What does the Parameter superb in LAPACKE_dgesvd(..) mean?

Question

Putting up questions like this one raises a bad conscience... nevertheless I find it surprisingly difficult to google this one. I am experimenting with

lapack_int LAPACKE_dgesvd(
  int matrix_order, char jobu, char jobvt,
  lapack_int m, lapack_int n, double* a,
  lapack_int lda, double* s, double* u, lapack_int ldu,
  double* vt, lapack_int ldvt, double* superb);

which promises a Singular Value Decomposition. Having already stopped to fear Fortran I found a gold mine of information here: http://www.netlib.no/netlib/lapack/double/dgesvd.f

Actually that link's target explains all parameters but the LAPACKE specific double* superb (well and the order parameter, but in FORTRAN all is COL_MAJOR).

Next, here http://software.intel.com/sites/products/documentation/doclib/mkl_sa/11/mkl_lapack_examples/lapacke_dgesvd_row.c.htm I found a program which seems to hint at 'this is some kind of worker cache'.

However, if that were true what is the reason for LAPACKE_dgesvd_work(..)?

In addition I have a second question: In the example they use min(M,N)-1 as a size for superb. Why?

Answer 1

According to http://www.netlib.no/netlib/lapack/double/dgesvd.f , about parameter WORK of the fortran version :

WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK; if INFO > 0, WORK(2:MIN(M,N)) contains the unconverged superdiagonal elements of an upper bidiagonal matrix B whose diagonal is in S (not necessarily sorted). B satisfies A = U * B * VT, so it has the same singular values as A, and singular vectors related by U and VT.

There is a chance that superb is the superdiagonal of this upper bidiagonal matrix B which has the same singular values as A. This also explain the length min(n,m)-1

A look at lapack-3.5.0/lapacke/src/lapacke_dgesvd.c downloaded from http://www.netlib.org/lapack/ confirms it.

The source code also shows that the high level function lapacke_dgesvd() calls the middle level interface lapacke_dgesvd_work(). If you use the high level interface, you don't have to care about the optimal size of WORK. It will be computed and WORK will be allocated in lapacke_dgesvd()

I wonder if there is any gain to use the middle level interface instead...Maybe when this function is called many times on little matrices of same sizes...

Bye,

Francis