nag_dbdsqr (f08mec) computes the singular value decomposition of a real upper or lower bidiagonal matrix, or of a real general matrix which has been reduced to bidiagonal form.

nag_dbdsqr (f08mec) computes the singular values and, optionally, the left or right singular vectors of a real upper or lower bidiagonal matrix . In other words, it can compute the singular value decomposition (SVD) of  as

Here  is a diagonal matrix with real diagonal elements  (the singular values of ), such that

  is an orthogonal matrix whose columns are the left singular vectors ;  is an orthogonal matrix whose rows are the right singular vectors . Thus

To compute  and/or , the arrays u and/or vt must be initialized to the unit matrix before nag_dbdsqr (f08mec) is called.

The function may also be used to compute the SVD of a real general matrix  which has been reduced to bidiagonal form by an orthogonal transformation: . If  is  by  with , then  is  by  and  is  by ; if  is  by  with , then  is  by  and  is  by . In this case, the matrices  and/or  must be formed explicitly by f08kfc (nag_dorgbr) and passed to nag_dbdsqr (f08mec) in the arrays u and/or vt respectively.

nag_dbdsqr (f08mec) also has the capability of forming , where  is an arbitrary real matrix; this is needed when using the SVD to solve linear least-squares problems.

nag_dbdsqr (f08mec) uses two different algorithms. If any singular vectors are required (i.e., if  or  or ), the bidiagonal  algorithm is used, switching between zero-shift and implicitly shifted forms to preserve the accuracy of small singular values, and switching between  and  variants in order to handle graded matrices effectively (see Demmel and Kahan (1990)). If only singular values are required (that is, if ), they are computed by the differential qd algorithm (see Fernando and Parlett (1994)), which is faster and can achieve even greater accuracy.

The singular vectors are normalized so that , but are determined only to within a factor .

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_BAD_PARAM"

On entry, argument  had an illegal value.

"NE_CONVERGENCE"

   off-diagonals did not converge. The arrays d and e contain the diagonal and off-diagonal elements, respectively, of a bidiagonal matrix orthogonally equivalent to .

"NE_INT"

On entry, . Constraint: .

On entry, . Constraint: .

On entry, . Constraint: .

On entry, . Constraint: .

"NE_INTERNAL_ERROR"

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please consult NAG for assistance.

Each singular value and singular vector is computed to high relative accuracy. However, the reduction to bidiagonal form (prior to calling the function) may exclude the possibility of obtaining high relative accuracy in the small singular values of the original matrix if its singular values vary widely in magnitude.

If  is an exact singular value of  and  is the corresponding computed value, then

where  is a modestly increasing function of  and , and  is the machine precision. If only singular values are computed, they are computed more accurately (i.e., the function  is smaller), than when some singular vectors are also computed.

If  is the corresponding exact left singular vector of , and  is the corresponding computed left singular vector, then the angle  between them is bounded as follows:

where  is the relative gap between  and the other singular values, defined by

A similar error bound holds for the right singular vectors.

The total number of floating-point operations is roughly proportional to  if only the singular values are computed. About  additional operations are required to compute the left singular vectors and about  to compute the right singular vectors. The operations to compute the singular values must all be performed in scalar mode; the additional operations to compute the singular vectors can be vectorized and on some machines may be performed much faster.

The complex analogue of this function is f08msc (nag_zbdsqr).