NAG CL Interface
f08kbc (dgesvd)

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1 Purpose

f08kbc computes the singular value decomposition (SVD) of a real m×n matrix A, optionally computing the left and/or right singular vectors.

2 Specification

#include <nag.h>
void  f08kbc (Nag_OrderType order, Nag_ComputeUType jobu, Nag_ComputeVTType jobvt, Integer m, Integer n, double a[], Integer pda, double s[], double u[], Integer pdu, double vt[], Integer pdvt, double work[], NagError *fail)
The function may be called by the names: f08kbc, nag_lapackeig_dgesvd or nag_dgesvd.

3 Description

The SVD is written as
A = UΣVT ,  
where Σ is an m×n matrix which is zero except for its min(m,n) diagonal elements, U is an m×m orthogonal matrix, and V is an n×n orthogonal matrix. The diagonal elements of Σ are the singular values of A; they are real and non-negative, and are returned in descending order. The first min(m,n) columns of U and V are the left and right singular vectors of A.
Note that the function returns VT, not V.

4 References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: order Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: jobu Nag_ComputeUType Input
On entry: specifies options for computing all or part of the matrix U.
jobu=Nag_AllU
All m columns of U are returned in array u.
jobu=Nag_SingularVecsU
The first min(m,n) columns of U (the left singular vectors) are returned in the array u.
jobu=Nag_Overwrite
The first min(m,n) columns of U (the left singular vectors) are overwritten on the array a.
jobu=Nag_NotU
No columns of U (no left singular vectors) are computed.
Constraint: jobu=Nag_AllU, Nag_SingularVecsU, Nag_Overwrite or Nag_NotU.
3: jobvt Nag_ComputeVTType Input
On entry: specifies options for computing all or part of the matrix VT.
jobvt=Nag_AllVT
All n rows of VT are returned in the array vt.
jobvt=Nag_SingularVecsVT
The first min(m,n) rows of VT (the right singular vectors) are returned in the array vt.
jobvt=Nag_OverwriteVT
The first min(m,n) rows of VT (the right singular vectors) are overwritten on the array a.
jobvt=Nag_NotVT
No rows of VT (no right singular vectors) are computed.
Constraints:
  • jobvt=Nag_AllVT, Nag_SingularVecsVT, Nag_OverwriteVT or Nag_NotVT;
  • If jobu=Nag_Overwrite, jobvt cannot be Nag_OverwriteVT.
4: m Integer Input
On entry: m, the number of rows of the matrix A.
Constraint: m0.
5: n Integer Input
On entry: n, the number of columns of the matrix A.
Constraint: n0.
6: a[dim] double Input/Output
Note: the dimension, dim, of the array a must be at least
  • max(1,pda×n) when order=Nag_ColMajor;
  • max(1,m×pda) when order=Nag_RowMajor.
The (i,j)th element of the matrix A is stored in
  • a[(j-1)×pda+i-1] when order=Nag_ColMajor;
  • a[(i-1)×pda+j-1] when order=Nag_RowMajor.
On entry: the m×n matrix A.
On exit: if jobu=Nag_Overwrite, a is overwritten with the first min(m,n) columns of U (the left singular vectors, stored column-wise).
If jobvt=Nag_OverwriteVT, a is overwritten with the first min(m,n) rows of VT (the right singular vectors, stored row-wise).
If jobuNag_Overwrite and jobvtNag_OverwriteVT, the contents of a are destroyed.
7: pda Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
  • if order=Nag_ColMajor, pdamax(1,m);
  • if order=Nag_RowMajor, pdamax(1,n).
8: s[dim] double Output
Note: the dimension, dim, of the array s must be at least max(1,min(m,n)) .
On exit: the singular values of A, sorted so that s[i-1]s[i].
9: u[dim] double Output
Note: the dimension, dim, of the array u must be at least
  • max(1,pdu×m) when jobu=Nag_AllU;
  • max(1,pdu×min(m,n)) when jobu=Nag_SingularVecsU and order=Nag_ColMajor;
  • max(1,m×pdu) when jobu=Nag_SingularVecsU and order=Nag_RowMajor;
  • max(1,m) otherwise.
The (i,j)th element of the matrix U is stored in
  • u[(j-1)×pdu+i-1] when order=Nag_ColMajor;
  • u[(i-1)×pdu+j-1] when order=Nag_RowMajor.
On exit: if jobu=Nag_AllU, u contains the m×m orthogonal matrix U.
If jobu=Nag_SingularVecsU, u contains the first min(m,n) columns of U (the left singular vectors, stored column-wise).
If jobu=Nag_NotU or Nag_Overwrite, u is not referenced.
10: pdu Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array u.
Constraints:
  • if order=Nag_ColMajor,
    • if jobu=Nag_AllU, pdu max(1,m) ;
    • if jobu=Nag_SingularVecsU, pdu max(1,m) ;
    • otherwise pdu1;
  • if order=Nag_RowMajor,
    • if jobu=Nag_AllU, pdumax(1,m);
    • if jobu=Nag_SingularVecsU, pdumax(1,min(m,n));
    • otherwise pdu1.
11: vt[dim] double Output
Note: the dimension, dim, of the array vt must be at least
  • max(1,pdvt×n) when jobvt=Nag_AllVT;
  • max(1,pdvt×n) when jobvt=Nag_SingularVecsVT and order=Nag_ColMajor;
  • max(1,min(m,n)×pdvt) when jobvt=Nag_SingularVecsVT and order=Nag_RowMajor;
  • max(1,min(m,n)) otherwise.
The (i,j)th element of the matrix is stored in
  • vt[(j-1)×pdvt+i-1] when order=Nag_ColMajor;
  • vt[(i-1)×pdvt+j-1] when order=Nag_RowMajor.
On exit: if jobvt=Nag_AllVT, vt contains the n×n orthogonal matrix VT.
If jobvt=Nag_SingularVecsVT, vt contains the first min(m,n) rows of VT (the right singular vectors, stored row-wise).
If jobvt=Nag_NotVT or Nag_OverwriteVT, vt is not referenced.
12: pdvt Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array vt.
Constraints:
  • if order=Nag_ColMajor,
    • if jobvt=Nag_AllVT, pdvt max(1,n) ;
    • if jobvt=Nag_SingularVecsVT, pdvt max(1,min(m,n)) ;
    • otherwise pdvt1;
  • if order=Nag_RowMajor,
    • if jobvt=Nag_AllVT, pdvtmax(1,n);
    • if jobvt=Nag_SingularVecsVT, pdvtmax(1,n);
    • otherwise pdvt1.
13: work[min(m,n)] double Output
On exit: if fail.code= NE_CONVERGENCE, WORK(2:min(m,n)) (using the notation described in Section 3.1.4 in the Introduction to the NAG Library CL Interface) contains the unconverged superdiagonal elements of an upper bidiagonal matrix B whose diagonal is in s (not necessarily sorted). B satisfies A=UBVT, so it has the same singular values as A, and singular vectors related by U and VT.
14: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_CONVERGENCE
If f08kbc did not converge, fail.errnum specifies how many superdiagonals of an intermediate bidiagonal form did not converge to zero.
NE_ENUM_INT_2
On entry, jobu=value, pdu=value and m=value.
Constraint: if jobu=Nag_AllU, pdu max(1,m) ;
if jobu=Nag_SingularVecsU, pdu max(1,m) ;
otherwise pdu1.
On entry, jobvt=value, pdvt=value and n=value.
Constraint: if jobvt=Nag_AllVT, pdvtmax(1,n);
if jobvt=Nag_SingularVecsVT, pdvtmax(1,n);
otherwise pdvt1.
NE_ENUM_INT_3
On entry, jobu=value, pdu=value, m=value and n=value.
Constraint: if jobu=Nag_AllU, pdumax(1,m);
if jobu=Nag_SingularVecsU, pdumax(1,min(m,n));
otherwise pdu1.
On entry, jobvt=value, pdvt=value, m=value and n=value.
Constraint: if jobvt=Nag_AllVT, pdvt max(1,n) ;
if jobvt=Nag_SingularVecsVT, pdvt max(1,min(m,n)) ;
otherwise pdvt1.
NE_INT
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdu=value.
Constraint: pdu>0.
On entry, pdvt=value.
Constraint: pdvt>0.
NE_INT_2
On entry, pda=value and m=value.
Constraint: pdamax(1,m).
On entry, pda=value and n=value.
Constraint: pdamax(1,n).
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 contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

The computed singular value decomposition is nearly the exact singular value decomposition for a nearby matrix (A+E) , where
E2 = O(ε) A2 ,  
and ε is the machine precision. In addition, the computed singular vectors are nearly orthogonal to working precision. See Section 4.9 of Anderson et al. (1999) for further details.

8 Parallelism and Performance

f08kbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08kbc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The total number of floating-point operations is approximately proportional to mn2 when m>n and m2n otherwise.
The singular values are returned in descending order.
The complex analogue of this function is f08kpc.

10 Example

This example finds the singular values and left and right singular vectors of the 6×4 matrix
A = ( 2.27 -1.54 1.15 -1.94 0.28 -1.67 0.94 -0.78 -0.48 -3.09 0.99 -0.21 1.07 1.22 0.79 0.63 -2.35 2.93 -1.45 2.30 0.62 -7.39 1.03 -2.57 ) ,  
together with approximate error bounds for the computed singular values and vectors.
The example program for f08kdc illustrates finding a singular value decomposition for the case mn.

10.1 Program Text

Program Text (f08kbce.c)

10.2 Program Data

Program Data (f08kbce.d)

10.3 Program Results

Program Results (f08kbce.r)