p?ggqrf
p?ggqrf
Computes the generalized QR factorization.
Syntax
void
psggqrf
(
MKL_INT
*n
,
MKL_INT
*m
,
MKL_INT
*p
,
float
*a
,
MKL_INT
*ia
,
MKL_INT
*ja
,
MKL_INT
*desca
,
float
*taua
,
float
*b
,
MKL_INT
*ib
,
MKL_INT
*jb
,
MKL_INT
*descb
,
float
*taub
,
float
*work
,
MKL_INT
*lwork
,
MKL_INT
*info
);
void
pdggqrf
(
MKL_INT
*n
,
MKL_INT
*m
,
MKL_INT
*p
,
double
*a
,
MKL_INT
*ia
,
MKL_INT
*ja
,
MKL_INT
*desca
,
double
*taua
,
double
*b
,
MKL_INT
*ib
,
MKL_INT
*jb
,
MKL_INT
*descb
,
double
*taub
,
double
*work
,
MKL_INT
*lwork
,
MKL_INT
*info
);
void
pcggqrf
(
MKL_INT
*n
,
MKL_INT
*m
,
MKL_INT
*p
,
MKL_Complex8
*a
,
MKL_INT
*ia
,
MKL_INT
*ja
,
MKL_INT
*desca
,
MKL_Complex8
*taua
,
MKL_Complex8
*b
,
MKL_INT
*ib
,
MKL_INT
*jb
,
MKL_INT
*descb
,
MKL_Complex8
*taub
,
MKL_Complex8
*work
,
MKL_INT
*lwork
,
MKL_INT
*info
);
void
pzggqrf
(
MKL_INT
*n
,
MKL_INT
*m
,
MKL_INT
*p
,
MKL_Complex16
*a
,
MKL_INT
*ia
,
MKL_INT
*ja
,
MKL_INT
*desca
,
MKL_Complex16
*taua
,
MKL_Complex16
*b
,
MKL_INT
*ib
,
MKL_INT
*jb
,
MKL_INT
*descb
,
MKL_Complex16
*taub
,
MKL_Complex16
*work
,
MKL_INT
*lwork
,
MKL_INT
*info
);
Include Files
- mkl_scalapack.h
Description
The
p?ggqrf
function
forms the generalized Q
R
factorization of an n
-by-m
matrix sub(
A
) = A
(ia
:ia
+n
-1, ja
:ja
+m
-1)and an
n
-by-p
matrix sub(
B
) = B
(ib
:ib
+n
-1, jb
:jb
+p
-1):as
sub(A) =
Q
*R
, sub(B
) = Q
*T
*Z
,where
Q
is an n
-by-n
orthogonal/unitary matrix, Z
is a p
-by-p
orthogonal/unitary matrix, and R
and T
assume one of the forms:If
n
≥ m

or if
n
< m

where is upper triangular, and
R
11


where or is an upper triangular matrix.
T
12
T
21
In particular, if sub(
B
) is square and nonsingular, the GQR
factorization of sub(A
) and sub(B
) implicitly gives the Q
R
factorization of inv (sub(B
))* sub (A
): inv(sub(*(inv(
B
))*sub(A
) = Z
H
T)
*R
)Input Parameters
- n
- (global) The number of rows in the distributed matrices sub (A) and sub(B)(.n≥0)
- m
- (global) The number of columns in the distributed matrix sub(A)(.m≥0)
- p
- The number of columns in the distributed matrix sub(B)(.p≥0)
- a
- (local)Pointer into the local memory to an array of size. Contains the local pieces of thelld_a*LOCc(ja+m-1)n-by-mmatrix sub(A) to be factored.
- ia,ja
- (global) The row and column indices in the global matrixAindicating the first row and the first column of the submatrixA, respectively.
- desca
- (global and local) array of sizedlen_. The array descriptor for the distributed matrixA.
- b
- (local)Pointer into the local memory to an array of size. Contains the local pieces of thelld_b*LOCc(jb+p-1)n-by-pmatrix sub(B) to be factored.
- ib,jb
- (global) The row and column indices in the global matrixBindicating the first row and the first column of the submatrixB, respectively.
- descb
- (global and local) array of sizedlen_. The array descriptor for the distributed matrix B.
- work
- (local)Workspace array of size oflwork.
- lwork
- (local or global) Sze ofwork, must be at least,lwork≥max(nb_a*(npa0+mqa0+nb_a),max((nb_a*(nb_a-1))/2, (pqb0+npb0)*nb_a)+nb_a*nb_a,mb_b*(npb0+pqb0+mb_b))where,iroffa=mod(ia-1,mb_A),icoffa=mod(ja-1,nb_a),iarow=indxg2p(ia,mb_a,MYROW,rsrc_a,NPROW),iacol=indxg2p(ja,nb_a,MYCOL,csrc_a,NPCOL),npa0=numroc(n+iroffa,mb_a,MYROW,iarow,NPROW)mqa0=numroc(m+icoffa,nb_a,MYCOL,iacol,NPCOL),iroffb=mod(ib-1,mb_b),icoffb=mod(jb-1,nb_b),ibrow=indxg2p(ib,mb_b,MYROW,rsrc_b,NPROW),ibcol=indxg2p(jb,nb_b,MYCOL,csrc_b,NPCOL),npb0=numroc(n+iroffa,mb_b,MYROW,Ibrow,NPROW)pqb0=numroc(m+icoffb,nb_b,MYCOL,ibcol,NPCOL)mod(is the integer remainder ofx,y).x/yandnumroc,indxg2pare ScaLAPACK tool functions;MYROW,MYCOL,NPROWandNPCOLcan be determined by calling thefunctionblacs_gridinfo.If, thenlwork= -1lworkis global input and a workspace query is assumed; thefunctiononly calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by pxerbla.
Output Parameters
- a
- On exit, the elements on and above the diagonal of sub (A) contain the min(n,m)-by-mupper trapezoidal matrixR(Ris upper triangular ifn≥m); the elements below the diagonal, with the arraytaua, represent the orthogonal/unitary matrixQas a product of min(n,m) elementary reflectors. (See Application Notes below).
- taua,taub
- (local)Arrays of sizeforLOCc(ja+min(n,m)-1)tauaandforLOCr(ib+n-1)taub.The arraytauacontains the scalar factors of the elementary reflectors which represent the orthogonal/unitary matrixQ.tauais tied to the distributed matrixA. (See Application Notes below).The arraytaubcontains the scalar factors of the elementary reflectors which represent the orthogonal/unitary matrixZ.taubis tied to the distributed matrixB.(See Application Notes below).
- work[0]
- On exitcontains the minimum value ofwork[0]lworkrequired for optimum performance.
- info
- (global)= 0: the execution is successful.< 0: if thei-th argument is an array and thej-th entry, indexedhad an illegal value, thenj- 1,info= -(i*100+j); if thei-th argument is a scalar and had an illegal value, theninfo=-i.
Application Notes
The matrix
Q
is represented as a product of elementary reflectors Q
= H
(ja
)*H
(ja
+1)*...*H
(ja
+k
-1),where
k
= min(
.n
,m
)Each
H
(i
) has the form H
(i
) = i
- taua
*v
*v'
where
taua
is a real/complex scalar, and v
is a real/complex vector with v
(1:i
-1) = 0 and v
(i
) = 1; v(i
+1:n
) is stored on exit in A
(ia
+i
:ia
+n
-1, ja
+i
-1) , and taua
in taua
[
.To form ja
+i
-2]Q
explicitly, use ScaLAPACK function
p?orgqr
/p?ungqr
. To use Q
to update another matrix, use ScaLAPACK function
p?ormqr
/p?unmqr
.The matrix
Z
is represented as a product of elementary reflectorsZ
= H
(ib
)*H
(ib
+1)*...*H
(ib
+k
-1), where k
= min(
.n
,p
)Each
H
(i
) has the form H
(i
) = i
- taub
*v
*v'
where
taub
is a real/complex scalar, and v
is a real/complex vector with v
(p
-k
+i
+1:p
) = 0 and v
(p
-k
+i
) = 1; v
(1:p
-k
+i
-1) is stored on exit in B
(ib
+n
-k
+i
-1,jb
:jb
+p
-k
+i
-2), and taub
in taub
[
. To form ib
+n
-k
+i
-2]Z
explicitly, use ScaLAPACK function
p?orgrq
/p?ungrq
. To use Z
to update another matrix, use ScaLAPACK function
p?ormrq
/p?unmrq
.