SymSmallestEigenVal

This function calculates the smallest eigenvalue of a real sym dense matrix.

• It calls ARPACK SSAUPD or DSAUPD routine with MODE=3
• Currently, there are four interfaces under this generic method.
• In this routine we use shift-inverted method to compute the smallest eigenvalue of a regular (standard) eigenvalue problem.
• This is because ARPACK is good at finding the largest eigenvalue.
• Internally this routine solves a system of linear equations: mat * y = x by using LU decomposition.
• In this routine we make a call to SymLUSolve and SymGetLU routine.

܀ Interface 1

INTERFACE
  MODULE FUNCTION SymSmallestEigenVal(mat, sigma, which, NCV, maxIter, tol) &
    & RESULT(ans)
    REAL(DFP), INTENT(IN) :: mat(:, :)
    !! dense matrix
    REAL(DFP), OPTIONAL, INTENT(IN) :: sigma
    !! Default value is 0.0
    CHARACTER(LEN=*), OPTIONAL, INTENT(IN) :: which
    !! `which = "SM"` ⇨ absolute smallest eigenvalue
    !! `which = "SA"` ⇨ algebraic smallest eigenvalue
    !! default is "SA"
    INTEGER(I4B), OPTIONAL, INTENT(IN) :: NCV
    !! Number of Lanczos vectors generated
    !! It must be greater than 1 and smaller than `size(mat,1)`
    !! Default is `NCV = MIN(n, 20)`
    INTEGER(I4B), OPTIONAL, INTENT(IN) :: maxIter
    !! Maximum number of iteration default = `N*10`
    REAL(DFP), OPTIONAL, INTENT(IN) :: tol
    !! tolerance, default = 0.0
    REAL(DFP) :: ans
    !! maximum eigenvalue
  END FUNCTION SymSmallestEigenVal
END INTERFACE

caution

This routine makes a copy of mat in mat0. Then, compute the LU decomposition of mat0.

• mat dense matrix
• sigma Default value is 0.0
• which = "SM" ⇨ absolute smallest eigenvalue
• which = "SA" ⇨ algebraic smallest eigenvalue
• default value of which is "SA"
• NCV Number of Lanczos vectors generated. It must be greater than 1 and smaller than size(mat,1). Default is NCV = MIN(n, 20)
• maxIter Maximum number of iteration default = N*10
• tol tolerance, default = 0.0
• ans maximum eigenvalue.

܀ Interface 2

INTERFACE
  MODULE FUNCTION SymSmallestEigenVal(mat, isFactor, ipiv, sigma, which, &
    & NCV, maxIter, tol) RESULT(ans)
    REAL(DFP), INTENT(INOUT) :: mat(:, :)
    !!
    !! Dense matrix
    !! If isFactor is false, then this matrix will change on return
    !!    in this case, it will contain LU decomposition
    !! If isFactor is true, then this matrix will not change
    !!
    LOGICAL(LGT), INTENT(INOUT) :: isFactor
    !! if mat is already factorized, the set isFactor to true
    !! if mat is not factorized, then set isFactor to false
    INTEGER(I4B), OPTIONAL, INTENT(IN) :: ipiv(:)
    !! When `isFactor` is true, then `mat` represents the
    !! `LU` factorization obtained by `SymGetLU` routine.
    !! In this case `ipiv` is returned by `SymGetLU`.
    REAL(DFP), OPTIONAL, INTENT(IN) :: sigma
    !! Default value is 0.0
    CHARACTER(LEN=*), OPTIONAL, INTENT(IN) :: which
    !! `which = "SM"` ⇨ absolute smallest eigenvalue
    !! `which = "SA"` ⇨ algebraic smallest eigenvalue
    !! default is "SA"
    INTEGER(I4B), OPTIONAL, INTENT(IN) :: NCV
    !! Number of Lanczos vectors generated
    !! It must be greater than 1 and smaller than `size(mat,1)`
    !! Default is `NCV = MIN(n, 20)`
    INTEGER(I4B), OPTIONAL, INTENT(IN) :: maxIter
    !! Maximum number of iteration default = `N*10`
    REAL(DFP), OPTIONAL, INTENT(IN) :: tol
    !! tolerance, default = 0.0
    REAL(DFP) :: ans
    !! maximum eigenvalue
  END FUNCTION SymSmallestEigenVal
END INTERFACE

• mat Dense matrix
• isFactor if mat is already factorized, then set isFactor to true. if mat is not factorized, then set isFactor to false. If isFactor is false, then this matrix will change on return. In this case, it will contain LU decomposition computed by SymGetLU. If isFactor is true, then this matrix will not change, and matrix mat should contain the LU decomposition computed from SymGetLU.
• ipiv When isFactor is true, then mat represents the LU factorization obtained by SymGetLU routine. In this case ipiv is returned by SymGetLU.
• sigma Default value is 0.0
• which = "SM" ⇨ absolute smallest eigenvalue
• which = "SA" ⇨ algebraic smallest eigenvalue
• default value of which is "SA"
• NCV Number of Lanczos vectors generated
• It must be greater than 1 and smaller than size(mat,1)
• Default value of NCV is NCV = MIN(n, 20)
• maxIter Maximum number of iteration default = N*10
• tol tolerance, default = 0.0
• ans, the smallest eigenvalue.

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