Extended Eigensolver RCI Interface Description

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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Date 3/22/2024

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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 G(σ )= (σB - A)^-1 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 Z_e :

Factorize the matrix (Z_e *B - A)
Solve the linear system (Z_e *B - A)X = Y or (Z_e *B - A)^HX = Y with multiple right hand sides, where H means transpose conjugate
Matrix-matrix multiply BX = Y or AX = 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)
       call  ?feast_srci(ijob, N, Ze, work1, work2, Aq,  Bq,
                   &fpm, epsout, loop, Emin, Emax, M0, E, lambda, q, res, info)

       select case(ijob)
       case(10) !! Factorize the complex matrix (ZeB-A)
 . . . . . . . . . . . . . . . . <<< user entry
     
       case(11) !! Solve the complex linear system (ZeB-A)x=work2(1:N,1:M0) result in work2
. . . . . . . . . . . . . . . . <<< user entry

       case(30) !! Perform multiplication A*q(1:N,i:j) result in work1(1:N,i:j)
                !!  where i=fpm(24) and j=fpm(24)+fpm(25)−1
. . . . . . . . . . . . . . . . <<< user entry
       case(40) !! Perform multiplication B*q(1:N,i:j) result in work1(1:N,i:j)
                !!  where i=fpm(24) and j=fpm(24)+fpm(25)−1
. . . . . . . . . . . . . . . . <<< user entry

    end select
end do

NOTE:

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
    do while (ijob/=0)
       call  ?feast_hrci(ijob, N, Ze, work1, work2, Aq,  Bq,
                   &fpm, epsout, loop, Emin, Emax, M0, E, lambda, q, res, info)

       select case(ijob)
       case(10) !! Factorize the complex matrix (ZeB-A)
 . . . . . . . . . . . . . . . . <<< user entry
       case (11)!! Solve the linear system (ZeB−A)y=work2 (1:N, 1:M0) result in work2
. . . . . . . . . . . . . . . . <<< user entry
       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)
. . . . . . . . . . . . . . . . <<< user entry
       case (21) !! Solve the linear system (ZeB−A)ˆHy=work2(1:N, 1:M0) result in  work2
	!!REMARK: case (20) becomes obsolete if this solve can be performed
	!! using the factorization in case (10)
. . . . . . . . . . . . . . . . <<< user entry
       case(30) !! Perform multiplication A*q(1:N,i:j) result in work1(1:N,i:j)
                !!  where i=fpm(24) and j=fpm(25)+fpm(24)−1
. . . . . . . . . . . . . . . . <<< user entry
       case(40) !! Perform multiplication B*q(1:N,i:j) result in work1(1:N,i:j)
                !!  where i=fpm(24) and j=fpm(25)+fpm(24)−1
. . . . . . . . . . . . . . . . <<< user entry

    end select
end do

NOTE:

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 ijob = 10 (and ijob = 20 if applicable) for a given value of Z_e, and the associated iterative solve would then be performed with ijob = 11 (and ijob = 21 if applicable).

Product and Performance Information
Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex. Notice revision #20201201

Product and Performance Information

Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.

Notice revision #20201201

Parent topic: Extended Eigensolver RCI Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

Extended Eigensolver RCI Interface Description