0];
     57:   for (j=1;j<n;j++) {
     58:     for (k=n-1;k>=j;k--) {
     59:       d = sx[k]-sx[k-j];
     60:       if (d == 0.0) sf[k] = 0.0;  /* assume that the derivative is 0.0 and apply the Hermite interpolation */
     61:       else sf[k] = (sf[k]-sf[k-1]) / d;
     62:     }
     63:     sa[j] = sf[j];
     64:   }
     65:   return(0);
     66: }

     68: /*
     69:    FILTLAN function HermiteBaseFilterInChebyshevBasis

     71:    compute a base filter P(z) which is a continuous, piecewise polynomial P(z) expanded
     72:    in a basis of `translated' (i.e. scale-and-shift) Chebyshev polynomials in each interval

     74:    The base filter P(z) equals P_j(z) for z in the j-th interval [intv(j), intv(j+1)), where
     75:    P_j(z) a Hermite interpolating polynomial

     77:    input:
     78:    intv is a vector which defines the intervals; the j-th interval is [intv(j), intv(j+1))
     79:    HiLowFlags determines the shape of the base filter P(z)
     80:    Consider the j-th interval [intv(j), intv(j+1)]
     81:    HighLowFlag[j-1]==1,  P(z)==1 for z in [intv(j), intv(j+1)]
     82:                    ==0,  P(z)==0 for z in [intv(j), intv(j+1)]
     83:                    ==-1, [intv(j), intv(j+1)] is a transition interval;
     84:                          P(intv(j)) and P(intv(j+1)) are defined such that P(z) is continuous
     85:    baseDeg is the degree of smoothness of the Hermite (piecewise polynomial) interpolation
     86:    to be precise, the i-th derivative of P(z) is zero, i.e. d^{i}P(z)/dz^i==0, at all interval
     87:    end points z=intv(j) for i=1,...,baseDeg

     89:    output:
     90:    P(z) expanded in a basis of `translated' (scale-and-shift) Chebyshev polynomials
     91:    to be precise, for z in the j-th interval [intv(j),intv(j+1)), P(z) equals
     92:        P_j(z) = pp(1,j)*S_0(z) + pp(2,j)*S_1(z) + ... + pp(n,j)*S_{n-1}(z),
     93:    where S_i(z) is the `translated' Chebyshev polynomial in that interval,
     94:        S_i((z-c)/h) = T_i(z),  c = (intv(j)+intv(j+1))) / 2,  h = (intv(j+1)-intv(j)) / 2,
     95:    with T_i(z) the Chebyshev polynomial of the first kind,
     96:        T_0(z) = 1, T_1(z) = z, and T_i(z) = 2*z*T_{i-1}(z) - T_{i-2}(z) for i>=2
     97:    the return matrix is the matrix of Chebyshev coefficients pp just described

     99:    note that the degree of P(z) in each interval is (at most) 2*baseDeg+1, with 2*baseDeg+2 coefficients
    100:    let n be the length of intv; then there are n-1 intervals
    101:    therefore the return matrix pp is of size (2*baseDeg+2)-by-(n-1)
    102: */
    103: static PetscErrorCode FILTLAN_HermiteBaseFilterInChebyshevBasis(PetscReal *baseFilter,PetscReal *intv,PetscInt npoints,const PetscInt *HighLowFlags,PetscInt baseDeg)
    104: {
    106:   PetscInt       m,ii,jj;
    107:   PetscReal      flag,flag0,flag2,aa,bb,*px,*py,*sx,*sy,*pp,*qq,*sq,*sbf,*work,*currentPoint = intv;
    108:   const PetscInt *hilo = HighLowFlags;

    111:   m = 2*baseDeg+2;
    112:   jj = npoints-1;  /* jj is initialized as the number of intervals */
    113:   PetscMalloc5(m,&px,m,&py,m,&pp,m,&qq,m,&work);
    114:   sbf = baseFilter;

    116:   while (jj--) {  /* the main loop to compute the Chebyshev coefficients */

    118:     flag = (PetscReal)(*hilo++);  /* get the flag of the current interval */
    119:     if (flag == -1.0) {  /* flag == -1 means that the current interval is a transition polynomial */

    121:       flag2 = (PetscReal)(*hilo);  /* get flag2, the flag of the next interval */
    122:       flag0 = 1.0-flag2;       /* the flag of the previous interval is 1-flag2 */

    124:       /* two pointers for traversing x[] and y[] */
    125:       sx = px;
    126:       sy = py;

    128:       /* find the current interval [aa,bb] */
    129:       aa = *currentPoint++;
    130:       bb = *currentPoint;

    132:       /* now left-hand side */
    133:       ii = baseDeg+1;
    134:       while (ii--) {
    135:         *sy++ = flag0;
    136:         *sx++ = aa;
    137:       }

    139:       /* now right-hand side */
    140:       ii = baseDeg+1;
    141:       while (ii--) {
    142:         *sy++ = flag2;
    143:         *sx++ = bb;
    144:       }

    146:       /* build a Newton polynomial (indeed, the generalized Hermite interpolating polynomial) with x[] and y[] */
    147:       FILTLAN_NewtonPolynomial(m,px,py,pp,work);

    149:       /* pp contains coefficients of the Newton polynomial P(z) in the current interval [aa,bb], where
    150:          P(z) = pp(1) + pp(2)*(z-px(1)) + pp(3)*(z-px(1))*(z-px(2)) + ... + pp(n)*(z-px(1))*...*(z-px(n-1)) */

    152:       /* translate the Newton coefficients to the Chebyshev coefficients */
    153:       FILTLAN_ExpandNewtonPolynomialInChebyshevBasis(m,aa,bb,pp,px,qq,work);
    154:       /* qq contains coefficients of the polynomial in [aa,bb] in the `translated' Chebyshev basis */

    156:       /* copy the Chebyshev coefficients to baseFilter
    157:          OCTAVE/MATLAB: B(:,j) = qq, where j = (npoints-1)-jj and B is the return matrix */
    158:       sq = qq;
    159:       ii = 2*baseDeg+2;
    160:       while (ii--) *sbf++ = *sq++;

    162:     } else {

    164:       /* a constant polynomial P(z)=flag, where either flag==0 or flag==1
    165:        OCTAVE/MATLAB: B(1,j) = flag, where j = (npoints-1)-jj and B is the return matrix */
    166:       *sbf++ = flag;

    168:       /* the other coefficients are all zero, since P(z) is a constant
    169:        OCTAVE/MATLAB: B(1,j) = 0, where j = (npoints-1)-jj and B is the return matrix */
    170:       ii = 2*baseDeg+1;
    171:       while (ii--) *sbf++ = 0.0;

    173:       /* for the next point */
    174:       currentPoint++;
    175:     }
    176:   }
    177:   PetscFree5(px,py,pp,qq,work);
    178:   return(0);
    179: }

    181: /* ////////////////////////////////////////////////////////////////////////////
    182:    //    Base Filter
    183:    //////////////////////////////////////////////////////////////////////////// */

    185: /*
    186:    FILTLAN function GetIntervals

    188:    this routine determines the intervals (including the transition one(s)) by an iterative process

    190:    frame is a vector consisting of 4 ordered elements:
    191:        [frame(1),frame(4)] is the interval which (tightly) contains all eigenvalues, and
    192:        [frame(2),frame(3)] is the interval in which the eigenvalues are sought
    193:    baseDeg is the left-and-right degree of the base filter for each interval
    194:    polyDeg is the (maximum possible) degree of s(z), with z*s(z) the polynomial filter
    195:    intv is the output; the j-th interval is [intv(j),intv(j+1))
    196:    opts is a collection of interval options

    198:    the base filter P(z) is a piecewise polynomial from Hermite interpolation with degree baseDeg
    199:    at each end point of intervals

    201:    the polynomial filter Q(z) is in the form z*s(z), i.e. Q(0)==0, such that ||(1-P(z))-z*s(z)||_w is
    202:    minimized with s(z) a polynomial of degree up to polyDeg

    204:    the resulting polynomial filter Q(z) satisfies Q(x)>=Q(y) for x in [frame[1],frame[2]], and
    205:    y in [frame[0],frame[3]] but not in [frame[1],frame[2]]

    207:    the routine fills a PolynomialFilterInfo struct which gives some information of the polynomial filter
    208: */
    209: static PetscErrorCode FILTLAN_GetIntervals(PetscReal *intervals,PetscReal *frame,PetscInt polyDeg,PetscInt baseDeg,FILTLAN_IOP opts,FILTLAN_PFI filterInfo)
    210: {
    211:   PetscErrorCode  ierr;
    212:   PetscReal       intv[6],x,y,z1,z2,c,c1,c2,fc,fc2,halfPlateau,leftDelta,rightDelta,gridSize;
    213:   PetscReal       yLimit,ySummit,yLeftLimit,yRightLimit,bottom,qIndex,*baseFilter,*polyFilter;
    214:   PetscReal       yLimitGap=0.0,yLeftSummit=0.0,yLeftBottom=0.0,yRightSummit=0.0,yRightBottom=0.0;
    215:   PetscInt        i,ii,npoints,numIter,numLeftSteps=1,numRightSteps=1,numMoreLooked=0;
    216:   PetscBool       leftBoundaryMet=PETSC_FALSE,rightBoundaryMet=PETSC_FALSE,stepLeft,stepRight;
    217:   const PetscReal a=frame[0],a1=frame[1],b1=frame[2],b=frame[3];
    218:   const PetscInt  HighLowFlags[5] = { 1, -1, 0, -1, 1 };  /* if filterType is 1, only first 3 elements will be used */
    219:   const PetscInt  numLookMore = 2*(PetscInt)(0.5+(PetscLogReal(2.0)/PetscLogReal(opts->shiftStepExpanRate)));

    222:   if (a>a1 || a1>b1 || b1>b) SETERRQ(PETSC_COMM_SELF,1,"Values in the frame vector should be non-decreasing");
    223:   if (a1 == b1) SETERRQ(PETSC_COMM_SELF,1,"The range of wanted eigenvalues cannot be of size zero");
    224:   filterInfo->filterType = 2;      /* mid-pass filter, for interior eigenvalues */
    225:   if (b == b1) {
    226:     if (a == a1) SETERRQ(PETSC_COMM_SELF,1,"A polynomial filter should not cover all eigenvalues");
    227:     filterInfo->filterType = 1;    /* high-pass filter, for largest eigenvalues */
    228:   } else if (a == a1) SETERRQ(PETSC_COMM_SELF,1,"filterType==3 for smallest eigenvalues should be pre-converted to filterType==1 for largest eigenvalues");

    230:   /* the following recipe follows Yousef Saad (2005, 2006) with a few minor adaptations / enhancements */
    231:   halfPlateau = 0.5*(b1-a1)*opts->initialPlateau;    /* half length of the "plateau" of the (dual) base filter */
    232:   leftDelta = (b1-a1)*opts->initialShiftStep;        /* initial left shift */
    233:   rightDelta = leftDelta;                            /* initial right shift */
    234:   opts->numGridPoints = PetscMax(opts->numGridPoints,(PetscInt)(2.0*(b-a)/halfPlateau));
    235:   gridSize = (b-a) / (PetscReal)(opts->numGridPoints);

    237:   for (i=0;i<6;i++) intv[i] = 0.0;
    238:   if (filterInfo->filterType == 2) {  /* for interior eigenvalues */
    239:     npoints = 6;
    240:     intv[0] = a;
    241:     intv[5] = b;
    242:     /* intv[1], intv[2], intv[3], and intv[4] to be determined */
    243:   } else { /* filterType == 1 (or 3 with conversion), for extreme eigenvalues */
    244:     npoints = 4;
    245:     intv[0] = a;
    246:     intv[3] = b;
    247:     /* intv[1], and intv[2] to be determined */
    248:   }
    249:   z1 = a1 - leftDelta;
    250:   z2 = b1 + rightDelta;
    251:   filterInfo->filterOK = 0;  /* not yet found any OK filter */

    253:   /* allocate matrices */
    254:   PetscMalloc2((polyDeg+2)*(npoints-1),&polyFilter,(2*baseDeg+2)*(npoints-1),&baseFilter);

    256:   /* initialize the intervals, mainly for the case opts->maxOuterIter == 0 */
    257:   intervals[0] = intv[0];
    258:   intervals[3] = intv[3];
    259:   intervals[5] = intv[5];
    260:   intervals[1] = z1;
    261:   if (filterInfo->filterType == 2) {  /* a mid-pass filter for interior eigenvalues */
    262:     intervals[4] = z2;
    263:     c = (a1+b1) / 2.0;
    264:     intervals[2] = c - halfPlateau;
    265:     intervals[3] = c + halfPlateau;
    266:   } else {  /* filterType == 1 (or 3 with conversion) for extreme eigenvalues */
    267:     intervals[2] = z1 + (b1-z1)*opts->transIntervalRatio;
    268:   }

    270:   /* the main loop */
    271:   for (numIter=1;numIter<=opts->maxOuterIter;numIter++) {
    272:     if (z1 <= a) {  /* outer loop updates z1 and z2 */
    273:       z1 = a;
    274:       leftBoundaryMet = PETSC_TRUE;
    275:     }
    276:     if (filterInfo->filterType == 2) {  /* a <= z1 < (a1) */
    277:       if (z2 >= b) {  /* a mid-pass filter for interior eigenvalues */
    278:         z2 = b;
    279:         rightBoundaryMet = PETSC_TRUE;
    280:       }
    281:       /* a <= z1 < c-h < c+h < z2 <= b, where h is halfPlateau */
    282:       /* [z1, c-h] and [c+h, z2] are transition interval */
    283:       intv[1] = z1;
    284:       intv[4] = z2;
    285:       c1 = z1 + halfPlateau;
    286:       intv[2] = z1;           /* i.e. c1 - halfPlateau */
    287:       intv[3] = c1 + halfPlateau;
    288:       FILTLAN_HermiteBaseFilterInChebyshevBasis(baseFilter,intv,6,HighLowFlags,baseDeg);
    289:       FILTLAN_FilteredConjugateResidualPolynomial(polyFilter,baseFilter,2*baseDeg+2,intv,6,opts->intervalWeights,polyDeg);
    290:       /* fc1 = FILTLAN_PiecewisePolynomialEvaluationInChebyshevBasis(polyFilter,polyDeg+2,intv,npoints,b1) - FILTLAN_PiecewisePolynomialEvaluationInChebyshevBasis(polyFilter,polyDeg+2,intv,npoints,a1); */
    291:       c2 = z2 - halfPlateau;
    292:       intv[2] = c2 - halfPlateau;
    293:       intv[3] = z2;           /* i.e. c2 + halfPlateau */
    294:       FILTLAN_HermiteBaseFilterInChebyshevBasis(baseFilter,intv,6,HighLowFlags,baseDeg);
    295:       FILTLAN_FilteredConjugateResidualPolynomial(polyFilter,baseFilter,2*baseDeg+2,intv,6,opts->intervalWeights,polyDeg);
    296:       fc2 = FILTLAN_PiecewisePolynomialEvaluationInChebyshevBasis(polyFilter,polyDeg+2,intv,npoints,b1) - FILTLAN_PiecewisePolynomialEvaluationInChebyshevBasis(polyFilter,polyDeg+2,intv,npoints,a1);
    297:       yLimitGap = PETSC_MAX_REAL;
    298:       ii = opts->maxInnerIter;
    299:       while (ii-- && !(yLimitGap <= opts->yLimitTol)) {
    300:         /* recursive bisection to get c such that p(a1) are p(b1) approximately the same */
    301:         c = (c1+c2) / 2.0;
    302:         intv[2] = c - halfPlateau;
    303:         intv[3] = c + halfPlateau;
    304:         FILTLAN_HermiteBaseFilterInChebyshevBasis(baseFilter,intv,6,HighLowFlags,baseDeg);
    305:         FILTLAN_FilteredConjugateResidualPolynomial(polyFilter,baseFilter,2*baseDeg+2,intv,6,opts->intervalWeights,polyDeg);
    306:         fc = FILTLAN_PiecewisePolynomialEvaluationInChebyshevBasis(polyFilter,polyDeg+2,intv,npoints,b1) - FILTLAN_PiecewisePolynomialEvaluationInChebyshevBasis(polyFilter,polyDeg+2,intv,npoints,a1);
    307:         if (fc*fc2 < 0.0) {
    308:           c1 = c;
    309:           /* fc1 = fc; */
    310:         } else {
    311:           c2 = c;
    312:           fc2 = fc;
    313:         }
    314:         yLimitGap = PetscAbsReal(fc);
    315:       }
    316:     } else {  /* filterType == 1 (or 3 with conversion) for extreme eigenvalues */
    317:       intv[1] = z1;
    318:       intv[2] = z1 + (b1-z1)*opts->transIntervalRatio;
    319:       FILTLAN_HermiteBaseFilterInChebyshevBasis(baseFilter,intv,4,HighLowFlags,baseDeg);
    320:       FILTLAN_FilteredConjugateResidualPolynomial(polyFilter,baseFilter,2*baseDeg+2,intv,4,opts->intervalWeights,polyDeg);
    321:     }
    322:     /* polyFilter contains the coefficients of the polynomial filter which approximates phi(x)
    323:        expanded in the `translated' Chebyshev basis */
    324:     /* psi(x) = 1.0 - phi(x) is the dual base filter approximated by a polynomial in the form x*p(x) */
    325:     yLeftLimit  = 1.0 - FILTLAN_PiecewisePolynomialEvaluationInChebyshevBasis(polyFilter,polyDeg+2,intv,npoints,a1);
    326:     yRightLimit = 1.0 - FILTLAN_PiecewisePolynomialEvaluationInChebyshevBasis(polyFilter,polyDeg+2,intv,npoints,b1);
    327:     yLimit  = (yLeftLimit < yRightLimit) ? yLeftLimit : yRightLimit;
    328:     ySummit = (yLeftLimit > yRightLimit) ? yLeftLimit : yRightLimit;
    329:     x = a1;
    330:     while ((x+=gridSize) < b1) {
    331:       y = 1.0 - FILTLAN_PiecewisePolynomialEvaluationInChebyshevBasis(polyFilter,polyDeg+2,intv,npoints,x);
    332:       if (y < yLimit)  yLimit  = y;
    333:       if (y > ySummit) ySummit = y;
    334:     }
    335:     /* now yLimit is the minimum of x*p(x) for x in [a1, b1] */
    336:     stepLeft  = PETSC_FALSE;
    337:     stepRight = PETSC_FALSE;
    338:     if ((yLimit < yLeftLimit && yLimit < yRightLimit) || yLimit < opts->yBottomLine) {
    339:       /* very bad, step to see what will happen */
    340:       stepLeft = PETSC_TRUE;
    341:       if (filterInfo->filterType == 2) stepRight = PETSC_TRUE;
    342:     } else if (filterInfo->filterType == 2) {
    343:       if (yLeftLimit < yRightLimit) {
    344:         if (yRightLimit-yLeftLimit > opts->yLimitTol) stepLeft = PETSC_TRUE;
    345:       } else if (yLeftLimit-yRightLimit > opts->yLimitTol) stepRight = PETSC_TRUE;
    346:     }
    347:     if (!stepLeft) {
    348:       yLeftBottom = yLeftLimit;
    349:       x = a1;
    350:       while ((x-=gridSize) >= a) {
    351:         y = 1.0 - FILTLAN_PiecewisePolynomialEvaluationInChebyshevBasis(polyFilter,polyDeg+2,intv,npoints,x);
    352:         if (y < yLeftBottom) yLeftBottom = y;
    353:         else if (y > yLeftBottom) break;
    354:       }
    355:       yLeftSummit = yLeftBottom;
    356:       while ((x-=gridSize) >= a) {
    357:         y = 1.0 - FILTLAN_PiecewisePolynomialEvaluationInChebyshevBasis(polyFilter,polyDeg+2,intv,npoints,x);
    358:         if (y > yLeftSummit) {
    359:           yLeftSummit = y;
    360:           if (yLeftSummit > yLimit*opts->yRippleLimit) {
    361:             stepLeft = PETSC_TRUE;
    362:             break;
    363:           }
    364:         }
    365:         if (y < yLeftBottom) yLeftBottom = y;
    366:       }
    367:     }
    368:     if (filterInfo->filterType == 2 && !stepRight) {
    369:       yRightBottom = yRightLimit;
    370:       x = b1;
    371:       while ((x+=gridSize) <= b) {
    372:         y = 1.0 - FILTLAN_PiecewisePolynomialEvaluationInChebyshevBasis(polyFilter,polyDeg+2,intv,npoints,x);
    373:         if (y < yRightBottom) yRightBottom = y;
    374:         else if (y > yRightBottom) break;
    375:       }
    376:       yRightSummit = yRightBottom;
    377:       while ((x+=gridSize) <= b) {
    378:         y = 1.0 - FILTLAN_PiecewisePolynomialEvaluationInChebyshevBasis(polyFilter,polyDeg+2,intv,npoints,x);
    379:         if (y > yRightSummit) {
    380:           yRightSummit = y;
    381:           if (yRightSummit > yLimit*opts->yRippleLimit) {
    382:             stepRight = PETSC_TRUE;
    383:             break;
    384:           }
    385:         }
    386:         if (y < yRightBottom) yRightBottom = y;
    387:       }
    388:     }
    389:     if (!stepLeft && !stepRight) {
    390:       if (filterInfo->filterType == 2) bottom = PetscMin(yLeftBottom,yRightBottom);
    391:       else bottom = yLeftBottom;
    392:       qIndex = 1.0 - (yLimit-bottom) / (ySummit-bottom);
    393:       if (filterInfo->filterOK == 0 || filterInfo->filterQualityIndex < qIndex) {
    394:         /* found the first OK filter or a better filter */
    395:         for (i=0;i<6;i++) intervals[i] = intv[i];
    396:         filterInfo->filterOK           = 1;
    397:         filterInfo->filterQualityIndex = qIndex;
    398:         filterInfo->numIter            = numIter;
    399:         filterInfo->yLimit             = yLimit;
    400:         filterInfo->ySummit            = ySummit;
    401:         filterInfo->numLeftSteps       = numLeftSteps;
    402:         filterInfo->yLeftSummit        = yLeftSummit;
    403:         filterInfo->yLeftBottom        = yLeftBottom;
    404:         if (filterInfo->filterType == 2) {
    405:           filterInfo->yLimitGap        = yLimitGap;
    406:           filterInfo->numRightSteps    = numRightSteps;
    407:           filterInfo->yRightSummit     = yRightSummit;
    408:           filterInfo->yRightBottom     = yRightBottom;
    409:         }
    410:         numMoreLooked = 0;
    411:       } else if (++numMoreLooked == numLookMore) {
    412:         /* filter has been optimal */
    413:         filterInfo->filterOK = 2;
    414:         break;
    415:       }
    416:       /* try stepping further to see whether it can improve */
    417:       stepLeft = PETSC_TRUE;
    418:       if (filterInfo->filterType == 2) stepRight = PETSC_TRUE;
    419:     }
    420:     /* check whether we can really "step" */
    421:     if (leftBoundaryMet) {
    422:       if (filterInfo->filterType == 1 || rightBoundaryMet) break;  /* cannot step further, so break the loop */
    423:       if (stepLeft) {
    424:         /* cannot step left, so try stepping right */
    425:         stepLeft  = PETSC_FALSE;
    426:         stepRight = PETSC_TRUE;
    427:       }
    428:     }
    429:     if (rightBoundaryMet && stepRight) {
    430:       /* cannot step right, so try stepping left */
    431:       stepRight = PETSC_FALSE;
    432:       stepLeft  = PETSC_TRUE;
    433:     }
    434:     /* now "step" */
    435:     if (stepLeft) {
    436:       numLeftSteps++;
    437:       if (filterInfo->filterType == 2) leftDelta *= opts->shiftStepExpanRate; /* expand the step for faster convergence */
    438:       z1 -= leftDelta;
    439:     }
    440:     if (stepRight) {
    441:       numRightSteps++;
    442:       rightDelta *= opts->shiftStepExpanRate;  /* expand the step for faster convergence */
    443:       z2 += rightDelta;
    444:     }
    445:     if (filterInfo->filterType == 2) {
    446:       /* shrink the "plateau" of the (dual) base filter */
    447:       if (stepLeft && stepRight) halfPlateau /= opts->plateauShrinkRate;
    448:       else halfPlateau /= PetscSqrtReal(opts->plateauShrinkRate);
    449:     }
    450:   }
    451:   if (!filterInfo->filterOK) SETERRQ(PETSC_COMM_SELF,1,"STFILTER cannot get the filter specified; please adjust your filter parameters (e.g. increasing the polynomial degree)");

    453:   filterInfo->totalNumIter = numIter;
    454:   PetscFree2(polyFilter,baseFilter);
    455:   return(0);
    456: }

    458: /* ////////////////////////////////////////////////////////////////////////////
    459:    //    Chebyshev Polynomials
    460:    //////////////////////////////////////////////////////////////////////////// */

    462: /*
    463:    FILTLAN function ExpandNewtonPolynomialInChebyshevBasis

    465:    translate the coefficients of a Newton polynomial to the coefficients
    466:    in a basis of the `translated' (scale-and-shift) Chebyshev polynomials

    468:    input:
    469:    a Newton polynomial defined by vectors a and x:
    470:        P(z) = a(1) + a(2)*(z-x(1)) + a(3)*(z-x(1))*(z-x(2)) + ... + a(n)*(z-x(1))*...*(z-x(n-1))
    471:    the interval [aa,bb] defines the `translated' Chebyshev polynomials S_i(z) = T_i((z-c)/h),
    472:        where c=(aa+bb)/2 and h=(bb-aa)/2, and T_i is the Chebyshev polynomial of the first kind
    473:    note that T_i is defined by T_0(z)=1, T_1(z)=z, and T_i(z)=2*z*T_{i-1}(z)+T_{i-2}(z) for i>=2

    475:    output:
    476:    a vector q containing the Chebyshev coefficients:
    477:        P(z) = q(1)*S_0(z) + q(2)*S_1(z) + ... + q(n)*S_{n-1}(z)
    478: */
    479: PETSC_STATIC_INLINE PetscErrorCode FILTLAN_ExpandNewtonPolynomialInChebyshevBasis(PetscInt n,PetscReal aa,PetscReal bb,PetscReal *a,PetscReal *x,PetscReal *q,PetscReal *q2)
    480: {
    481:   PetscInt  m,mm;
    482:   PetscReal *sa,*sx,*sq,*sq2,c,c2,h,h2;

    485:   sa = a+n;    /* pointers for traversing a and x */
    486:   sx = x+n-1;
    487:   *q = *--sa;  /* set q[0] = a(n) */

    489:   c = (aa+bb)/2.0;
    490:   h = (bb-aa)/2.0;
    491:   h2 = h/2.0;

    493:   for (m=1;m<=n-1;m++) {  /* the main loop for translation */

    495:     /* compute q2[0:m-1] = (c-x[n-m-1])*q[0:m-1] */
    496:     mm = m;
    497:     sq = q;
    498:     sq2 = q2;
    499:     c2 = c-(*--sx);
    500:     while (mm--) *(sq2++) = c2*(*sq++);
    501:     *sq2 = 0.0;         /* q2[m] = 0.0 */
    502:     *(q2+1) += h*(*q);  /* q2[1] = q2[1] + h*q[0] */

    504:     /* compute q2[0:m-2] = q2[0:m-2] + h2*q[1:m-1] */
    505:     mm = m-1;
    506:     sq2 = q2;
    507:     sq = q+1;
    508:     while (mm--) *(sq2++) += h2*(*sq++);

    510:     /* compute q2[2:m] = q2[2:m] + h2*q[1:m-1] */
    511:     mm = m-1;
    512:     sq2 = q2+2;
    513:     sq = q+1;
    514:     while (mm--) *(sq2++) += h2*(*sq++);

    516:     /* compute q[0:m] = q2[0:m] */
    517:     mm = m+1;
    518:     sq2 = q2;
    519:     sq = q;
    520:     while (mm--) *sq++ = *sq2++;
    521:     *q += (*--sa);      /* q[0] = q[0] + p[n-m-1] */
    522:   }
    523:   return(0);
    524: }

    526: /*
    527:    FILTLAN function PolynomialEvaluationInChebyshevBasis

    529:    evaluate P(z) at z=z0, where P(z) is a polynomial expanded in a basis of
    530:    the `translated' (i.e. scale-and-shift) Chebyshev polynomials

    532:    input:
    533:    c is a vector of Chebyshev coefficients which defines the polynomial
    534:        P(z) = c(1)*S_0(z) + c(2)*S_1(z) + ... + c(n)*S_{n-1}(z),
    535:    where S_i is the `translated' Chebyshev polynomial S_i((z-c)/h) = T_i(z), with
    536:        c = (intv(j)+intv(j+1)) / 2,  h = (intv(j+1)-intv(j)) / 2
    537:    note that T_i(z) is the Chebyshev polynomial of the first kind,
    538:        T_0(z) = 1, T_1(z) = z, and T_i(z) = 2*z*T_{i-1}(z) - T_{i-2}(z) for i>=2

    540:    output:
    541:    the evaluated value of P(z) at z=z0
    542: */
    543: PETSC_STATIC_INLINE PetscReal FILTLAN_PolynomialEvaluationInChebyshevBasis(PetscReal *pp,PetscInt m,PetscInt idx,PetscReal z0,PetscReal aa,PetscReal bb)
    544: {
    545:   PetscInt  ii,deg1;
    546:   PetscReal y,zz,t0,t1,t2,*sc;

    549:   deg1 = m;
    550:   if (aa==-1.0 && bb==1.0) zz = z0;  /* treat it as a special case to reduce rounding errors */
    551:   else zz = (aa==bb)? 0.0 : -1.0+2.0*(z0-aa)/(bb-aa);

    553:   /* compute y = P(z0), where we utilize the Chebyshev recursion */
    554:   sc = pp+(idx-1)*m;   /* sc points to column idx of pp */
    555:   y = *sc++;
    556:   t1 = 1.0; t2 = zz;
    557:   ii = deg1-1;
    558:   while (ii--) {
    559:     /* Chebyshev recursion: T_0(zz)=1, T_1(zz)=zz, and T_{i+1}(zz) = 2*zz*T_i(zz) + T_{i-1}(zz) for i>=2
    560:        the values of T_{i+1}(zz), T_i(zz), T_{i-1}(zz) are stored in t0, t1, t2, respectively */
    561:     t0 = 2*zz*t1 - t2;
    562:     /* it also works for the base case / the first iteration, where t0 equals 2*zz*1-zz == zz which is T_1(zz) */
    563:     t2 = t1;
    564:     t1 = t0;
    565:     y += (*sc++)*t0;
    566:   }
    567:   PetscFunctionReturn(y);
    568: }

    570: #define basisTranslated PETSC_TRUE
    571: /*
    572:    FILTLAN function PiecewisePolynomialEvaluationInChebyshevBasis

    574:    evaluate P(z) at z=z0, where P(z) is a piecewise polynomial expanded
    575:    in a basis of the (optionally translated, i.e. scale-and-shift) Chebyshev polynomials for each interval

    577:    input:
    578:    intv is a vector which defines the intervals; the j-th interval is [intv(j), intv(j+1))
    579:    pp is a matrix of Chebyshev coefficients which defines a piecewise polynomial P(z)
    580:    in a basis of the `translated' Chebyshev polynomials in each interval
    581:    the polynomial P_j(z) in the j-th interval, i.e. when z is in [intv(j), intv(j+1)), is defined by the j-th column of pp:
    582:        if basisTranslated == false, then
    583:            P_j(z) = pp(1,j)*T_0(z) + pp(2,j)*T_1(z) + ... + pp(n,j)*T_{n-1}(z),
    584:        where T_i(z) is the Chebyshev polynomial of the first kind,
    585:            T_0(z) = 1, T_1(z) = z, and T_i(z) = 2*z*T_{i-1}(z) - T_{i-2}(z) for i>=2
    586:        if basisTranslated == true, then
    587:            P_j(z) = pp(1,j)*S_0(z) + pp(2,j)*S_1(z) + ... + pp(n,j)*S_{n-1}(z),
    588:        where S_i is the `translated' Chebyshev polynomial S_i((z-c)/h) = T_i(z), with
    589:            c = (intv(j)+intv(j+1)) / 2,  h = (intv(j+1)-intv(j)) / 2

    591:    output:
    592:    the evaluated value of P(z) at z=z0

    594:    note that if z0 falls below the first interval, then the polynomial in the first interval will be used to evaluate P(z0)
    595:              if z0 flies over  the last  interval, then the polynomial in the last  interval will be used to evaluate P(z0)
    596: */
    597: PETSC_STATIC_INLINE PetscReal FILTLAN_PiecewisePolynomialEvaluationInChebyshevBasis(PetscReal *pp,PetscInt m,PetscReal *intv,PetscInt npoints,PetscReal z0)
    598: {
    599:   PetscReal *sintv,aa,bb,resul;
    600:   PetscInt  idx;

    603:   /* determine the interval which contains z0 */
    604:   sintv = &intv[1];
    605:   idx = 1;
    606:   if (npoints>2 && z0 >= *sintv) {
    607:     sintv++;
    608:     while (++idx < npoints-1) {
    609:       if (z0 < *sintv) break;
    610:       sintv++;
    611:     }
    612:   }
    613:   /* idx==1 if npoints<=2; otherwise idx satisfies:
    614:          intv(idx) <= z0 < intv(idx+1),  if 2 <= idx <= npoints-2
    615:          z0 < intv(idx+1),               if idx == 1
    616:          intv(idx) <= z0,                if idx == npoints-1
    617:      in addition, sintv points to &intv(idx+1) */
    618:   if (basisTranslated) {
    619:     /* the basis consists of `translated' Chebyshev polynomials */
    620:     /* find the interval of concern, [aa,bb] */
    621:     aa = *(sintv-1);
    622:     bb = *sintv;
    623:     resul = FILTLAN_PolynomialEvaluationInChebyshevBasis(pp,m,idx,z0,aa,bb);
    624:   } else {
    625:     /* the basis consists of standard Chebyshev polynomials, with interval [-1.0,1.0] for integration */
    626:     resul = FILTLAN_PolynomialEvaluationInChebyshevBasis(pp,m,idx,z0,-1.0,1.0);
    627:   }
    628:   PetscFunctionReturn(resul);
    629: }

    631: /*
    632:    FILTLAN function PiecewisePolynomialInnerProductInChebyshevBasis

    634:    compute the weighted inner product of two piecewise polynomials expanded
    635:    in a basis of `translated' (i.e. scale-and-shift) Chebyshev polynomials for each interval

    637:    pp and qq are two matrices of Chebyshev coefficients which define the piecewise polynomials P(z) and Q(z), respectively
    638:    for z in the j-th interval, P(z) equals
    639:        P_j(z) = pp(1,j)*S_0(z) + pp(2,j)*S_1(z) + ... + pp(n,j)*S_{n-1}(z),
    640:    and Q(z) equals
    641:        Q_j(z) = qq(1,j)*S_0(z) + qq(2,j)*S_1(z) + ... + qq(n,j)*S_{n-1}(z),
    642:    where S_i(z) is the `translated' Chebyshev polynomial in that interval,
    643:        S_i((z-c)/h) = T_i(z),  c = (aa+bb)) / 2,  h = (bb-aa) / 2,
    644:    with T_i(z) the Chebyshev polynomial of the first kind
    645:        T_0(z) = 1, T_1(z) = z, and T_i(z) = 2*z*T_{i-1}(z) - T_{i-2}(z) for i>=2

    647:    the (scaled) j-th interval inner product is defined by
    648:        <P_j,Q_j> = (Pi/2)*( pp(1,j)*qq(1,j) + sum_{k} pp(k,j)*qq(k,j) ),
    649:    which comes from the property
    650:        <T_0,T_0>=pi, <T_i,T_i>=pi/2 for i>=1, and <T_i,T_j>=0 for i!=j

    652:    the weighted inner product is <P,Q> = sum_{j} intervalWeights(j)*<P_j,Q_j>,
    653:    which is the return value

    655:    note that for unit weights, pass an empty vector of intervalWeights (i.e. of length 0)
    656: */
    657: PETSC_STATIC_INLINE PetscReal FILTLAN_PiecewisePolynomialInnerProductInChebyshevBasis(PetscReal *pp,PetscInt prows,PetscInt pcols,PetscInt ldp,PetscReal *qq,PetscInt qrows,PetscInt qcols,PetscInt ldq,const PetscReal *intervalWeights)
    658: {
    659:   PetscInt  nintv,deg1,i,k;
    660:   PetscReal *sp,*sq,ans=0.0,ans2;

    663:   deg1 = PetscMin(prows,qrows);  /* number of effective coefficients, one more than the effective polynomial degree */
    664:   if (!deg1) PetscFunctionReturn(0.0);
    665:   nintv = PetscMin(pcols,qcols);  /* number of intervals */

    667:   /* scaled by intervalWeights(i) in the i-th interval (we assume intervalWeights[] are always provided).
    668:      compute ans = sum_{i=1,...,nintv} intervalWeights(i)*[ pp(1,i)*qq(1,i) + sum_{k=1,...,deg} pp(k,i)*qq(k,i) ] */
    669:   for (i=0;i<nintv;i++) {   /* compute ans2 = pp(1,i)*qq(1,i) + sum_{k=1,...,deg} pp(k,i)*qq(k,i) */
    670:     sp = pp+i*ldp;
    671:     sq = qq+i*ldq;
    672:     ans2 = (*sp) * (*sq);  /* the first term pp(1,i)*qq(1,i) is being added twice */
    673:     for (k=0;k<deg1;k++) ans2 += (*sp++)*(*sq++);  /* add pp(k,i)*qq(k,i) */
    674:     ans += ans2*intervalWeights[i];  /* compute ans += ans2*intervalWeights(i) */
    675:   }
    676:   PetscFunctionReturn(ans*PETSC_PI/2.0);
    677: }

    679: /*
    680:    FILTLAN function PiecewisePolynomialInChebyshevBasisMultiplyX

    682:    compute Q(z) = z*P(z), where P(z) and Q(z) are piecewise polynomials expanded
    683:    in a basis of `translated' (i.e. scale-and-shift) Chebyshev polynomials for each interval

    685:    P(z) and Q(z) are stored as matrices of Chebyshev coefficients pp and qq, respectively

    687:    For z in the j-th interval, P(z) equals
    688:        P_j(z) = pp(1,j)*S_0(z) + pp(2,j)*S_1(z) + ... + pp(n,j)*S_{n-1}(z),
    689:    and Q(z) equals
    690:        Q_j(z) = qq(1,j)*S_0(z) + qq(2,j)*S_1(z) + ... + qq(n,j)*S_{n-1}(z),
    691:    where S_i(z) is the `translated' Chebyshev polynomial in that interval,
    692:        S_i((z-c)/h) = T_i(z),  c = (intv(j)+intv(j+1))) / 2,  h = (intv(j+1)-intv(j)) / 2,
    693:    with T_i(z) the Chebyshev polynomial of the first kind
    694:        T_0(z) = 1, T_1(z) = z, and T_i(z) = 2*z*T_{i-1}(z) - T_{i-2}(z) for i>=2

    696:    the returned matrix is qq which represents Q(z) = z*P(z)
    697: */
    698: PETSC_STATIC_INLINE PetscErrorCode FILTLAN_PiecewisePolynomialInChebyshevBasisMultiplyX(PetscReal *pp,PetscInt deg1,PetscInt ldp,PetscReal *intv,PetscInt nintv,PetscReal *qq,PetscInt ldq)
    699: {
    700:   PetscInt  i,j;
    701:   PetscReal c,h,h2,tmp,*sp,*sq,*sp2,*sq2;

    704:   for (j=0;j<nintv;j++) {    /* consider interval between intv(j) and intv(j+1) */
    705:     sp = pp+j*ldp;
    706:     sq = qq+j*ldq;
    707:     sp2 = sp;
    708:     sq2 = sq+1;
    709:     c = 0.5*(intv[j] + intv[j+1]);   /* compute c = (intv(j) + intv(j+1))/2 */
    710:     h = 0.5*(intv[j+1] - intv[j]);   /* compute h = (intv(j+1) - intv(j))/2  and  h2 = h/2 */
    711:     h2 = 0.5*h;
    712:     i = deg1;
    713:     while (i--) *sq++ = c*(*sp++);    /* compute q(1:deg1,j) = c*p(1:deg1,j) */
    714:     *sq++ = 0.0;                      /* set q(deg1+1,j) = 0.0 */
    715:     *(sq2++) += h*(*sp2++);           /* compute q(2,j) = q(2,j) + h*p(1,j) */
    716:     i = deg1-1;
    717:     while (i--) {       /* compute q(3:deg1+1,j) += h2*p(2:deg1,j) and then q(1:deg1-1,j) += h2*p(2:deg1,j) */
    718:       tmp = h2*(*sp2++);
    719:       *(sq2-2) += tmp;
    720:       *(sq2++) += tmp;
    721:     }
    722:   }
    723:   return(0);
    724: }

    726: /* ////////////////////////////////////////////////////////////////////////////
    727:    //    Conjugate Residual Method in the Polynomial Space
    728:    //////////////////////////////////////////////////////////////////////////// */

    730: /*
    731:     B := alpha*A + beta*B

    733:     A,B are nxk
    734: */
    735: PETSC_STATIC_INLINE PetscErrorCode Mat_AXPY_BLAS(PetscInt n,PetscInt k,PetscReal alpha,const PetscReal *A,PetscInt lda,PetscReal beta,PetscReal *B,PetscInt ldb)
    736: {
    738:   PetscInt       i,j;

    741:   if (beta==(PetscReal)1.0) {
    742:     for (j=0;j<k;j++) for (i=0;i<n;i++) B[i+j*ldb] += alpha*A[i+j*lda];
    743:     PetscLogFlops(2.0*n*k);
    744:   } else {
    745:     for (j=0;j<k;j++) for (i=0;i<n;i++) B[i+j*ldb] = alpha*A[i+j*lda] + beta*B[i+j*ldb];
    746:     PetscLogFlops(3.0*n*k);
    747:   }
    748:   return(0);
    749: }

    751: /*
    752:    FILTLAN function FilteredConjugateResidualPolynomial

    754:    ** Conjugate Residual Method in the Polynomial Space

    756:    this routine employs a conjugate-residual-type algorithm in polynomial space to minimize ||P(z)-Q(z)||_w,
    757:    where P(z), the base filter, is the input piecewise polynomial, and
    758:          Q(z) is the output polynomial satisfying Q(0)==1, i.e. the constant term of Q(z) is 1
    759:    niter is the number of conjugate-residual iterations; therefore, the degree of Q(z) is up to niter+1
    760:    both P(z) and Q(z) are expanded in the `translated' (scale-and-shift) Chebyshev basis for each interval,
    761:    and presented as matrices of Chebyshev coefficients, denoted by pp and qq, respectively

    763:    input:
    764:    intv is a vector which defines the intervals; the j-th interval is [intv(j),intv(j+1))
    765:    w is a vector of Chebyshev weights; the weight of j-th interval is w(j)
    766:        the interval weights define the inner product of two continuous functions and then
    767:        the derived w-norm ||P(z)-Q(z)||_w
    768:    pp is a matrix of Chebyshev coefficients which defines the piecewise polynomial P(z)
    769:    to be specific, for z in [intv(j), intv(j+1)), P(z) equals
    770:        P_j(z) = pp(1,j)*S_0(z) + pp(2,j)*S_1(z) + ... + pp(niter+2,j)*S_{niter+1}(z),
    771:    where S_i(z) is the `translated' Chebyshev polynomial in that interval,
    772:        S_i((z-c)/h) = T_i(z),  c = (intv(j)+intv(j+1))) / 2,  h = (intv(j+1)-intv(j)) / 2,
    773:    with T_i(z) the Chebyshev polynomial of the first kind,
    774:        T_0(z) = 1, T_1(z) = z, and T_i(z) = 2*z*T_{i-1}(z) - T_{i-2}(z) for i>=2

    776:    output:
    777:    the return matrix, denoted by qq, represents a polynomial Q(z) with degree up to 1+niter
    778:    and satisfying Q(0)==1, such that ||P(z))-Q(z)||_w is minimized
    779:    this polynomial Q(z) is expanded in the `translated' Chebyshev basis for each interval
    780:    to be precise, considering z in [intv(j), intv(j+1)), Q(z) equals
    781:        Q_j(z) = qq(1,j)*S_0(z) + qq(2,j)*S_1(z) + ... + qq(niter+2,j)*S_{niter+1}(z)

    783:    note:
    784:    1. since Q(0)==1, P(0)==1 is expected; if P(0)!=1, one can translate P(z)
    785:       for example, if P(0)==0, one can use 1-P(z) as input instead of P(z)
    786:    2. typically, the base filter, defined by pp and intv, is from Hermite interpolation
    787:       in intervals [intv(j),intv(j+1)) for j=1,...,nintv, with nintv the number of intervals
    788: */
    789: static PetscErrorCode FILTLAN_FilteredConjugateResidualPolynomial(PetscReal *cpol,PetscReal *baseFilter,PetscInt nbase,PetscReal *intv,PetscInt m,PetscReal *intervalWeights,PetscInt niter)
    790: {
    792:   PetscInt       i,j,srpol,scpol,sarpol,sppol,sappol,ld,nintv;
    793:   PetscReal      rho,rho0,rho1,den,bet,alp,alp0,*ppol,*rpol,*appol,*arpol;

    796:   nintv = m-1;
    797:   ld = niter+2;  /* leading dimension */
    798:   PetscCalloc4(ld*nintv,&ppol,ld*nintv,&rpol,ld*nintv,&appol,ld*nintv,&arpol);
    799:   PetscMemzero(cpol,ld*nintv*sizeof(PetscReal));
    800:   /* initialize polynomial ppol to be 1 (i.e. multiplicative identity) in all intervals */
    801:   sppol = 2;
    802:   srpol = 2;
    803:   scpol = 2;
    804:   for (j=0;j<nintv;j++) {
    805:     ppol[j*ld] = 1.0;
    806:     rpol[j*ld] = 1.0;
    807:     cpol[j*ld] = 1.0;
    808:   }
    809:   /* ppol is the initial p-polynomial (corresponding to the A-conjugate vector p in CG)
    810:      rpol is the r-polynomial (corresponding to the residual vector r in CG)
    811:      cpol is the "corrected" residual polynomial (result of this function) */
    812:   sappol = 3;
    813:   sarpol = 3;
    814:   FILTLAN_PiecewisePolynomialInChebyshevBasisMultiplyX(ppol,sppol,ld,intv,nintv,appol,ld);
    815:   for (i=0;i<3;i++) for (j=0;j<nintv;j++) arpol[i+j*ld] = appol[i+j*ld];
    816:   rho = FILTLAN_PiecewisePolynomialInnerProductInChebyshevBasis(rpol,srpol,nintv,ld,arpol,sarpol,nintv,ld,intervalWeights);
    817:   for (i=0;i<niter;i++) {
    818:     den = FILTLAN_PiecewisePolynomialInnerProductInChebyshevBasis(appol,sappol,nintv,ld,appol,sappol,nintv,ld,intervalWeights);
    819:     alp0 = rho/den;
    820:     rho1 = FILTLAN_PiecewisePolynomialInnerProductInChebyshevBasis(baseFilter,nbase,nintv,nbase,appol,sappol,nintv,ld,intervalWeights);
    821:     alp  = (rho-rho1)/den;
    822:     srpol++;
    823:     scpol++;
    824:     Mat_AXPY_BLAS(srpol,nintv,-alp0,appol,ld,1.0,rpol,ld);
    825:     Mat_AXPY_BLAS(scpol,nintv,-alp,appol,ld,1.0,cpol,ld);
    826:     if (i+1 == niter) break;
    827:     sarpol++;
    828:     FILTLAN_PiecewisePolynomialInChebyshevBasisMultiplyX(rpol,srpol,ld,intv,nintv,arpol,ld);
    829:     rho0 = rho;
    830:     rho = FILTLAN_PiecewisePolynomialInnerProductInChebyshevBasis(rpol,srpol,nintv,ld,arpol,sarpol,nintv,ld,intervalWeights);
    831:     bet  = rho / rho0;
    832:     sppol++;
    833:     sappol++;
    834:     Mat_AXPY_BLAS(sppol,nintv,1.0,rpol,ld,bet,ppol,ld);
    835:     Mat_AXPY_BLAS(sappol,nintv,1.0,arpol,ld,bet,appol,ld);
    836:   }
    837:   PetscFree4(ppol,rpol,appol,arpol);
    838:   return(0);
    839: }

    841: /*
    842:    FILTLAN function FilteredConjugateResidualMatrixPolynomialVectorProduct

    844:    this routine employs a conjugate-residual-type algorithm in polynomial space to compute
    845:    x = x0 + s(A)*r0 with r0 = b - A*x0, such that ||(1-P(z))-z*s(z)||_w is minimized, where
    846:    P(z) is a given piecewise polynomial, called the base filter,
    847:    s(z) is a polynomial of degree up to niter, the number of conjugate-residual iterations,
    848:    and b and x0 are given vectors

    850:    note that P(z) is expanded in the `translated' (scale-and-shift) Chebyshev basis for each interval,
    851:    and presented as a matrix of Chebyshev coefficients pp

    853:    input:
    854:    A is a sparse matrix
    855:    x0, b are vectors
    856:    niter is the number of conjugate-residual iterations
    857:    intv is a vector which defines the intervals; the j-th interval is [intv(j),intv(j+1))
    858:    w is a vector of Chebyshev weights; the weight of j-th interval is w(j)
    859:        the interval weights define the inner product of two continuous functions and then
    860:        the derived w-norm ||P(z)-Q(z)||_w
    861:    pp is a matrix of Chebyshev coefficients which defines the piecewise polynomial P(z)
    862:    to be specific, for z in [intv(j), intv(j+1)), P(z) equals
    863:        P_j(z) = pp(1,j)*S_0(z) + pp(2,j)*S_1(z) + ... + pp(niter+2,j)*S_{niter+1}(z),
    864:    where S_i(z) is the `translated' Chebyshev polynomial in that interval,
    865:        S_i((z-c)/h) = T_i(z),  c = (intv(j)+intv(j+1))) / 2,  h = (intv(j+1)-intv(j)) / 2,
    866:    with T_i(z) the Chebyshev polynomial of the first kind,
    867:        T_0(z) = 1, T_1(z) = z, and T_i(z) = 2*z*T_{i-1}(z) - T_{i-2}(z) for i>=2
    868:    tol is the tolerance; if the residual polynomial in z-norm is dropped by a factor lower
    869:        than tol, then stop the conjugate-residual iteration

    871:    output:
    872:    the return vector is x = x0 + s(A)*r0 with r0 = b - A*x0, such that ||(1-P(z))-z*s(z)||_w is minimized,
    873:    subject to that s(z) is a polynomial of degree up to niter, where P(z) is the base filter
    874:    in short, z*s(z) approximates 1-P(z)

    876:    note:
    877:    1. since z*s(z) approximates 1-P(z), P(0)==1 is expected; if P(0)!=1, one can translate P(z)
    878:       for example, if P(0)==0, one can use 1-P(z) as input instead of P(z)
    879:    2. typically, the base filter, defined by pp and intv, is from Hermite interpolation
    880:       in intervals [intv(j),intv(j+1)) for j=1,...,nintv, with nintv the number of intervals
    881:    3. a common application is to compute R(A)*b, where R(z) approximates 1-P(z)
    882:       in this case, one can set x0 = 0 and then the return vector is x = s(A)*b, where
    883:       z*s(z) approximates 1-P(z); therefore, A*x is the wanted R(A)*b
    884: */
    885: static PetscErrorCode FILTLAN_FilteredConjugateResidualMatrixPolynomialVectorProduct(Mat A,Vec b,Vec x,PetscReal *baseFilter,PetscInt nbase,PetscReal *intv,PetscInt nintv,PetscReal *intervalWeights,PetscInt niter,Vec *work)
    886: {
    888:   PetscInt       i,j,srpol,scpol,sarpol,sppol,sappol,ld;
    889:   PetscReal      rho,rho0,rho00,rho1,den,bet,alp,alp0,*cpol,*ppol,*rpol,*appol,*arpol,tol=0.0;
    890:   Vec            r=work[0],p=work[1],ap=work[2],w=work[3];
    891:   PetscScalar    alpha;

    894:   ld = niter+3;  /* leading dimension */
    895:   PetscCalloc5(ld*nintv,&ppol,ld*nintv,&rpol,ld*nintv,&cpol,ld*nintv,&appol,ld*nintv,&arpol);
    896:   /* initialize polynomial ppol to be 1 (i.e. multiplicative identity) in all intervals */
    897:   sppol = 2;
    898:   srpol = 2;
    899:   scpol = 2;
    900:   for (j=0;j<nintv;j++) {
    901:     ppol[j*ld] = 1.0;
    902:     rpol[j*ld] = 1.0;
    903:     cpol[j*ld] = 1.0;
    904:   }
    905:   /* ppol is the initial p-polynomial (corresponding to the A-conjugate vector p in CG)
    906:      rpol is the r-polynomial (corresponding to the residual vector r in CG)
    907:      cpol is the "corrected" residual polynomial */
    908:   sappol = 3;
    909:   sarpol = 3;
    910:   FILTLAN_PiecewisePolynomialInChebyshevBasisMultiplyX(ppol,sppol,ld,intv,nintv,appol,ld);
    911:   for (i=0;i<3;i++) for (j=0;j<nintv;j++) arpol[i+j*ld] = appol[i+j*ld];
    912:   rho00 = FILTLAN_PiecewisePolynomialInnerProductInChebyshevBasis(rpol,srpol,nintv,ld,arpol,sarpol,nintv,ld,intervalWeights);
    913:   rho = rho00;

    915:   /* corrected CR in vector space */
    916:   /* we assume x0 is always 0 */
    917:   VecSet(x,0.0);
    918:   VecCopy(b,r);     /* initial residual r = b-A*x0 */
    919:   VecCopy(r,p);     /* p = r */
    920:   MatMult(A,p,ap);  /* ap = A*p */

    922:   for (i=0;i<niter;i++) {
    923:     /* iteration in the polynomial space */
    924:     den = FILTLAN_PiecewisePolynomialInnerProductInChebyshevBasis(appol,sappol,nintv,ld,appol,sappol,nintv,ld,intervalWeights);
    925:     alp0 = rho/den;
    926:     rho1 = FILTLAN_PiecewisePolynomialInnerProductInChebyshevBasis(baseFilter,nbase,nintv,nbase,appol,sappol,nintv,ld,intervalWeights);
    927:     alp  = (rho-rho1)/den;
    928:     srpol++;
    929:     scpol++;
    930:     Mat_AXPY_BLAS(srpol,nintv,-alp0,appol,ld,1.0,rpol,ld);
    931:     Mat_AXPY_BLAS(scpol,nintv,-alp,appol,ld,1.0,cpol,ld);
    932:     sarpol++;
    933:     FILTLAN_PiecewisePolynomialInChebyshevBasisMultiplyX(rpol,srpol,ld,intv,nintv,arpol,ld);
    934:     rho0 = rho;
    935:     rho = FILTLAN_PiecewisePolynomialInnerProductInChebyshevBasis(rpol,srpol,nintv,ld,arpol,sarpol,nintv,ld,intervalWeights);

    937:     /* update x in the vector space */
    938:     alpha = alp;
    939:     VecAXPY(x,alpha,p);   /* x += alp*p */
    940:     if (rho < tol*rho00) break;

    942:     /* finish the iteration in the polynomial space */
    943:     bet = rho / rho0;
    944:     sppol++;
    945:     sappol++;
    946:     Mat_AXPY_BLAS(sppol,nintv,1.0,rpol,ld,bet,ppol,ld);
    947:     Mat_AXPY_BLAS(sappol,nintv,1.0,arpol,ld,bet,appol,ld);

    949:     /* finish the iteration in the vector space */
    950:     alpha = -alp0;
    951:     VecAXPY(r,alpha,ap);    /* r -= alp0*ap */
    952:     alpha = bet;
    953:     VecAYPX(p,alpha,r);     /* p = r + bet*p */
    954:     MatMult(A,r,w);         /* ap = A*r + bet*ap */
    955:     VecAYPX(ap,alpha,w);
    956:   }
    957:   PetscFree5(ppol,rpol,cpol,appol,arpol);
    958:   return(0);
    959: }

    961: /*
    962:    Gateway to FILTLAN for evaluating y=p(A)*x
    963: */
    964: PetscErrorCode STFilter_FILTLAN_Apply(ST st,Vec x,Vec y)
    965: {
    967:   ST_FILTER      *ctx = (ST_FILTER*)st->data;
    968:   PetscInt       npoints;

    971:   npoints = (ctx->filterInfo->filterType == 2)? 6: 4;
    972:   FILTLAN_FilteredConjugateResidualMatrixPolynomialVectorProduct(st->T[0],x,y,ctx->baseFilter,2*ctx->baseDegree+2,ctx->intervals,npoints-1,ctx->opts->intervalWeights,ctx->polyDegree,st->work);
    973:   VecCopy(y,st->work[0]);
    974:   MatMult(st->T[0],st->work[0],y);
    975:   return(0);
    976: }

    978: /*
    979:    FILTLAN function PolynomialFilterInterface::setFilter

    981:    Creates the shifted (and scaled) matrix and the base filter P(z)
    982: */
    983: PetscErrorCode STFilter_FILTLAN_setFilter(ST st)
    984: {
    986:   ST_FILTER      *ctx = (ST_FILTER*)st->data;
    987:   PetscInt       i,npoints;
    988:   PetscReal      frame2[4];
    989:   PetscScalar    alpha;
    990:   const PetscInt HighLowFlags[5] = { 1, -1, 0, -1, 1 };

    993:   MatDestroy(&st->T[0]);
    994:   if (ctx->frame[0] == ctx->frame[1]) {  /* low pass filter, convert it to high pass filter */
    995:     /* T = frame[3]*eye(n) - A */
    996:     MatDuplicate(st->A[0],MAT_COPY_VALUES,&st->T[0]);
    997:     MatScale(st->T[0],-1.0);
    998:     alpha = ctx->frame[3];
    999:     MatShift(st->T[0],alpha);
    1000:     for (i=0;i<4;i++) frame2[i] = ctx->frame[3] - ctx->frame[3-i];
    1001:     FILTLAN_GetIntervals(ctx->intervals,frame2,ctx->polyDegree,ctx->baseDegree,ctx->opts,ctx->filterInfo);
    1002:     /* translate the intervals back */
    1003:     for (i=0;i<4;i++) ctx->intervals2[i] = ctx->frame[3] - ctx->intervals[3-i];
    1004:   } else {  /* it can be a mid-pass filter or a high-pass filter */
    1005:       if (ctx->frame[0] == 0.0) {
    1006:       PetscObjectReference((PetscObject)st->A[0]);
    1007:       st->T[0] = st->A[0];
    1008:       FILTLAN_GetIntervals(ctx->intervals,ctx->frame,ctx->polyDegree,ctx->baseDegree,ctx->opts,ctx->filterInfo);
    1009:       for (i=0;i<6;i++) ctx->intervals2[i] = ctx->intervals[i];
    1010:     } else {
    1011:       /* T = A - frame[0]*eye(n) */
    1012:       MatDuplicate(st->A[0],MAT_COPY_VALUES,&st->T[0]);
    1013:       alpha = -ctx->frame[0];
    1014:       MatShift(st->T[0],alpha);
    1015:       for (i=0;i<4;i++) frame2[i] = ctx->frame[i] - ctx->frame[0];
    1016:       FILTLAN_GetIntervals(ctx->intervals,frame2,ctx->polyDegree,ctx->baseDegree,ctx->opts,ctx->filterInfo);
    1017:       /* translate the intervals back */
    1018:       for (i=0;i<6;i++) ctx->intervals2[i] = ctx->intervals[i] + ctx->frame[0];
    1019:     }
    1020:   }
    1021:   npoints = (ctx->filterInfo->filterType == 2)? 6: 4;
    1022:   PetscFree(ctx->baseFilter);
    1023:   PetscMalloc1((2*ctx->baseDegree+2)*(npoints-1),&ctx->baseFilter);
    1024:   FILTLAN_HermiteBaseFilterInChebyshevBasis(ctx->baseFilter,ctx->intervals,npoints,HighLowFlags,ctx->baseDegree);
    1025:   PetscInfo1(st,"Computed value of yLimit = %g\n",ctx->filterInfo->yLimit);
    1026:   return(0);
    1027: }