sytrd_sy2sb

ssytrd_sy2sb

Functions

void ssytrd_sy2sb(
    const char* uplo,
    const INT   n,
    const INT   kd,
          f32*  A,
    const INT   lda,
          f32*  AB,
    const INT   ldab,
          f32*  tau,
          f32*  work,
    const INT   lwork,
          INT*  info
);

void ssytrd_sy2sb(const char *uplo, const INT n, const INT kd, f32 *A, const INT lda, f32 *AB, const INT ldab, f32 *tau, f32 *work, const INT lwork, INT *info)

SSYTRD_SY2SB reduces a real symmetric matrix A to real symmetric band-diagonal form AB by an orthogonal similarity transformation: Q**T * A * Q = AB.

Further Details:

Implemented by Azzam Haidar.

All details are available on technical report, SC11, SC13 papers.

Azzam Haidar, Hatem Ltaief, and Jack Dongarra. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC ‘11), New York, NY, USA, Article 8 , 11 pages. https://doi.org/10.1145/2063384.2063394

A. Haidar, J. Kurzak, P. Luszczek, 2013. An improved parallel singular value algorithm and its implementation for multicore hardware, In Proceedings of 2013 International Conference for High Performance Computing, Networking, Storage and Analysis (SC ‘13). Denver, Colorado, USA, 2013. Article 90, 12 pages. https://doi.org/10.1145/2503210.2503292

A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. A novel hybrid CPU-GPU generalized eigensolver for electronic structure calculations based on fine-grained memory aware tasks. International Journal of High Performance Computing Applications. Volume 28 Issue 2, Pages 196-209, May 2014. https://doi.org/10.1177/1094342013502097

If uplo = ‘U’, the matrix Q is represented as a product of elementary reflectors

Q = H(k-1)**T . . . H(1)**T H(0)**T, where k = n-kd.

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with
v[0:i+kd-1] = 0 and v[i+kd] = 1; conjg(v[i+kd+1:n-1]) is stored on
exit in A[i, i+kd+1:n-1], and tau in tau[i].

If uplo = ‘L’, the matrix Q is represented as a product of elementary reflectors

Q = H(0) H(1) . . . H(k-1), where k = n-kd.

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with
v[kd:i] = 0 and v[i+kd+1] = 1; v[i+kd+2:n-1] is stored on exit in
A[i+kd+2:n-1, i], and tau in tau[i].

The contents of A on exit are illustrated by the following examples with n = 5:

if UPLO = 'U':                       if UPLO = 'L':

  (  ab  ab/v0  v0     v0     v0    )  (  ab                            )
  (      ab     ab/v1  v1     v1    )  (  ab/v0  ab                     )
  (             ab     ab/v2  v2    )  (  v0     ab/v1  ab              )
  (                    ab     ab/v3 )  (  v0     v1     ab/v2  ab       )
  (                           ab    )  (  v0     v1     v2     ab/v3 ab )

where d and e denote diagonal and off-diagonal elements of T, and vi
denotes an element of the vector defining H(i).

Parameters

in

uplo

'U': Upper triangle of A is stored; 'L': Lower triangle of A is stored.

in

n

The order of the matrix A. n>=0.

in

kd

The number of superdiagonals of the reduced matrix if uplo='U', or the number of subdiagonals if uplo='L'. kd>=0. The reduced matrix is stored in the array AB.

inout

A

On entry, the symmetric matrix A. If uplo='U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If uplo='L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, if uplo='U', the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array tau, represent the orthogonal matrix Q as a product of elementary reflectors; if uplo='L', the diagonal and first subdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array tau, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details.

in

lda

The leading dimension of the array A. lda>=max(1,n).

out

AB

On exit, the upper or lower triangle of the symmetric band matrix A, stored in the first kd+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if uplo='U', AB[kd+i-j,j]=A[i,j] for max(0,j-kd)<=i<=j; if uplo='L', AB[i-j,j]=A[i,j] for j<=i<=min(n-1,j+kd).

in

ldab

The leading dimension of the array AB. ldab>=kd+1.

out

tau

Array of dimension (n-kd). The scalar factors of the elementary reflectors (see Further Details).

out

work

Array of dimension (max(1,lwork)). On exit, if info=0, or if lwork=-1, work[0] returns the size of lwork.

in

lwork

The dimension of the array work which should be calculated by a workspace query. If n<=kd+1, lwork>=1, else lwork = max(1, LWORK_QUERY). If lwork=-1, then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array. LWORK_QUERY = n*kd + n*max(kd,FACTOPTNB) + 2*kd*kd where FACTOPTNB is the blocking used by the QR or LQ algorithm, usually FACTOPTNB=128 is a good choice otherwise putting lwork=-1 will provide the size of work.

out

info

• info=0: successful exit

• info<0: if info=-i, the i-th argument had an illegal value

dsytrd_sy2sb

Functions

void dsytrd_sy2sb(
    const char* uplo,
    const INT   n,
    const INT   kd,
          f64*  A,
    const INT   lda,
          f64*  AB,
    const INT   ldab,
          f64*  tau,
          f64*  work,
    const INT   lwork,
          INT*  info
);

void dsytrd_sy2sb(const char *uplo, const INT n, const INT kd, f64 *A, const INT lda, f64 *AB, const INT ldab, f64 *tau, f64 *work, const INT lwork, INT *info)

DSYTRD_SY2SB reduces a real symmetric matrix A to real symmetric band-diagonal form AB by an orthogonal similarity transformation: Q**T * A * Q = AB.

Further Details:

Implemented by Azzam Haidar.

All details are available on technical report, SC11, SC13 papers.

Azzam Haidar, Hatem Ltaief, and Jack Dongarra. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC ‘11), New York, NY, USA, Article 8 , 11 pages. https://doi.org/10.1145/2063384.2063394

A. Haidar, J. Kurzak, P. Luszczek, 2013. An improved parallel singular value algorithm and its implementation for multicore hardware, In Proceedings of 2013 International Conference for High Performance Computing, Networking, Storage and Analysis (SC ‘13). Denver, Colorado, USA, 2013. Article 90, 12 pages. https://doi.org/10.1145/2503210.2503292

A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. A novel hybrid CPU-GPU generalized eigensolver for electronic structure calculations based on fine-grained memory aware tasks. International Journal of High Performance Computing Applications. Volume 28 Issue 2, Pages 196-209, May 2014. https://doi.org/10.1177/1094342013502097

If uplo = ‘U’, the matrix Q is represented as a product of elementary reflectors

Q = H(k-1)**T . . . H(1)**T H(0)**T, where k = n-kd.

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with
v[0:i+kd-1] = 0 and v[i+kd] = 1; conjg(v[i+kd+1:n-1]) is stored on
exit in A[i, i+kd+1:n-1], and tau in tau[i].

If uplo = ‘L’, the matrix Q is represented as a product of elementary reflectors

Q = H(0) H(1) . . . H(k-1), where k = n-kd.

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with
v[kd:i] = 0 and v[i+kd+1] = 1; v[i+kd+2:n-1] is stored on exit in
A[i+kd+2:n-1, i], and tau in tau[i].

The contents of A on exit are illustrated by the following examples with n = 5:

if UPLO = 'U':                       if UPLO = 'L':

  (  ab  ab/v0  v0     v0     v0    )  (  ab                            )
  (      ab     ab/v1  v1     v1    )  (  ab/v0  ab                     )
  (             ab     ab/v2  v2    )  (  v0     ab/v1  ab              )
  (                    ab     ab/v3 )  (  v0     v1     ab/v2  ab       )
  (                           ab    )  (  v0     v1     v2     ab/v3 ab )

where d and e denote diagonal and off-diagonal elements of T, and vi
denotes an element of the vector defining H(i).

Parameters

in

uplo

'U': Upper triangle of A is stored; 'L': Lower triangle of A is stored.

in

n

The order of the matrix A. n>=0.

in

kd

The number of superdiagonals of the reduced matrix if uplo='U', or the number of subdiagonals if uplo='L'. kd>=0. The reduced matrix is stored in the array AB.

inout

A

On entry, the symmetric matrix A. If uplo='U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If uplo='L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, if uplo='U', the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array tau, represent the orthogonal matrix Q as a product of elementary reflectors; if uplo='L', the diagonal and first subdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array tau, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details.

in

lda

The leading dimension of the array A. lda>=max(1,n).

out

AB

On exit, the upper or lower triangle of the symmetric band matrix A, stored in the first kd+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if uplo='U', AB[kd+i-j,j]=A[i,j] for max(0,j-kd)<=i<=j; if uplo='L', AB[i-j,j]=A[i,j] for j<=i<=min(n-1,j+kd).

in

ldab

The leading dimension of the array AB. ldab>=kd+1.

out

tau

Array of dimension (n-kd). The scalar factors of the elementary reflectors (see Further Details).

out

work

Array of dimension (max(1,lwork)). On exit, if info=0, or if lwork=-1, work[0] returns the size of lwork.

in

lwork

The dimension of the array work which should be calculated by a workspace query. If n<=kd+1, lwork>=1, else lwork = max(1, LWORK_QUERY). If lwork=-1, then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array. LWORK_QUERY = n*kd + n*max(kd,FACTOPTNB) + 2*kd*kd where FACTOPTNB is the blocking used by the QR or LQ algorithm, usually FACTOPTNB=128 is a good choice otherwise putting lwork=-1 will provide the size of work.

out

info

• info=0: successful exit

• info<0: if info=-i, the i-th argument had an illegal value