sytrd_sb2st

ssytrd_sb2st

Functions

void ssytrd_sb2st(
    const char* stage1,
    const char* vect,
    const char* uplo,
    const INT   n,
    const INT   kd,
          f32*  AB,
    const INT   ldab,
          f32*  D,
          f32*  E,
          f32*  hous,
    const INT   lhous,
          f32*  work,
    const INT   lwork,
          INT*  info
);

void ssytrd_sb2st(const char *stage1, const char *vect, const char *uplo, const INT n, const INT kd, f32 *AB, const INT ldab, f32 *D, f32 *E, f32 *hous, const INT lhous, f32 *work, const INT lwork, INT *info)

SSYTRD_SB2ST reduces a real symmetric band matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**T * A * Q = T.

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

Parameters

in

stage1

'N': “No”: to mention that the stage 1 of the reduction from dense to band using the ssytrd_sy2sb routine was not called before this routine to reproduce AB. In other term this routine is called as standalone. 'Y': “Yes”: to mention that the stage 1 of the reduction from dense to band using the ssytrd_sy2sb routine has been called to produce AB (e.g., AB is the output of ssytrd_sy2sb).

in

vect

'N': No need for the Householder representation, and thus lhous is of size max(1,4*n); 'V': the Householder representation is needed to either generate or to apply Q later on, then lhous is to be queried and computed. (NOT AVAILABLE IN THIS RELEASE).

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 matrix A if uplo='U', or the number of subdiagonals if uplo='L'. kd>=0.

inout

AB

On entry, 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). On exit, the diagonal elements of AB are overwritten by the diagonal elements of the tridiagonal matrix T; if kd>0, the elements on the first superdiagonal (if uplo='U') or the first subdiagonal (if uplo='L') are overwritten by the off-diagonal elements of T; the rest of AB is overwritten by values generated during the reduction.

in

ldab

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

out

D

Array of dimension (n). The diagonal elements of the tridiagonal matrix T.

out

E

Array of dimension (n-1). The off-diagonal elements of the tridiagonal matrix T: E[i]=T[i,i+1] if uplo='U'; E[i]=T[i+1,i] if uplo='L'.

out

hous

Array of dimension (max(1,lhous)). Stores the Householder representation.

in

lhous

The dimension of the array hous. If n=0 or kd<=1, lhous>=1, else lhous = max(1, dimension). If lwork=-1, or lhous=-1, then a query is assumed; the routine only calculates the optimal size of the hous array, returns this value as the first entry of the hous array. lhous = max(1, dimension) where dimension = 4*n if vect='N', not available now if vect='H'.

out

work

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

in

lwork

The dimension of the array work. If n=0 or kd<=1, lwork>=1, else lwork = max(1, dimension). If lwork=-1, or lhous=-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 = max(1, dimension) where dimension = (2*kd+1)*n + kd*NTHREADS, where kd is the blocking size of the reduction, FACTOPTNB is the blocking used by the QR or LQ algorithm, usually FACTOPTNB=128 is a good choice, NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1.

out

info

• info=0: successful exit

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

dsytrd_sb2st

Functions

void dsytrd_sb2st(
    const char* stage1,
    const char* vect,
    const char* uplo,
    const INT   n,
    const INT   kd,
          f64*  AB,
    const INT   ldab,
          f64*  D,
          f64*  E,
          f64*  hous,
    const INT   lhous,
          f64*  work,
    const INT   lwork,
          INT*  info
);

void dsytrd_sb2st(const char *stage1, const char *vect, const char *uplo, const INT n, const INT kd, f64 *AB, const INT ldab, f64 *D, f64 *E, f64 *hous, const INT lhous, f64 *work, const INT lwork, INT *info)

DSYTRD_SB2ST reduces a real symmetric band matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**T * A * Q = T.

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

Parameters

in

stage1

'N': “No”: to mention that the stage 1 of the reduction from dense to band using the dsytrd_sy2sb routine was not called before this routine to reproduce AB. In other term this routine is called as standalone. 'Y': “Yes”: to mention that the stage 1 of the reduction from dense to band using the dsytrd_sy2sb routine has been called to produce AB (e.g., AB is the output of dsytrd_sy2sb).

in

vect

'N': No need for the Householder representation, and thus lhous is of size max(1,4*n); 'V': the Householder representation is needed to either generate or to apply Q later on, then lhous is to be queried and computed. (NOT AVAILABLE IN THIS RELEASE).

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 matrix A if uplo='U', or the number of subdiagonals if uplo='L'. kd>=0.

inout

AB

On entry, 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). On exit, the diagonal elements of AB are overwritten by the diagonal elements of the tridiagonal matrix T; if kd>0, the elements on the first superdiagonal (if uplo='U') or the first subdiagonal (if uplo='L') are overwritten by the off-diagonal elements of T; the rest of AB is overwritten by values generated during the reduction.

in

ldab

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

out

D

Array of dimension (n). The diagonal elements of the tridiagonal matrix T.

out

E

Array of dimension (n-1). The off-diagonal elements of the tridiagonal matrix T: E[i]=T[i,i+1] if uplo='U'; E[i]=T[i+1,i] if uplo='L'.

out

hous

Array of dimension (max(1,lhous)). Stores the Householder representation.

in

lhous

The dimension of the array hous. If n=0 or kd<=1, lhous>=1, else lhous = max(1, dimension). If lwork=-1, or lhous=-1, then a query is assumed; the routine only calculates the optimal size of the hous array, returns this value as the first entry of the hous array. lhous = max(1, dimension) where dimension = 4*n if vect='N', not available now if vect='H'.

out

work

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

in

lwork

The dimension of the array work. If n=0 or kd<=1, lwork>=1, else lwork = max(1, dimension). If lwork=-1, or lhous=-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 = max(1, dimension) where dimension = (2*kd+1)*n + kd*NTHREADS, where kd is the blocking size of the reduction, FACTOPTNB is the blocking used by the QR or LQ algorithm, usually FACTOPTNB=128 is a good choice, NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1.

out

info

• info=0: successful exit

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