.TH "geqr" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
geqr \- geqr: QR factor, flexible
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "subroutine \fBcgeqr\fP (m, n, a, lda, t, tsize, work, lwork, info)"
.br
.RI "\fBCGEQR\fP "
.ti -1c
.RI "subroutine \fBdgeqr\fP (m, n, a, lda, t, tsize, work, lwork, info)"
.br
.RI "\fBDGEQR\fP "
.ti -1c
.RI "subroutine \fBsgeqr\fP (m, n, a, lda, t, tsize, work, lwork, info)"
.br
.RI "\fBSGEQR\fP "
.ti -1c
.RI "subroutine \fBzgeqr\fP (m, n, a, lda, t, tsize, work, lwork, info)"
.br
.RI "\fBZGEQR\fP "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "subroutine cgeqr (integer m, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) t, integer tsize, complex, dimension( * ) work, integer lwork, integer info)"

.PP
\fBCGEQR\fP 
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 CGEQR computes a QR factorization of a complex M-by-N matrix A:

    A = Q * ( R ),
            ( 0 )

 where:

    Q is a M-by-M orthogonal matrix;
    R is an upper-triangular N-by-N matrix;
    0 is a (M-N)-by-N zero matrix, if M > N\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIM\fP 
.PP
.nf
          M is INTEGER
          The number of rows of the matrix A\&.  M >= 0\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The number of columns of the matrix A\&.  N >= 0\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is COMPLEX array, dimension (LDA,N)
          On entry, the M-by-N matrix A\&.
          On exit, the elements on and above the diagonal of the array
          contain the min(M,N)-by-N upper trapezoidal matrix R
          (R is upper triangular if M >= N);
          the elements below the diagonal are used to store part of the 
          data structure to represent Q\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max(1,M)\&.
.fi
.PP
.br
\fIT\fP 
.PP
.nf
          T is COMPLEX array, dimension (MAX(5,TSIZE))
          On exit, if INFO = 0, T(1) returns optimal (or either minimal 
          or optimal, if query is assumed) TSIZE\&. See TSIZE for details\&.
          Remaining T contains part of the data structure used to represent Q\&.
          If one wants to apply or construct Q, then one needs to keep T 
          (in addition to A) and pass it to further subroutines\&.
.fi
.PP
.br
\fITSIZE\fP 
.PP
.nf
          TSIZE is INTEGER
          If TSIZE >= 5, the dimension of the array T\&.
          If TSIZE = -1 or -2, then a workspace query is assumed\&. The routine
          only calculates the sizes of the T and WORK arrays, returns these
          values as the first entries of the T and WORK arrays, and no error
          message related to T or WORK is issued by XERBLA\&.
          If TSIZE = -1, the routine calculates optimal size of T for the 
          optimum performance and returns this value in T(1)\&.
          If TSIZE = -2, the routine calculates minimal size of T and 
          returns this value in T(1)\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          (workspace) COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) contains optimal (or either minimal
          or optimal, if query was assumed) LWORK\&.
          See LWORK for details\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          LWORK is INTEGER
          The dimension of the array WORK\&.
          If LWORK = -1 or -2, then a workspace query is assumed\&. The routine
          only calculates the sizes of the T and WORK arrays, returns these
          values as the first entries of the T and WORK arrays, and no error
          message related to T or WORK is issued by XERBLA\&.
          If LWORK = -1, the routine calculates optimal size of WORK for the
          optimal performance and returns this value in WORK(1)\&.
          If LWORK = -2, the routine calculates minimal size of WORK and 
          returns this value in WORK(1)\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details\fP
.RS 4

.PP
.nf
 The goal of the interface is to give maximum freedom to the developers for
 creating any QR factorization algorithm they wish\&. The triangular 
 (trapezoidal) R has to be stored in the upper part of A\&. The lower part of A
 and the array T can be used to store any relevant information for applying or
 constructing the Q factor\&. The WORK array can safely be discarded after exit\&.

 Caution: One should not expect the sizes of T and WORK to be the same from one 
 LAPACK implementation to the other, or even from one execution to the other\&.
 A workspace query (for T and WORK) is needed at each execution\&. However, 
 for a given execution, the size of T and WORK are fixed and will not change 
 from one query to the next\&.
.fi
.PP
.RE
.PP
\fBFurther Details particular to this LAPACK implementation:\fP
.RS 4

.PP
.nf
 These details are particular for this LAPACK implementation\&. Users should not 
 take them for granted\&. These details may change in the future, and are not likely
 true for another LAPACK implementation\&. These details are relevant if one wants
 to try to understand the code\&. They are not part of the interface\&.

 In this version,

          T(2): row block size (MB)
          T(3): column block size (NB)
          T(6:TSIZE): data structure needed for Q, computed by
                           CLATSQR or CGEQRT

  Depending on the matrix dimensions M and N, and row and column
  block sizes MB and NB returned by ILAENV, CGEQR will use either
  CLATSQR (if the matrix is tall-and-skinny) or CGEQRT to compute
  the QR factorization\&.
.fi
.PP
 
.RE
.PP

.SS "subroutine dgeqr (integer m, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) t, integer tsize, double precision, dimension( * ) work, integer lwork, integer info)"

.PP
\fBDGEQR\fP 
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 DGEQR computes a QR factorization of a real M-by-N matrix A:

    A = Q * ( R ),
            ( 0 )

 where:

    Q is a M-by-M orthogonal matrix;
    R is an upper-triangular N-by-N matrix;
    0 is a (M-N)-by-N zero matrix, if M > N\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIM\fP 
.PP
.nf
          M is INTEGER
          The number of rows of the matrix A\&.  M >= 0\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The number of columns of the matrix A\&.  N >= 0\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix A\&.
          On exit, the elements on and above the diagonal of the array
          contain the min(M,N)-by-N upper trapezoidal matrix R
          (R is upper triangular if M >= N);
          the elements below the diagonal are used to store part of the 
          data structure to represent Q\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max(1,M)\&.
.fi
.PP
.br
\fIT\fP 
.PP
.nf
          T is DOUBLE PRECISION array, dimension (MAX(5,TSIZE))
          On exit, if INFO = 0, T(1) returns optimal (or either minimal 
          or optimal, if query is assumed) TSIZE\&. See TSIZE for details\&.
          Remaining T contains part of the data structure used to represent Q\&.
          If one wants to apply or construct Q, then one needs to keep T 
          (in addition to A) and pass it to further subroutines\&.
.fi
.PP
.br
\fITSIZE\fP 
.PP
.nf
          TSIZE is INTEGER
          If TSIZE >= 5, the dimension of the array T\&.
          If TSIZE = -1 or -2, then a workspace query is assumed\&. The routine
          only calculates the sizes of the T and WORK arrays, returns these
          values as the first entries of the T and WORK arrays, and no error
          message related to T or WORK is issued by XERBLA\&.
          If TSIZE = -1, the routine calculates optimal size of T for the 
          optimum performance and returns this value in T(1)\&.
          If TSIZE = -2, the routine calculates minimal size of T and 
          returns this value in T(1)\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) contains optimal (or either minimal
          or optimal, if query was assumed) LWORK\&.
          See LWORK for details\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          LWORK is INTEGER
          The dimension of the array WORK\&.
          If LWORK = -1 or -2, then a workspace query is assumed\&. The routine
          only calculates the sizes of the T and WORK arrays, returns these
          values as the first entries of the T and WORK arrays, and no error
          message related to T or WORK is issued by XERBLA\&.
          If LWORK = -1, the routine calculates optimal size of WORK for the
          optimal performance and returns this value in WORK(1)\&.
          If LWORK = -2, the routine calculates minimal size of WORK and 
          returns this value in WORK(1)\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details\fP
.RS 4

.PP
.nf
 The goal of the interface is to give maximum freedom to the developers for
 creating any QR factorization algorithm they wish\&. The triangular 
 (trapezoidal) R has to be stored in the upper part of A\&. The lower part of A
 and the array T can be used to store any relevant information for applying or
 constructing the Q factor\&. The WORK array can safely be discarded after exit\&.

 Caution: One should not expect the sizes of T and WORK to be the same from one 
 LAPACK implementation to the other, or even from one execution to the other\&.
 A workspace query (for T and WORK) is needed at each execution\&. However, 
 for a given execution, the size of T and WORK are fixed and will not change 
 from one query to the next\&.
.fi
.PP
.RE
.PP
\fBFurther Details particular to this LAPACK implementation:\fP
.RS 4

.PP
.nf
 These details are particular for this LAPACK implementation\&. Users should not 
 take them for granted\&. These details may change in the future, and are not likely
 true for another LAPACK implementation\&. These details are relevant if one wants
 to try to understand the code\&. They are not part of the interface\&.

 In this version,

          T(2): row block size (MB)
          T(3): column block size (NB)
          T(6:TSIZE): data structure needed for Q, computed by
                           DLATSQR or DGEQRT

  Depending on the matrix dimensions M and N, and row and column
  block sizes MB and NB returned by ILAENV, DGEQR will use either
  DLATSQR (if the matrix is tall-and-skinny) or DGEQRT to compute
  the QR factorization\&.
.fi
.PP
 
.RE
.PP

.SS "subroutine sgeqr (integer m, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) t, integer tsize, real, dimension( * ) work, integer lwork, integer info)"

.PP
\fBSGEQR\fP 
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 SGEQR computes a QR factorization of a real M-by-N matrix A:

    A = Q * ( R ),
            ( 0 )

 where:

    Q is a M-by-M orthogonal matrix;
    R is an upper-triangular N-by-N matrix;
    0 is a (M-N)-by-N zero matrix, if M > N\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIM\fP 
.PP
.nf
          M is INTEGER
          The number of rows of the matrix A\&.  M >= 0\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The number of columns of the matrix A\&.  N >= 0\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix A\&.
          On exit, the elements on and above the diagonal of the array
          contain the min(M,N)-by-N upper trapezoidal matrix R
          (R is upper triangular if M >= N);
          the elements below the diagonal are used to store part of the 
          data structure to represent Q\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max(1,M)\&.
.fi
.PP
.br
\fIT\fP 
.PP
.nf
          T is REAL array, dimension (MAX(5,TSIZE))
          On exit, if INFO = 0, T(1) returns optimal (or either minimal 
          or optimal, if query is assumed) TSIZE\&. See TSIZE for details\&.
          Remaining T contains part of the data structure used to represent Q\&.
          If one wants to apply or construct Q, then one needs to keep T 
          (in addition to A) and pass it to further subroutines\&.
.fi
.PP
.br
\fITSIZE\fP 
.PP
.nf
          TSIZE is INTEGER
          If TSIZE >= 5, the dimension of the array T\&.
          If TSIZE = -1 or -2, then a workspace query is assumed\&. The routine
          only calculates the sizes of the T and WORK arrays, returns these
          values as the first entries of the T and WORK arrays, and no error
          message related to T or WORK is issued by XERBLA\&.
          If TSIZE = -1, the routine calculates optimal size of T for the 
          optimum performance and returns this value in T(1)\&.
          If TSIZE = -2, the routine calculates minimal size of T and 
          returns this value in T(1)\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          (workspace) REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) contains optimal (or either minimal
          or optimal, if query was assumed) LWORK\&.
          See LWORK for details\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          LWORK is INTEGER
          The dimension of the array WORK\&.
          If LWORK = -1 or -2, then a workspace query is assumed\&. The routine
          only calculates the sizes of the T and WORK arrays, returns these
          values as the first entries of the T and WORK arrays, and no error
          message related to T or WORK is issued by XERBLA\&.
          If LWORK = -1, the routine calculates optimal size of WORK for the
          optimal performance and returns this value in WORK(1)\&.
          If LWORK = -2, the routine calculates minimal size of WORK and 
          returns this value in WORK(1)\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details\fP
.RS 4

.PP
.nf
 The goal of the interface is to give maximum freedom to the developers for
 creating any QR factorization algorithm they wish\&. The triangular 
 (trapezoidal) R has to be stored in the upper part of A\&. The lower part of A
 and the array T can be used to store any relevant information for applying or
 constructing the Q factor\&. The WORK array can safely be discarded after exit\&.

 Caution: One should not expect the sizes of T and WORK to be the same from one 
 LAPACK implementation to the other, or even from one execution to the other\&.
 A workspace query (for T and WORK) is needed at each execution\&. However, 
 for a given execution, the size of T and WORK are fixed and will not change 
 from one query to the next\&.
.fi
.PP
.RE
.PP
\fBFurther Details particular to this LAPACK implementation:\fP
.RS 4

.PP
.nf
 These details are particular for this LAPACK implementation\&. Users should not 
 take them for granted\&. These details may change in the future, and are not likely
 true for another LAPACK implementation\&. These details are relevant if one wants
 to try to understand the code\&. They are not part of the interface\&.

 In this version,

          T(2): row block size (MB)
          T(3): column block size (NB)
          T(6:TSIZE): data structure needed for Q, computed by
                           SLATSQR or SGEQRT

  Depending on the matrix dimensions M and N, and row and column
  block sizes MB and NB returned by ILAENV, SGEQR will use either
  SLATSQR (if the matrix is tall-and-skinny) or SGEQRT to compute
  the QR factorization\&.
.fi
.PP
 
.RE
.PP

.SS "subroutine zgeqr (integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) t, integer tsize, complex*16, dimension( * ) work, integer lwork, integer info)"

.PP
\fBZGEQR\fP 
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 ZGEQR computes a QR factorization of a complex M-by-N matrix A:

    A = Q * ( R ),
            ( 0 )

 where:

    Q is a M-by-M orthogonal matrix;
    R is an upper-triangular N-by-N matrix;
    0 is a (M-N)-by-N zero matrix, if M > N\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIM\fP 
.PP
.nf
          M is INTEGER
          The number of rows of the matrix A\&.  M >= 0\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The number of columns of the matrix A\&.  N >= 0\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is COMPLEX*16 array, dimension (LDA,N)
          On entry, the M-by-N matrix A\&.
          On exit, the elements on and above the diagonal of the array
          contain the min(M,N)-by-N upper trapezoidal matrix R
          (R is upper triangular if M >= N);
          the elements below the diagonal are used to store part of the 
          data structure to represent Q\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max(1,M)\&.
.fi
.PP
.br
\fIT\fP 
.PP
.nf
          T is COMPLEX*16 array, dimension (MAX(5,TSIZE))
          On exit, if INFO = 0, T(1) returns optimal (or either minimal 
          or optimal, if query is assumed) TSIZE\&. See TSIZE for details\&.
          Remaining T contains part of the data structure used to represent Q\&.
          If one wants to apply or construct Q, then one needs to keep T 
          (in addition to A) and pass it to further subroutines\&.
.fi
.PP
.br
\fITSIZE\fP 
.PP
.nf
          TSIZE is INTEGER
          If TSIZE >= 5, the dimension of the array T\&.
          If TSIZE = -1 or -2, then a workspace query is assumed\&. The routine
          only calculates the sizes of the T and WORK arrays, returns these
          values as the first entries of the T and WORK arrays, and no error
          message related to T or WORK is issued by XERBLA\&.
          If TSIZE = -1, the routine calculates optimal size of T for the 
          optimum performance and returns this value in T(1)\&.
          If TSIZE = -2, the routine calculates minimal size of T and 
          returns this value in T(1)\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          (workspace) COMPLEX*16 array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) contains optimal (or either minimal
          or optimal, if query was assumed) LWORK\&.
          See LWORK for details\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          LWORK is INTEGER
          The dimension of the array WORK\&.
          If LWORK = -1 or -2, then a workspace query is assumed\&. The routine
          only calculates the sizes of the T and WORK arrays, returns these
          values as the first entries of the T and WORK arrays, and no error
          message related to T or WORK is issued by XERBLA\&.
          If LWORK = -1, the routine calculates optimal size of WORK for the
          optimal performance and returns this value in WORK(1)\&.
          If LWORK = -2, the routine calculates minimal size of WORK and 
          returns this value in WORK(1)\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details\fP
.RS 4

.PP
.nf
 The goal of the interface is to give maximum freedom to the developers for
 creating any QR factorization algorithm they wish\&. The triangular 
 (trapezoidal) R has to be stored in the upper part of A\&. The lower part of A
 and the array T can be used to store any relevant information for applying or
 constructing the Q factor\&. The WORK array can safely be discarded after exit\&.

 Caution: One should not expect the sizes of T and WORK to be the same from one 
 LAPACK implementation to the other, or even from one execution to the other\&.
 A workspace query (for T and WORK) is needed at each execution\&. However, 
 for a given execution, the size of T and WORK are fixed and will not change 
 from one query to the next\&.
.fi
.PP
.RE
.PP
\fBFurther Details particular to this LAPACK implementation:\fP
.RS 4

.PP
.nf
 These details are particular for this LAPACK implementation\&. Users should not 
 take them for granted\&. These details may change in the future, and are not likely
 true for another LAPACK implementation\&. These details are relevant if one wants
 to try to understand the code\&. They are not part of the interface\&.

 In this version,

          T(2): row block size (MB)
          T(3): column block size (NB)
          T(6:TSIZE): data structure needed for Q, computed by
                           ZLATSQR or ZGEQRT

  Depending on the matrix dimensions M and N, and row and column
  block sizes MB and NB returned by ILAENV, ZGEQR will use either
  ZLATSQR (if the matrix is tall-and-skinny) or ZGEQRT to compute
  the QR factorization\&.
.fi
.PP
 
.RE
.PP

.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.