# p?sygs2/p?hegs2

Reduces a symmetric/Hermitian positive-definite generalized eigenproblem to standard form, using the factorization results obtained from p?potrf (local unblocked algorithm).

## Syntax

call pssygs2(ibtype, uplo, n, a, ia, ja, desca, b, ib, jb, descb, info)

call pdsygs2(ibtype, uplo, n, a, ia, ja, desca, b, ib, jb, descb, info)

call pchegs2(ibtype, uplo, n, a, ia, ja, desca, b, ib, jb, descb, info)

call pzhegs2(ibtype, uplo, n, a, ia, ja, desca, b, ib, jb, descb, info)

## Description

The p?sygs2/p?hegs2 routine reduces a real symmetric-definite or a complex Hermitian positive-definite generalized eigenproblem to standard form.

Here `sub(A)` denotes `A(ia:ia+n-1`, `ja:ja+n-1)`, and `sub(B)` denotes `B(ib:ib+n-1`, `jb:jb+n-1)`.

If `ibtype = 1`, the problem is

`sub(A)*x = λ*sub(B)*x`

and `sub(A)` is overwritten by

`inv(UT)*sub(A)*inv(U)` or `inv(L)*sub(A)*inv(LT)` - for real flavors, and

`inv(UH)*sub(A)*inv(U)` or `inv(L)*sub(A)*inv(LH)` - for complex flavors.

If `ibtype = 2` or `3`, the problem is

``sub(A)*sub(B)x = λ*x` or sub(B)*sub(A)x =λ*x`

and `sub(A)` is overwritten by

`U*sub(A)*UT` or `L**T*sub(A)*L` - for real flavors and

`U*sub(A)*UH` or `L**H*sub(A)*L` - for complex flavors.

The matrix `sub(B)` must have been previously factorized as `UT*U` or L*LT (for real flavors), or as `UH*U` or L*LH (for complex flavors) by p?potrf.

## Input Parameters

ibtype

(global) INTEGER.

= 1:

compute `inv(UT)*sub(A)*inv(U)`, or `inv(L)*sub(A)*inv(LT)` for real subroutines,

and `inv(UH)*sub(A)*inv(U)`, or `inv(L)*sub(A)*inv(LH)` for complex subroutines;

= 2 or 3:

compute `U*sub(A)*UT`, or `LT*sub(A)*L` for real subroutines,

and `U*sub(A)*UH` or `LH*sub(A)*L` for complex subroutines.

uplo

(global) CHARACTER

Specifies whether the upper or lower triangular part of the symmetric/Hermitian matrix sub(A) is stored, and how sub(B) is factorized.

= 'U': Upper triangular of sub(A) is stored and sub(B) is factorized as UT*U (for real subroutines) or as UH*U (for complex subroutines).

= 'L': Lower triangular of sub(A) is stored and sub(B) is factorized as L*LT (for real subroutines) or as L*LH (for complex subroutines)

n

(global) INTEGER.

The order of the matrices sub(A) and sub(B). `n ≥ 0`.

a

(local)

REAL for pssygs2

DOUBLE PRECISION for pdsygs2

COMPLEX for pchegs2

COMPLEX*16 for pzhegs2.

Pointer into the local memory to an array of size `(lld_a, LOCc(ja+n-1))`.

On entry, this array contains the local pieces of the n-by-n symmetric/Hermitian distributed matrix sub(A).

If `uplo = 'U'`, the leading n-by-n upper triangular part of sub(A) contains the upper triangular part of the matrix, and the strictly lower triangular part of sub(A) is not referenced.

If `uplo = 'L'`, the leading n-by-n lower triangular part of sub(A) contains the lower triangular part of the matrix, and the strictly upper triangular part of sub(A) is not referenced.

ia, ja

(global) INTEGER.

The row and column indices in the global matrix A indicating the first row and the first column of the sub(A), respectively.

desca

(global and local) INTEGER array of size dlen_. The array descriptor for the distributed matrix A.

B

(local)

REAL for pssygs2

DOUBLE PRECISION for pdsygs2

COMPLEX for pchegs2

COMPLEX*16 for pzhegs2.

Pointer into the local memory to an array of size `(lld_b, LOCc(jb+n-1))`.

On entry, this array contains the local pieces of the triangular factor from the Cholesky factorization of sub(B) as returned by p?potrf.

ib, jb

(global) INTEGER.

The row and column indices in the global matrix B indicating the first row and the first column of the sub(B), respectively.

descb

(global and local) INTEGER array of size dlen_. The array descriptor for the distributed matrix B.

## Output Parameters

a

(local)

On exit, if `info = 0`, the transformed matrix is stored in the same format as sub(A).

info

INTEGER.

= 0: successful exit.

< 0: if the i-th argument is an array and the j-th entry had an illegal value,

then info = - (i*100+ j),

if the i-th argument is a scalar and had an illegal value,

then info = -i.