/*
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   SLEPc - Scalable Library for Eigenvalue Problem Computations
   Copyright (c) 2002-2020, Universitat Politecnica de Valencia, Spain

   This file is part of SLEPc.
   SLEPc is distributed under a 2-clause BSD license (see LICENSE).
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
*/
/*
   Square root function  sqrt(x)
*/

#include <slepc/private/fnimpl.h>      /*I "slepcfn.h" I*/
#include <slepcblaslapack.h>

PetscErrorCode FNEvaluateFunction_Sqrt(FN fn,PetscScalar x,PetscScalar *y)
{
  PetscFunctionBegin;
#if !defined(PETSC_USE_COMPLEX)
  if (x<0.0) SETERRQ(PETSC_COMM_SELF,1,"Function not defined in the requested value");
#endif
  *y = PetscSqrtScalar(x);
  PetscFunctionReturn(0);
}

PetscErrorCode FNEvaluateDerivative_Sqrt(FN fn,PetscScalar x,PetscScalar *y)
{
  PetscFunctionBegin;
  if (x==0.0) SETERRQ(PETSC_COMM_SELF,1,"Derivative not defined in the requested value");
#if !defined(PETSC_USE_COMPLEX)
  if (x<0.0) SETERRQ(PETSC_COMM_SELF,1,"Derivative not defined in the requested value");
#endif
  *y = 1.0/(2.0*PetscSqrtScalar(x));
  PetscFunctionReturn(0);
}

PetscErrorCode FNEvaluateFunctionMat_Sqrt_Schur(FN fn,Mat A,Mat B)
{
  PetscErrorCode ierr;
  PetscBLASInt   n = 0;
  PetscScalar    *T;
  PetscInt       m;

  PetscFunctionBegin;
  if (A!=B) { ierr = MatCopy(A,B,SAME_NONZERO_PATTERN);CHKERRQ(ierr); }
  ierr = MatDenseGetArray(B,&T);CHKERRQ(ierr);
  ierr = MatGetSize(A,&m,NULL);CHKERRQ(ierr);
  ierr = PetscBLASIntCast(m,&n);CHKERRQ(ierr);
  ierr = SlepcSqrtmSchur(n,T,n,PETSC_FALSE);CHKERRQ(ierr);
  ierr = MatDenseRestoreArray(B,&T);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode FNEvaluateFunctionMatVec_Sqrt_Schur(FN fn,Mat A,Vec v)
{
  PetscErrorCode ierr;
  PetscBLASInt   n = 0;
  PetscScalar    *T;
  PetscInt       m;
  Mat            B;

  PetscFunctionBegin;
  ierr = FN_AllocateWorkMat(fn,A,&B);CHKERRQ(ierr);
  ierr = MatDenseGetArray(B,&T);CHKERRQ(ierr);
  ierr = MatGetSize(A,&m,NULL);CHKERRQ(ierr);
  ierr = PetscBLASIntCast(m,&n);CHKERRQ(ierr);
  ierr = SlepcSqrtmSchur(n,T,n,PETSC_TRUE);CHKERRQ(ierr);
  ierr = MatDenseRestoreArray(B,&T);CHKERRQ(ierr);
  ierr = MatGetColumnVector(B,v,0);CHKERRQ(ierr);
  ierr = FN_FreeWorkMat(fn,&B);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode FNEvaluateFunctionMat_Sqrt_DBP(FN fn,Mat A,Mat B)
{
  PetscErrorCode ierr;
  PetscBLASInt   n = 0;
  PetscScalar    *T;
  PetscInt       m;

  PetscFunctionBegin;
  if (A!=B) { ierr = MatCopy(A,B,SAME_NONZERO_PATTERN);CHKERRQ(ierr); }
  ierr = MatDenseGetArray(B,&T);CHKERRQ(ierr);
  ierr = MatGetSize(A,&m,NULL);CHKERRQ(ierr);
  ierr = PetscBLASIntCast(m,&n);CHKERRQ(ierr);
  ierr = SlepcSqrtmDenmanBeavers(n,T,n,PETSC_FALSE);CHKERRQ(ierr);
  ierr = MatDenseRestoreArray(B,&T);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode FNEvaluateFunctionMat_Sqrt_NS(FN fn,Mat A,Mat B)
{
  PetscErrorCode ierr;
  PetscBLASInt   n = 0;
  PetscScalar    *Ba;
  PetscInt       m;

  PetscFunctionBegin;
  if (A!=B) { ierr = MatCopy(A,B,SAME_NONZERO_PATTERN);CHKERRQ(ierr); }
  ierr = MatDenseGetArray(B,&Ba);CHKERRQ(ierr);
  ierr = MatGetSize(A,&m,NULL);CHKERRQ(ierr);
  ierr = PetscBLASIntCast(m,&n);CHKERRQ(ierr);
  ierr = SlepcSqrtmNewtonSchulz(n,Ba,n,PETSC_FALSE);CHKERRQ(ierr);
  ierr = MatDenseRestoreArray(B,&Ba);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#define MAXIT 50

/*
   Computes the principal square root of the matrix A using the
   Sadeghi iteration. A is overwritten with sqrtm(A).
 */
static PetscErrorCode SlepcSqrtmSadeghi(PetscBLASInt n,PetscScalar *A,PetscBLASInt ld)
{
  PetscScalar        *M,*M2,*G,*X=A,*work,work1,alpha,sqrtnrm;
  PetscScalar        szero=0.0,sone=1.0,smfive=-5.0,s1d16=1.0/16.0;
  PetscReal          tol,Mres=0.0,nrm,rwork[1];
  PetscBLASInt       N,i,it,*piv=NULL,info,lwork,query=-1;
  const PetscBLASInt one=1;
  PetscBool          converged=PETSC_FALSE;
  PetscErrorCode     ierr;
  unsigned int       ftz;

  PetscFunctionBegin;
  N = n*n;
  tol = PetscSqrtReal((PetscReal)n)*PETSC_MACHINE_EPSILON/2;
  ierr = SlepcSetFlushToZero(&ftz);CHKERRQ(ierr);

  /* query work size */
  PetscStackCallBLAS("LAPACKgetri",LAPACKgetri_(&n,A,&ld,piv,&work1,&query,&info));
  ierr = PetscBLASIntCast((PetscInt)PetscRealPart(work1),&lwork);CHKERRQ(ierr);

  ierr = PetscMalloc5(N,&M,N,&M2,N,&G,lwork,&work,n,&piv);CHKERRQ(ierr);
  ierr = PetscArraycpy(M,A,N);CHKERRQ(ierr);

  /* scale M */
  nrm = LAPACKlange_("fro",&n,&n,M,&n,rwork);
  if (nrm>1.0) {
    sqrtnrm = PetscSqrtReal(nrm);
    alpha = 1.0/nrm;
    PetscStackCallBLAS("BLASscal",BLASscal_(&N,&alpha,M,&one));
    tol *= nrm;
  }
  ierr = PetscInfo2(NULL,"||A||_F = %g, new tol: %g\n",(double)nrm,(double)tol);CHKERRQ(ierr);

  /* X = I */
  ierr = PetscArrayzero(X,N);CHKERRQ(ierr);
  for (i=0;i<n;i++) X[i+i*ld] = 1.0;

  for (it=0;it<MAXIT && !converged;it++) {

    /* G = (5/16)*I + (1/16)*M*(15*I-5*M+M*M) */
    PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&sone,M,&ld,M,&ld,&szero,M2,&ld));
    PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&N,&smfive,M,&one,M2,&one));
    for (i=0;i<n;i++) M2[i+i*ld] += 15.0;
    PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&s1d16,M,&ld,M2,&ld,&szero,G,&ld));
    for (i=0;i<n;i++) G[i+i*ld] += 5.0/16.0;

    /* X = X*G */
    ierr = PetscArraycpy(M2,X,N);CHKERRQ(ierr);
    PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&sone,M2,&ld,G,&ld,&szero,X,&ld));

    /* M = M*inv(G*G) */
    PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&sone,G,&ld,G,&ld,&szero,M2,&ld));
    PetscStackCallBLAS("LAPACKgetrf",LAPACKgetrf_(&n,&n,M2,&ld,piv,&info));
    SlepcCheckLapackInfo("getrf",info);
    PetscStackCallBLAS("LAPACKgetri",LAPACKgetri_(&n,M2,&ld,piv,work,&lwork,&info));
    SlepcCheckLapackInfo("getri",info);

    ierr = PetscArraycpy(G,M,N);CHKERRQ(ierr);
    PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&sone,G,&ld,M2,&ld,&szero,M,&ld));

    /* check ||I-M|| */
    ierr = PetscArraycpy(M2,M,N);CHKERRQ(ierr);
    for (i=0;i<n;i++) M2[i+i*ld] -= 1.0;
    Mres = LAPACKlange_("fro",&n,&n,M2,&n,rwork);
    ierr = PetscIsNanReal(Mres);CHKERRQ(ierr);
    if (Mres<=tol) converged = PETSC_TRUE;
    ierr = PetscInfo2(NULL,"it: %D res: %g\n",it,(double)Mres);CHKERRQ(ierr);
    ierr = PetscLogFlops(8.0*n*n*n+2.0*n*n+2.0*n*n*n/3.0+4.0*n*n*n/3.0+2.0*n*n*n+2.0*n*n);CHKERRQ(ierr);
  }

  if (Mres>tol) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_LIB,"SQRTM not converged after %d iterations",MAXIT);

  /* undo scaling */
  if (nrm>1.0) PetscStackCallBLAS("BLASscal",BLASscal_(&N,&sqrtnrm,A,&one));

  ierr = PetscFree5(M,M2,G,work,piv);CHKERRQ(ierr);
  ierr = SlepcResetFlushToZero(&ftz);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode FNEvaluateFunctionMat_Sqrt_Sadeghi(FN fn,Mat A,Mat B)
{
  PetscErrorCode ierr;
  PetscBLASInt   n = 0;
  PetscScalar    *Ba;
  PetscInt       m;

  PetscFunctionBegin;
  if (A!=B) { ierr = MatCopy(A,B,SAME_NONZERO_PATTERN);CHKERRQ(ierr); }
  ierr = MatDenseGetArray(B,&Ba);CHKERRQ(ierr);
  ierr = MatGetSize(A,&m,NULL);CHKERRQ(ierr);
  ierr = PetscBLASIntCast(m,&n);CHKERRQ(ierr);
  ierr = SlepcSqrtmSadeghi(n,Ba,n);CHKERRQ(ierr);
  ierr = MatDenseRestoreArray(B,&Ba);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode FNView_Sqrt(FN fn,PetscViewer viewer)
{
  PetscErrorCode ierr;
  PetscBool      isascii;
  char           str[50];
  const char     *methodname[] = {
                  "Schur method for the square root",
                  "Denman-Beavers (product form)",
                  "Newton-Schulz iteration",
                  "Sadeghi iteration"
  };
  const int      nmeth=sizeof(methodname)/sizeof(methodname[0]);

  PetscFunctionBegin;
  ierr = PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&isascii);CHKERRQ(ierr);
  if (isascii) {
    if (fn->beta==(PetscScalar)1.0) {
      if (fn->alpha==(PetscScalar)1.0) {
        ierr = PetscViewerASCIIPrintf(viewer,"  Square root: sqrt(x)\n");CHKERRQ(ierr);
      } else {
        ierr = SlepcSNPrintfScalar(str,sizeof(str),fn->alpha,PETSC_TRUE);CHKERRQ(ierr);
        ierr = PetscViewerASCIIPrintf(viewer,"  Square root: sqrt(%s*x)\n",str);CHKERRQ(ierr);
      }
    } else {
      ierr = SlepcSNPrintfScalar(str,sizeof(str),fn->beta,PETSC_TRUE);CHKERRQ(ierr);
      if (fn->alpha==(PetscScalar)1.0) {
        ierr = PetscViewerASCIIPrintf(viewer,"  Square root: %s*sqrt(x)\n",str);CHKERRQ(ierr);
      } else {
        ierr = PetscViewerASCIIPrintf(viewer,"  Square root: %s",str);CHKERRQ(ierr);
        ierr = PetscViewerASCIIUseTabs(viewer,PETSC_FALSE);CHKERRQ(ierr);
        ierr = SlepcSNPrintfScalar(str,sizeof(str),fn->alpha,PETSC_TRUE);CHKERRQ(ierr);
        ierr = PetscViewerASCIIPrintf(viewer,"*sqrt(%s*x)\n",str);CHKERRQ(ierr);
        ierr = PetscViewerASCIIUseTabs(viewer,PETSC_TRUE);CHKERRQ(ierr);
      }
    }
    if (fn->method<nmeth) {
      ierr = PetscViewerASCIIPrintf(viewer,"  computing matrix functions with: %s\n",methodname[fn->method]);CHKERRQ(ierr);
    }
  }
  PetscFunctionReturn(0);
}

SLEPC_EXTERN PetscErrorCode FNCreate_Sqrt(FN fn)
{
  PetscFunctionBegin;
  fn->ops->evaluatefunction          = FNEvaluateFunction_Sqrt;
  fn->ops->evaluatederivative        = FNEvaluateDerivative_Sqrt;
  fn->ops->evaluatefunctionmat[0]    = FNEvaluateFunctionMat_Sqrt_Schur;
  fn->ops->evaluatefunctionmat[1]    = FNEvaluateFunctionMat_Sqrt_DBP;
  fn->ops->evaluatefunctionmat[2]    = FNEvaluateFunctionMat_Sqrt_NS;
  fn->ops->evaluatefunctionmat[3]    = FNEvaluateFunctionMat_Sqrt_Sadeghi;
  fn->ops->evaluatefunctionmatvec[0] = FNEvaluateFunctionMatVec_Sqrt_Schur;
  fn->ops->view                      = FNView_Sqrt;
  PetscFunctionReturn(0);
}

