Extended Eigensolver RCI Interface
Description
The Extended Eigensolver RCI interfaces can be used to solve standard or
generalized eigenvalue problems, and are independent of the format of the
matrices. As mentioned earlier, the Extended Eigensolver algorithm is based on
the contour integration techniques of the matrix resolvent
over a circle. For solving a
generalized eigenvalue problem, Extended Eigensolver has to perform one or more
of the following operations at each contour point denoted below by
:
G
(σ
)=
(σ
B
-
A
)-1
Z
e
- Factorize the matrix(*ZeB-A)
- Solve the linear system(or*ZeB-A)X=Y(with multiple right hand sides, where H means transpose conjugate*ZeB-A)HX=Y
- Matrix-matrix multiplyorBX=YAX=Y
For solving a standard eigenvalue problem, replace the matrix
B
with the identity matrix
I
.
The primary aim of RCI interfaces is to isolate these operations: the
linear system solver, factorization of the matrix resolvent at each contour
point, and matrix-matrix multiplication. This gives universality to RCI
interfaces as they are independent of data structures and the specific
implementation of the operations like matrix-vector multiplication or inner
system solvers. However, this approach requires some additional effort when
calling the interface. In particular, operations listed above are performed by
routines that you supply on data structures that you find most appropriate for
the problem at hand.
To initialize an Extended Eigensolver RCI routine, set the job indicator
(
ijob
) parameter to the value -1. When the routine
requires the results of an operation, it generates a special value of
ijob
to indicate the operation that needs to be
performed. The routine also returns
ze
, the coordinate along the complex contour, the
values of array
work
or
workc
, and the number of columns to be used. Your
subroutine then must perform the operation at the given contour point
ze
, store the results in prescribed array, and
return control to the Extended Eigensolver RCI routine.
The following pseudocode shows the general scheme for using the Extended
Eigensolver RCI functionality for a real symmetric problem:
ijob=-1; // initialization do while (ijob!=0) { ?feast_srci(&ijob, &N, &Ze, work, workc, Aq, Bq, fpm, &epsout, &loop, &Emin, &Emax, &M0, E, lambda, &q, res, &info); switch(ijob) { case 10: // Factorize the complex matrix (ZeB-A) break; case 11: // Solve the complex linear system (ZeB-A)x=workc // Put result in workc break; case 30: // Perform multiplication A by Qi..Qjcolumns of QNxM0// where i = fpm[23] and j = fpm[23]+fpm[24]−1 // Qi..Qjlocated in q starting from q+N*(i-1) break; case 40: // Perform multiplication B by Qi..Qjcolumns of QNxM0// where i = fpm[23] and j = fpm[23]+fpm[24]−1 // Qi..Qjlocated in q starting from q+N*(i-1) // Result is stored in work+N*(i-1) break; } }
The
?
option in
?feast
in the pseudocode given above should be
replaced by either
s
or
d
, depending on the matrix data type of the
eigenvalue system.
The next pseudocode shows the general scheme for using the Extended
Eigensolver RCI functionality for a complex Hermitian problem:
ijob=-1; // initialization while (ijob!=0) { ?feast_hrci(&ijob, &N, &Ze, work, workc, Aq, Bq, fpm, &epsout, &loop, &Emin, &Emax, &M0, E, lambda, &q, res, &info); switch (ijob) { case 10: // Factorize the complex matrix (ZeB-A) break; case 11: // Solve the linear system (ZeB−A)y=workc // Put result in workc break; case 20: // Factorize (if needed by case 21) the complex matrix (ZeB−A)ˆH // ATTENTION: This option requires additional memory storage // (i.e . the resulting matrix from case 10 cannot be overwritten) break; case 21: // Solve the linear system (ZeB−A)ˆHy=workc // Put result in workc // REMARK: case 20 becomes obsolete if this solve can be performed // using the factorization in case 10 break; case 30: // Multiply A by Qi..Qjcolumns of QNxM0, // where i = fpm[23] and j = fpm[23]+fpm[24]−1 // Qi..Qjlocated in q starting from q+N*(i-1) // Result is stored in work+N*(i-1) break; case 40: // Perform multiplication B by Qi..Qjcolumns of QNxM0// where i = fpm[23] and j = fpm[23]+fpm[24]−1 // Qi..Qjlocated in q starting from q+N*(i-1) // Result is stored in work+N*(i-1) break; } } end do
The
?
option in
?feast
in the pseudocode given above should be
replaced by either
c
or
z
, depending on the matrix data type of the
eigenvalue system.
If
case 20
can be avoided, performance
could be up to twice as fast, and Extended Eigensolver functionality would use
half of the memory.
If an iterative solver is used along with a preconditioner, the
factorization of the preconditioner could be performed with
(and
if applicable) for a given value
of
, and the
associated iterative solve would then be performed with
(and
if applicable).
ijob
= 10ijob
= 20Z
e
ijob
= 11ijob
= 21Optimization Notice
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Intel's compilers may or may not optimize to the same degree for non-Intel microprocessors for optimizations that are not unique to Intel microprocessors. These optimizations include SSE2, SSE3, and SSSE3 instruction sets and other optimizations. Intel does not guarantee the availability, functionality, or effectiveness of any optimization on microprocessors not manufactured by Intel. Microprocessor-dependent optimizations in this product are intended for use with Intel microprocessors. Certain optimizations not specific to Intel microarchitecture are reserved for Intel microprocessors. Please refer to the applicable product User and Reference Guides for more information regarding the specific instruction sets covered by this notice.
Notice revision #20110804
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This notice covers the following instruction sets: SSE2, SSE4.2, AVX2, AVX-512.