gelqt#

Functions

void sgelqt(
    const INT           m,
    const INT           n,
    const INT           mb,
          f32* restrict A,
    const INT           lda,
          f32* restrict T,
    const INT           ldt,
          f32* restrict work,
          INT*          info
);
void sgelqt(const INT m, const INT n, const INT mb, f32 *restrict A, const INT lda, f32 *restrict T, const INT ldt, f32 *restrict work, INT *info)#

SGELQT computes a blocked LQ factorization of a real M-by-N matrix A using the compact WY representation of Q.

The matrix V stores the elementary reflectors H(i) in the i-th row above the diagonal. For example, if M=3 and N=5, the matrix V is 

         V = (  1  v1 v1 v1 v1 )
             (     1  v2 v2 v2 )
             (         1 v3 v3 )
where the vi’s represent the vectors which define H(i), which are returned in the matrix A. The 1’s along the diagonal of V are not stored in A.

Let K=MIN(M,N). The number of blocks is B = ceiling(K/MB), where each block is of order MB except for the last block, which is of order IB = K - (B-1)*MB. For each of the B blocks, an upper triangular block reflector factor is computed: T1, T2, …, TB. The MB-by-MB (and IB-by-IB for the last block) T’s are stored in the MB-by-K matrix T as 

         T = (T1 T2 ... TB).

Parameters

in
m

The number of rows of the matrix A. m >= 0.

in
n

The number of columns of the matrix A. n >= 0.

in
mb

The block size to be used in the blocked LQ. min(m,n) >= mb >= 1.

inout
A

Double precision array, dimension (lda, n). On entry, the M-by-N matrix A. On exit, the elements on and below the diagonal of the array contain the M-by-min(M,N) lower trapezoidal matrix L (L is lower triangular if M <= N); the elements above the diagonal are the rows of V.

in
lda

The leading dimension of the array A. lda >= max(1, m).

out
T

Double precision array, dimension (ldt, min(m,n)). The upper triangular block reflectors stored in compact form as a sequence of upper triangular blocks.

in
ldt

The leading dimension of the array T. ldt >= mb.

out
work

Double precision array, dimension (mb*m).

out
info

  • = 0: successful exit

  • < 0: if info = -i, the i-th argument had an illegal value.

Functions

void dgelqt(
    const INT           m,
    const INT           n,
    const INT           mb,
          f64* restrict A,
    const INT           lda,
          f64* restrict T,
    const INT           ldt,
          f64* restrict work,
          INT*          info
);
void dgelqt(const INT m, const INT n, const INT mb, f64 *restrict A, const INT lda, f64 *restrict T, const INT ldt, f64 *restrict work, INT *info)#

DGELQT computes a blocked LQ factorization of a real M-by-N matrix A using the compact WY representation of Q.

The matrix V stores the elementary reflectors H(i) in the i-th row above the diagonal. For example, if M=3 and N=5, the matrix V is 

         V = (  1  v1 v1 v1 v1 )
             (     1  v2 v2 v2 )
             (         1 v3 v3 )
where the vi’s represent the vectors which define H(i), which are returned in the matrix A. The 1’s along the diagonal of V are not stored in A.

Let K=MIN(M,N). The number of blocks is B = ceiling(K/MB), where each block is of order MB except for the last block, which is of order IB = K - (B-1)*MB. For each of the B blocks, an upper triangular block reflector factor is computed: T1, T2, …, TB. The MB-by-MB (and IB-by-IB for the last block) T’s are stored in the MB-by-K matrix T as 

         T = (T1 T2 ... TB).

Parameters

in
m

The number of rows of the matrix A. m >= 0.

in
n

The number of columns of the matrix A. n >= 0.

in
mb

The block size to be used in the blocked LQ. min(m,n) >= mb >= 1.

inout
A

Double precision array, dimension (lda, n). On entry, the M-by-N matrix A. On exit, the elements on and below the diagonal of the array contain the M-by-min(M,N) lower trapezoidal matrix L (L is lower triangular if M <= N); the elements above the diagonal are the rows of V.

in
lda

The leading dimension of the array A. lda >= max(1, m).

out
T

Double precision array, dimension (ldt, min(m,n)). The upper triangular block reflectors stored in compact form as a sequence of upper triangular blocks.

in
ldt

The leading dimension of the array T. ldt >= mb.

out
work

Double precision array, dimension (mb*m).

out
info

  • = 0: successful exit

  • < 0: if info = -i, the i-th argument had an illegal value.

Functions

void cgelqt(
    const INT           m,
    const INT           n,
    const INT           mb,
          c64* restrict A,
    const INT           lda,
          c64* restrict T,
    const INT           ldt,
          c64* restrict work,
          INT*          info
);
void cgelqt(const INT m, const INT n, const INT mb, c64 *restrict A, const INT lda, c64 *restrict T, const INT ldt, c64 *restrict work, INT *info)#

CGELQT computes a blocked LQ factorization of a complex M-by-N matrix A using the compact WY representation of Q.

The matrix V stores the elementary reflectors H(i) in the i-th row above the diagonal. For example, if M=3 and N=5, the matrix V is 

         V = (  1  v1 v1 v1 v1 )
             (     1  v2 v2 v2 )
             (         1 v3 v3 )
where the vi’s represent the vectors which define H(i), which are returned in the matrix A. The 1’s along the diagonal of V are not stored in A.

Let K=MIN(M,N). The number of blocks is B = ceiling(K/MB), where each block is of order MB except for the last block, which is of order IB = K - (B-1)*MB. For each of the B blocks, an upper triangular block reflector factor is computed: T1, T2, …, TB. The MB-by-MB (and IB-by-IB for the last block) T’s are stored in the MB-by-K matrix T as 

         T = (T1 T2 ... TB).

Parameters

in
m

The number of rows of the matrix A. m >= 0.

in
n

The number of columns of the matrix A. n >= 0.

in
mb

The block size to be used in the blocked LQ. min(m,n) >= mb >= 1.

inout
A

Single complex array, dimension (lda, n). On entry, the M-by-N matrix A. On exit, the elements on and below the diagonal of the array contain the M-by-min(M,N) lower trapezoidal matrix L (L is lower triangular if M <= N); the elements above the diagonal are the rows of V.

in
lda

The leading dimension of the array A. lda >= max(1, m).

out
T

Single complex array, dimension (ldt, min(m,n)). The upper triangular block reflectors stored in compact form as a sequence of upper triangular blocks.

in
ldt

The leading dimension of the array T. ldt >= mb.

out
work

Single complex array, dimension (mb*m).

out
info

  • = 0: successful exit

  • < 0: if info = -i, the i-th argument had an illegal value.

Functions

void zgelqt(
    const INT            m,
    const INT            n,
    const INT            mb,
          c128* restrict A,
    const INT            lda,
          c128* restrict T,
    const INT            ldt,
          c128* restrict work,
          INT*           info
);
void zgelqt(const INT m, const INT n, const INT mb, c128 *restrict A, const INT lda, c128 *restrict T, const INT ldt, c128 *restrict work, INT *info)#

ZGELQT computes a blocked LQ factorization of a complex M-by-N matrix A using the compact WY representation of Q.

The matrix V stores the elementary reflectors H(i) in the i-th row above the diagonal. For example, if M=3 and N=5, the matrix V is 

         V = (  1  v1 v1 v1 v1 )
             (     1  v2 v2 v2 )
             (         1 v3 v3 )
where the vi’s represent the vectors which define H(i), which are returned in the matrix A. The 1’s along the diagonal of V are not stored in A.

Let K=MIN(M,N). The number of blocks is B = ceiling(K/MB), where each block is of order MB except for the last block, which is of order IB = K - (B-1)*MB. For each of the B blocks, an upper triangular block reflector factor is computed: T1, T2, …, TB. The MB-by-MB (and IB-by-IB for the last block) T’s are stored in the MB-by-K matrix T as 

         T = (T1 T2 ... TB).

Parameters

in
m

The number of rows of the matrix A. m >= 0.

in
n

The number of columns of the matrix A. n >= 0.

in
mb

The block size to be used in the blocked LQ. min(m,n) >= mb >= 1.

inout
A

Double complex array, dimension (lda, n). On entry, the M-by-N matrix A. On exit, the elements on and below the diagonal of the array contain the M-by-min(M,N) lower trapezoidal matrix L (L is lower triangular if M <= N); the elements above the diagonal are the rows of V.

in
lda

The leading dimension of the array A. lda >= max(1, m).

out
T

Double complex array, dimension (ldt, min(m,n)). The upper triangular block reflectors stored in compact form as a sequence of upper triangular blocks.

in
ldt

The leading dimension of the array T. ldt >= mb.

out
work

Double complex array, dimension (mb*m).

out
info

  • = 0: successful exit

  • < 0: if info = -i, the i-th argument had an illegal value.

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