/*@
SVDComputeRelativeError - Computes the relative error bound associated
with the i-th singular triplet.
Collective on SVD
Input Parameter:
+ svd - the singular value solver context
- i - the solution index
Output Parameter:
. error - the relative error bound, computed as sqrt(n1^2+n2^2)/sigma
where n1 = ||A*v-sigma*u||_2 , n2 = ||A^T*u-sigma*v||_2 , sigma is the singular value,
u and v are the left and right singular vectors.
If sigma is too small the relative error is computed as sqrt(n1^2+n2^2).
Level: beginner
.seealso: SVDSolve(), SVDComputeResidualNorms()
@*/
PetscErrorCode SVDComputeRelativeError(SVD svd,PetscInt i,PetscReal *error)
{
PetscErrorCode ierr;
PetscReal sigma,norm1,norm2;
PetscFunctionBegin;
PetscValidHeaderSpecific(svd,SVD_CLASSID,1);
PetscValidLogicalCollectiveInt(svd,i,2);
PetscValidPointer(error,3);
ierr = SVDGetSingularTriplet(svd,i,&sigma,NULL,NULL);CHKERRQ(ierr);
ierr = SVDComputeResidualNorms(svd,i,&norm1,&norm2);CHKERRQ(ierr);
*error = PetscSqrtReal(norm1*norm1+norm2*norm2);
if (sigma>*error) *error /= sigma;
PetscFunctionReturn(0);
}
/*@
SVDComputeResidualNorms - Computes the norms of the residual vectors associated with
the i-th computed singular triplet.
Collective on SVD
Input Parameters:
+ svd - the singular value solver context
- i - the solution index
Output Parameters:
+ norm1 - the norm ||A*v-sigma*u||_2 where sigma is the
singular value, u and v are the left and right singular vectors.
- norm2 - the norm ||A^T*u-sigma*v||_2 with the same sigma, u and v
Note:
The index i should be a value between 0 and nconv-1 (see SVDGetConverged()).
Both output parameters can be NULL on input if not needed.
Level: beginner
.seealso: SVDSolve(), SVDGetConverged(), SVDComputeRelativeError()
@*/
PetscErrorCode SVDComputeResidualNorms(SVD svd,PetscInt i,PetscReal *norm1,PetscReal *norm2)
{
PetscErrorCode ierr;
Vec u,v,x = NULL,y = NULL;
PetscReal sigma;
PetscInt M,N;
PetscFunctionBegin;
PetscValidHeaderSpecific(svd,SVD_CLASSID,1);
PetscValidLogicalCollectiveInt(svd,i,2);
if (svd->reason == SVD_CONVERGED_ITERATING) SETERRQ(PetscObjectComm((PetscObject)svd),PETSC_ERR_ARG_WRONGSTATE,"SVDSolve must be called first");
if (i<0 || i>=svd->nconv) SETERRQ(PetscObjectComm((PetscObject)svd),PETSC_ERR_ARG_OUTOFRANGE,"Argument 2 out of range");
ierr = MatGetVecs(svd->OP,&v,&u);CHKERRQ(ierr);
ierr = SVDGetSingularTriplet(svd,i,&sigma,u,v);CHKERRQ(ierr);
if (norm1) {
ierr = VecDuplicate(u,&x);CHKERRQ(ierr);
ierr = MatMult(svd->OP,v,x);CHKERRQ(ierr);
ierr = VecAXPY(x,-sigma,u);CHKERRQ(ierr);
ierr = VecNorm(x,NORM_2,norm1);CHKERRQ(ierr);
}
if (norm2) {
ierr = VecDuplicate(v,&y);CHKERRQ(ierr);
if (svd->A && svd->AT) {
ierr = MatGetSize(svd->OP,&M,&N);CHKERRQ(ierr);
if (M<N) {
ierr = MatMult(svd->A,u,y);CHKERRQ(ierr);
} else {
ierr = MatMult(svd->AT,u,y);CHKERRQ(ierr);
}
} else {
#if defined(PETSC_USE_COMPLEX)
ierr = MatMultHermitianTranspose(svd->OP,u,y);CHKERRQ(ierr);
#else
ierr = MatMultTranspose(svd->OP,u,y);CHKERRQ(ierr);
#endif
}
ierr = VecAXPY(y,-sigma,v);CHKERRQ(ierr);
ierr = VecNorm(y,NORM_2,norm2);CHKERRQ(ierr);
}
ierr = VecDestroy(&v);CHKERRQ(ierr);
ierr = VecDestroy(&u);CHKERRQ(ierr);
ierr = VecDestroy(&x);CHKERRQ(ierr);
ierr = VecDestroy(&y);CHKERRQ(ierr);
PetscFunctionReturn(0);
}
int main( int argc, char **argv )
{
PetscErrorCode ierr;
Mat A; /* Grcar matrix */
SVD svd; /* singular value solver context */
PetscInt N=30, Istart, Iend, i, col[5], nconv1, nconv2;
PetscScalar value[] = { -1, 1, 1, 1, 1 };
PetscReal sigma_1, sigma_n;
SlepcInitialize(&argc,&argv,(char*)0,help);
ierr = PetscOptionsGetInt(PETSC_NULL,"-n",&N,PETSC_NULL);CHKERRQ(ierr);
ierr = PetscPrintf(PETSC_COMM_WORLD,"\nEstimate the condition number of a Grcar matrix, n=%d\n\n",N);CHKERRQ(ierr);
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Generate the matrix
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
ierr = MatCreate(PETSC_COMM_WORLD,&A);CHKERRQ(ierr);
ierr = MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,N,N);CHKERRQ(ierr);
ierr = MatSetFromOptions(A);CHKERRQ(ierr);
ierr = MatGetOwnershipRange(A,&Istart,&Iend);CHKERRQ(ierr);
for( i=Istart; i<Iend; i++ ) {
col[0]=i-1; col[1]=i; col[2]=i+1; col[3]=i+2; col[4]=i+3;
if (i==0) {
ierr = MatSetValues(A,1,&i,4,col+1,value+1,INSERT_VALUES);CHKERRQ(ierr);
}
else {
ierr = MatSetValues(A,1,&i,PetscMin(5,N-i+1),col,value,INSERT_VALUES);CHKERRQ(ierr);
}
}
ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
ierr = MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Create the singular value solver and set the solution method
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
/*
Create singular value context
*/
ierr = SVDCreate(PETSC_COMM_WORLD,&svd);CHKERRQ(ierr);
/*
Set operator
*/
ierr = SVDSetOperator(svd,A);CHKERRQ(ierr);
/*
Set solver parameters at runtime
*/
ierr = SVDSetFromOptions(svd);CHKERRQ(ierr);
ierr = SVDSetDimensions(svd,1,PETSC_IGNORE,PETSC_IGNORE);CHKERRQ(ierr);
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Solve the eigensystem
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
/*
First request an eigenvalue from one end of the spectrum
*/
ierr = SVDSetWhichSingularTriplets(svd,SVD_LARGEST);CHKERRQ(ierr);
ierr = SVDSolve(svd);CHKERRQ(ierr);
/*
Get number of converged singular values
*/
ierr = SVDGetConverged(svd,&nconv1);CHKERRQ(ierr);
/*
Get converged singular values: largest singular value is stored in sigma_1.
In this example, we are not interested in the singular vectors
*/
if (nconv1 > 0) {
ierr = SVDGetSingularTriplet(svd,0,&sigma_1,PETSC_NULL,PETSC_NULL);CHKERRQ(ierr);
} else {
ierr = PetscPrintf(PETSC_COMM_WORLD," Unable to compute large singular value!\n\n");CHKERRQ(ierr);
}
/*
Request an eigenvalue from the other end of the spectrum
*/
ierr = SVDSetWhichSingularTriplets(svd,SVD_SMALLEST);CHKERRQ(ierr);
ierr = SVDSolve(svd);CHKERRQ(ierr);
/*
Get number of converged eigenpairs
*/
ierr = SVDGetConverged(svd,&nconv2);CHKERRQ(ierr);
/*
Get converged singular values: smallest singular value is stored in sigma_n.
As before, we are not interested in the singular vectors
*/
if (nconv2 > 0) {
ierr = SVDGetSingularTriplet(svd,0,&sigma_n,PETSC_NULL,PETSC_NULL);CHKERRQ(ierr);
} else {
ierr = PetscPrintf(PETSC_COMM_WORLD," Unable to compute small singular value!\n\n");CHKERRQ(ierr);
}
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Display solution and clean up
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
if (nconv1 > 0 && nconv2 > 0) {
ierr = PetscPrintf(PETSC_COMM_WORLD," Computed singular values: sigma_1=%6f, sigma_n=%6f\n",sigma_1,sigma_n);CHKERRQ(ierr);
ierr = PetscPrintf(PETSC_COMM_WORLD," Estimated condition number: sigma_1/sigma_n=%6f\n\n",sigma_1/sigma_n);CHKERRQ(ierr);
}
/*
Free work space
*/
ierr = SVDDestroy(svd);CHKERRQ(ierr);
ierr = MatDestroy(A);CHKERRQ(ierr);
ierr = SlepcFinalize();CHKERRQ(ierr);
return 0;
}
int main(int argc,char **argv)
{
Mat A; /* operator matrix */
Vec u,v; /* left and right singular vectors */
SVD svd; /* singular value problem solver context */
SVDType type;
PetscReal error,tol,sigma,mu=PETSC_SQRT_MACHINE_EPSILON;
PetscInt n=100,i,j,Istart,Iend,nsv,maxit,its,nconv;
PetscErrorCode ierr;
SlepcInitialize(&argc,&argv,(char*)0,help);
ierr = PetscOptionsGetInt(NULL,"-n",&n,NULL);CHKERRQ(ierr);
ierr = PetscOptionsGetReal(NULL,"-mu",&mu,NULL);CHKERRQ(ierr);
ierr = PetscPrintf(PETSC_COMM_WORLD,"\nLauchli singular value decomposition, (%D x %D) mu=%g\n\n",n+1,n,(double)mu);CHKERRQ(ierr);
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Build the Lauchli matrix
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
ierr = MatCreate(PETSC_COMM_WORLD,&A);CHKERRQ(ierr);
ierr = MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,n+1,n);CHKERRQ(ierr);
ierr = MatSetFromOptions(A);CHKERRQ(ierr);
ierr = MatSetUp(A);CHKERRQ(ierr);
ierr = MatGetOwnershipRange(A,&Istart,&Iend);CHKERRQ(ierr);
for (i=Istart;i<Iend;i++) {
if (i == 0) {
for (j=0;j<n;j++) {
ierr = MatSetValue(A,0,j,1.0,INSERT_VALUES);CHKERRQ(ierr);
}
} else {
ierr = MatSetValue(A,i,i-1,mu,INSERT_VALUES);CHKERRQ(ierr);
}
}
ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
ierr = MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
ierr = MatGetVecs(A,&v,&u);CHKERRQ(ierr);
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Create the singular value solver and set various options
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
/*
Create singular value solver context
*/
ierr = SVDCreate(PETSC_COMM_WORLD,&svd);CHKERRQ(ierr);
/*
Set operator
*/
ierr = SVDSetOperator(svd,A);CHKERRQ(ierr);
/*
Use thick-restart Lanczos as default solver
*/
ierr = SVDSetType(svd,SVDTRLANCZOS);CHKERRQ(ierr);
/*
Set solver parameters at runtime
*/
ierr = SVDSetFromOptions(svd);CHKERRQ(ierr);
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Solve the singular value system
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
ierr = SVDSolve(svd);CHKERRQ(ierr);
ierr = SVDGetIterationNumber(svd,&its);CHKERRQ(ierr);
ierr = PetscPrintf(PETSC_COMM_WORLD," Number of iterations of the method: %D\n",its);CHKERRQ(ierr);
/*
Optional: Get some information from the solver and display it
*/
ierr = SVDGetType(svd,&type);CHKERRQ(ierr);
ierr = PetscPrintf(PETSC_COMM_WORLD," Solution method: %s\n\n",type);CHKERRQ(ierr);
ierr = SVDGetDimensions(svd,&nsv,NULL,NULL);CHKERRQ(ierr);
ierr = PetscPrintf(PETSC_COMM_WORLD," Number of requested singular values: %D\n",nsv);CHKERRQ(ierr);
ierr = SVDGetTolerances(svd,&tol,&maxit);CHKERRQ(ierr);
ierr = PetscPrintf(PETSC_COMM_WORLD," Stopping condition: tol=%.4g, maxit=%D\n",(double)tol,maxit);CHKERRQ(ierr);
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Display solution and clean up
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
/*
Get number of converged singular triplets
*/
ierr = SVDGetConverged(svd,&nconv);CHKERRQ(ierr);
ierr = PetscPrintf(PETSC_COMM_WORLD," Number of converged approximate singular triplets: %D\n\n",nconv);CHKERRQ(ierr);
if (nconv>0) {
/*
Display singular values and relative errors
*/
ierr = PetscPrintf(PETSC_COMM_WORLD,
" sigma relative error\n"
" --------------------- ------------------\n");CHKERRQ(ierr);
for (i=0;i<nconv;i++) {
/*
Get converged singular triplets: i-th singular value is stored in sigma
*/
ierr = SVDGetSingularTriplet(svd,i,&sigma,u,v);CHKERRQ(ierr);
/*
Compute the error associated to each singular triplet
*/
ierr = SVDComputeRelativeError(svd,i,&error);CHKERRQ(ierr);
ierr = PetscPrintf(PETSC_COMM_WORLD," % 6f ",(double)sigma);CHKERRQ(ierr);
ierr = PetscPrintf(PETSC_COMM_WORLD," % 12g\n",(double)error);CHKERRQ(ierr);
}
ierr = PetscPrintf(PETSC_COMM_WORLD,"\n");CHKERRQ(ierr);
}
/*
Free work space
*/
ierr = SVDDestroy(&svd);CHKERRQ(ierr);
ierr = MatDestroy(&A);CHKERRQ(ierr);
ierr = VecDestroy(&u);CHKERRQ(ierr);
ierr = VecDestroy(&v);CHKERRQ(ierr);
ierr = SlepcFinalize();
return 0;
}
