NAG Library Function Document

nag_real_partial_svd (f02wgc)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

1:     order – Nag_OrderTypeInput

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 $\mathbf{order}=\mathrm{Nag\_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint: $\mathbf{order}=\mathrm{Nag\_RowMajor}$ or Nag_ColMajor.

2:     m – IntegerInput

On entry: $m$, the number of rows of the matrix $A$.

Constraint: $\mathbf{m}\ge 0$.

If $\mathbf{m}=0$, an immediate return is effected.

3:     n – IntegerInput

On entry: $n$, the number of columns of the matrix $A$.

Constraint: $\mathbf{n}\ge 0$.

If $\mathbf{n}=0$, an immediate return is effected.

4:     k – IntegerInput

On entry: $k$, the number of singular values to be computed.

Constraint: $0<\mathbf{k}<\min \left(\mathbf{m},\mathbf{n}\right)-1$.

5:     ncv – IntegerInput

On entry: the dimension of the arrays sigma and resid. This is the number of Lanczos basis vectors to use during the computation of the largest eigenvalues of $A^{\mathrm{T}}A$ ($m\ge n$) or $A{A}^{\mathrm{T}}$ ($m<n$).

At present there is no a priori analysis to guide the selection of ncv relative to k. However, it is recommended that $\mathbf{ncv}\ge 2\times \mathbf{k}+1$. If many problems of the same type are to be solved, you should experiment with varying ncv while keeping k fixed for a given test problem. This will usually decrease the required number of matrix-vector operations but it also increases the internal storage required to maintain the orthogonal basis vectors. The optimal ‘cross-over’ with respect to CPU time is problem dependent and must be determined empirically.

Constraint: $\mathbf{k}<\mathbf{ncv}\le \min \left(\mathbf{m},\mathbf{n}\right)$.

6:     av – function, supplied by the userExternal Function

av must return the vector result of the matrix-vector product $Ax$ or $A^{\mathrm{T}}x$, as indicated by the input value of iflag, for the given vector $x$.

The specification of av is:

void  av (Integer *iflag, Integer m, Integer n, const double x[], double ax[], Nag_Comm *comm)

1:     iflag – Integer *Input/Output

On entry: if $\mathbf{iflag}=1$, ax must return the $m$-vector result of the matrix-vector product $Ax$.

If $\mathbf{iflag}=2$, ax must return the $n$-vector result of the matrix-vector product $A^{\mathrm{T}}x$.

On exit: may be used as a flag to indicate a failure in the computation of $Ax$ or $A^{\mathrm{T}}x$. If iflag is negative on exit from av, nag_real_partial_svd (f02wgc) will exit immediately with fail set to iflag.

2:     m – IntegerInput

On entry: the number of rows of the matrix $A$.

3:     n – IntegerInput

On entry: the number of columns of the matrix $A$.

4:     x[$\mathit{dim}$] – const doubleInput

Note: the dimension of the array x will be

• $\mathbf{n}$ when $\mathbf{iflag}=1$;
• $\mathbf{m}$ when $\mathbf{iflag}=2$.

On entry: the vector to be pre-multiplied by the matrix $A$ or $A^{\mathrm{T}}$.

5:     ax[$\mathit{dim}$] – doubleOutput

Note: the dimension of the array ax will be

• $\mathbf{m}$ when $\mathbf{iflag}=1$;
• $\mathbf{n}$ when $\mathbf{iflag}=2$.

On exit: if $\mathbf{iflag}=1$, contains the $m$-vector result of the matrix-vector product $Ax$.

If $\mathbf{iflag}=2$, contains the $n$-vector result of the matrix-vector product $A^{\mathrm{T}}x$.

6:     comm – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to av.

user – double *

iuser – Integer *

p – Pointer 

The type Pointer will be void *. Before calling nag_real_partial_svd (f02wgc) you may allocate memory and initialize these pointers with various quantities for use by av when called from nag_real_partial_svd (f02wgc) (see Section 3.2.1 in the Essential Introduction).

7:     nconv – Integer *Output

On exit: the number of converged singular values found.

8:     sigma[ncv] – doubleOutput

On exit: the nconv converged singular values are stored in the first nconv elements of sigma.

9:     u[$\mathit{dim}$] – doubleOutput

Note: the dimension, dim, of the array u must be at least

• $\max \left(1,\mathbf{pdu}\times \mathbf{ncv}\right)$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\max \left(1,\mathbf{m}\times \mathbf{pdu}\right)$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

Where $\mathbf{U}\left(i,j\right)$ appears in this document, it refers to the array element

• $\mathbf{u}\left[\left(j-1\right)\times \mathbf{pdu}+i-1\right]$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\mathbf{u}\left[\left(i-1\right)\times \mathbf{pdu}+j-1\right]$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

On exit: the left singular vectors corresponding to the singular values stored in sigma.

The $\mathit{i}$th element of the $\mathit{j}$th left singular vector $u_{\mathit{j}}$ is stored in $\mathbf{U}\left(\mathit{i},\mathit{j}\right)$, for $\mathit{i}=1,2,\dots ,m$ and $\mathit{j}=1,2,\dots ,\mathbf{nconv}$.

10:   pdu – IntegerInput

On entry: the stride used in the array u.

Constraints:

• if $\mathbf{order}=\mathrm{Nag\_ColMajor}$, $\mathbf{pdu}\ge \max \left(1,\mathbf{m}\right)$;
• if $\mathbf{order}=\mathrm{Nag\_RowMajor}$, $\mathbf{pdu}\ge \mathbf{ncv}$.

11:   v[$\mathit{dim}$] – doubleOutput

Note: the dimension, dim, of the array v must be at least

• $\max \left(1,\mathbf{pdv}\times \mathbf{ncv}\right)$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\max \left(1,\mathbf{n}\times \mathbf{pdv}\right)$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

Where $\mathbf{V}\left(i,j\right)$ appears in this document, it refers to the array element

• $\mathbf{v}\left[\left(j-1\right)\times \mathbf{pdv}+i-1\right]$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\mathbf{v}\left[\left(i-1\right)\times \mathbf{pdv}+j-1\right]$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

On exit: the right singular vectors corresponding to the singular values stored in sigma.

The $\mathit{i}$th element of the $\mathit{j}$th right singular vector $v_{\mathit{j}}$ is stored in $\mathbf{V}\left(\mathit{i},\mathit{j}\right)$, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,\mathbf{nconv}$.

12:   pdv – IntegerInput

On entry: the stride used in the array v.

Constraints:

• if $\mathbf{order}=\mathrm{Nag\_ColMajor}$, $\mathbf{pdv}\ge \max \left(1,\mathbf{n}\right)$;
• if $\mathbf{order}=\mathrm{Nag\_RowMajor}$, $\mathbf{pdv}\ge \mathbf{ncv}$.

13:   resid[ncv] – doubleOutput

On exit: the residual $‖A{v}_j-{\sigma }_j{u}_j‖$, for $m\ge n$, or $‖A^{\mathrm{T}}{u}_j-{\sigma }_j{v}_j‖$, for $m<n$, for each of the converged singular values ${\sigma }_j$ and corresponding left and right singular vectors $u_j$ and $v_j$.

14:   comm – Nag_Comm *Communication Structure

The NAG communication argument (see Section 3.2.1.1 in the Essential Introduction).

15:   fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

nag_real_partial_svd (f02wgc) returns with $\mathbf{fail}\mathbf{.}\mathbf{code}=$NE_NOERROR if at least $k$ singular values have converged and the corresponding left and right singular vectors have been computed.

6  Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.

NE_BAD_PARAM

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.

NE_EIGENVALUES

The number of eigenvalues found to sufficient accuracy is zero.

NE_INT

On entry, $\mathbf{k}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{k}>0$.

On entry, $\mathbf{m}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{m}\ge 0$.

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}\ge 0$.

On entry, $\mathbf{pdu}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdu}>0$.

On entry, $\mathbf{pdv}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdv}>0$.

NE_INT_2

On entry, $\mathbf{pdu}=〈\mathit{\text{value}}〉$ and $\mathbf{m}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdu}\ge \mathbf{m}$.

On entry, $\mathbf{pdu}=〈\mathit{\text{value}}〉$ and $\mathbf{ncv}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdu}\ge \mathbf{ncv}$.

On entry, $\mathbf{pdv}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdv}\ge \mathbf{n}$.

On entry, $\mathbf{pdv}=〈\mathit{\text{value}}〉$ and $\mathbf{ncv}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdv}\ge \mathbf{ncv}$.

NE_INT_3

On entry, $\mathbf{m}=〈\mathit{\text{value}}〉$, $\mathbf{n}=〈\mathit{\text{value}}〉$ and $\mathbf{k}=〈\mathit{\text{value}}〉$.
Constraint: $0<\mathbf{k}<\min \left(\mathbf{m},\mathbf{n}\right)-1$.

NE_INT_4

On entry, $\mathbf{k}=〈\mathit{\text{value}}〉$, $\mathbf{ncv}=〈\mathit{\text{value}}〉$, $\mathbf{m}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{k}<\mathbf{ncv}\le \min \left(\mathbf{m},\mathbf{n}\right)$.

NE_INTERNAL_ERROR

An error occurred during an internal call. Consider increasing the size of ncv relative to k.

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.

NE_LANCZOS_ITERATION

No shifts could be applied during a cycle of the implicitly restarted Lanczos iteration.

NE_MAX_ITER

The maximum number of iterations has been reached. The maximum number of iterations $=〈\mathit{\text{value}}〉$. The number of converged eigenvalues $=〈\mathit{\text{value}}〉$.

NE_NO_LANCZOS_FAC

Could not build a full Lanczos factorization.

NE_USER_STOP

On output from user-defined function av, iflag was set to a negative value, $\mathbf{iflag}=〈\mathit{\text{value}}〉$.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (f02wgce.c)

9.2  Program Data

Program Data (f02wgce.d)

9.3  Program Results

Program Results (f02wgce.r)