## table of contents

variantsGEcomputational(3) | LAPACK | variantsGEcomputational(3) |

# NAME¶

variantsGEcomputational - Variants Computational routines

# SYNOPSIS¶

## Functions¶

subroutine **cgetrf** (M, N, A, LDA, IPIV, INFO)

**CGETRF** VARIANT: Crout Level 3 BLAS version of the algorithm. subroutine
**dgetrf** (M, N, A, LDA, IPIV, INFO)

**DGETRF** VARIANT: Crout Level 3 BLAS version of the algorithm. subroutine
**sgetrf** (M, N, A, LDA, IPIV, INFO)

**SGETRF** VARIANT: Crout Level 3 BLAS version of the algorithm. subroutine
**zgetrf** (M, N, A, LDA, IPIV, INFO)

**ZGETRF** VARIANT: Crout Level 3 BLAS version of the algorithm. subroutine
**cgeqrf** (M, N, A, LDA, TAU, WORK, LWORK, INFO)

**CGEQRF** VARIANT: left-looking Level 3 BLAS version of the algorithm.
subroutine **dgeqrf** (M, N, A, LDA, TAU, WORK, LWORK, INFO)

**DGEQRF** VARIANT: left-looking Level 3 BLAS version of the algorithm.
subroutine **sgeqrf** (M, N, A, LDA, TAU, WORK, LWORK, INFO)

**SGEQRF** VARIANT: left-looking Level 3 BLAS version of the algorithm.
subroutine **zgeqrf** (M, N, A, LDA, TAU, WORK, LWORK, INFO)

**ZGEQRF** VARIANT: left-looking Level 3 BLAS of the algorithm.

# Detailed Description¶

This is the group of Variants Computational routines

# Function Documentation¶

## subroutine cgeqrf (integer M, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK, integer LWORK, integer INFO)¶

**CGEQRF** VARIANT: left-looking Level 3 BLAS version of the
algorithm. **Purpose:**

CGEQRF computes a QR factorization of a real M-by-N matrix A:

A = Q * R.

This is the left-looking Level 3 BLAS version of the algorithm.

**Parameters**

*M*

M is INTEGER

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

*N*

N is INTEGER

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

*A*

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,

with the array TAU, represent the orthogonal matrix Q as a

product of min(m,n) elementary reflectors (see Further

Details).

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*TAU*

TAU is COMPLEX array, dimension (min(M,N))

The scalar factors of the elementary reflectors (see Further

Details).

*WORK*

WORK is COMPLEX array, dimension (MAX(1,LWORK))

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK is INTEGER

The dimension of the array WORK. The dimension can be divided into three parts.

1) The part for the triangular factor T. If the very last T is not bigger

than any of the rest, then this part is NB x ceiling(K/NB), otherwise,

NB x (K-NT), where K = min(M,N) and NT is the dimension of the very last T

2) The part for the very last T when T is bigger than any of the rest T.

The size of this part is NT x NT, where NT = K - ceiling ((K-NX)/NB) x NB,

where K = min(M,N), NX is calculated by

NX = MAX( 0, ILAENV( 3, 'CGEQRF', ' ', M, N, -1, -1 ) )

3) The part for dlarfb is of size max((N-M)*K, (N-M)*NB, K*NB, NB*NB)

So LWORK = part1 + part2 + part3

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further** **Details**

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v'

where tau is a real scalar, and v is a real vector with

v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),

and tau in TAU(i).

## subroutine cgetrf (integer M, integer N, complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)¶

**CGETRF** VARIANT: Crout Level 3 BLAS version of the
algorithm. **CGETRF** VARIANT: iterative version of Sivan Toledo's
recursive LU algorithm

**CGETRF** VARIANT: left-looking Level 3 BLAS version of the
algorithm.

**Purpose:**

CGETRF computes an LU factorization of a general M-by-N matrix A

using partial pivoting with row interchanges.

The factorization has the form

A = P * L * U

where P is a permutation matrix, L is lower triangular with unit

diagonal elements (lower trapezoidal if m > n), and U is upper

triangular (upper trapezoidal if m < n).

This is the Crout Level 3 BLAS version of the algorithm.

**Parameters**

*M*

M is INTEGER

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

*N*

N is INTEGER

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

*A*

A is COMPLEX array, dimension (LDA,N)

On entry, the M-by-N matrix to be factored.

On exit, the factors L and U from the factorization

A = P*L*U; the unit diagonal elements of L are not stored.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*IPIV*

IPIV is INTEGER array, dimension (min(M,N))

The pivot indices; for 1 <= i <= min(M,N), row i of the

matrix was interchanged with row IPIV(i).

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, U(i,i) is exactly zero. The factorization

has been completed, but the factor U is exactly

singular, and division by zero will occur if it is used

to solve a system of equations.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Purpose:**

CGETRF computes an LU factorization of a general M-by-N matrix A

using partial pivoting with row interchanges.

The factorization has the form

A = P * L * U

where P is a permutation matrix, L is lower triangular with unit

diagonal elements (lower trapezoidal if m > n), and U is upper

triangular (upper trapezoidal if m < n).

This is the left-looking Level 3 BLAS version of the algorithm.

**Parameters**

*M*

M is INTEGER

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

*N*

N is INTEGER

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

*A*

A is COMPLEX array, dimension (LDA,N)

On entry, the M-by-N matrix to be factored.

On exit, the factors L and U from the factorization

A = P*L*U; the unit diagonal elements of L are not stored.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*IPIV*

IPIV is INTEGER array, dimension (min(M,N))

The pivot indices; for 1 <= i <= min(M,N), row i of the

matrix was interchanged with row IPIV(i).

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, U(i,i) is exactly zero. The factorization

has been completed, but the factor U is exactly

singular, and division by zero will occur if it is used

to solve a system of equations.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Purpose:**

CGETRF computes an LU factorization of a general M-by-N matrix A

using partial pivoting with row interchanges.

The factorization has the form

A = P * L * U

where P is a permutation matrix, L is lower triangular with unit

diagonal elements (lower trapezoidal if m > n), and U is upper

triangular (upper trapezoidal if m < n).

This code implements an iterative version of Sivan Toledo's recursive

LU algorithm[1]. For square matrices, this iterative versions should

be within a factor of two of the optimum number of memory transfers.

The pattern is as follows, with the large blocks of U being updated

in one call to DTRSM, and the dotted lines denoting sections that

have had all pending permutations applied:

1 2 3 4 5 6 7 8

+-+-+---+-------+------

| |1| | |

|.+-+ 2 | |

| | | | |

|.|.+-+-+ 4 |

| | | |1| |

| | |.+-+ |

| | | | | |

|.|.|.|.+-+-+---+ 8

| | | | | |1| |

| | | | |.+-+ 2 |

| | | | | | | |

| | | | |.|.+-+-+

| | | | | | | |1|

| | | | | | |.+-+

| | | | | | | | |

|.|.|.|.|.|.|.|.+-----

| | | | | | | | |

The 1-2-1-4-1-2-1-8-... pattern is the position of the last 1 bit in

the binary expansion of the current column. Each Schur update is

applied as soon as the necessary portion of U is available.

[1] Toledo, S. 1997. Locality of Reference in LU Decomposition with

Partial Pivoting. SIAM J. Matrix Anal. Appl. 18, 4 (Oct. 1997),

1065-1081. http://dx.doi.org/10.1137/S0895479896297744

**Parameters**

*M*

M is INTEGER

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

*N*

N is INTEGER

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

*A*

A is COMPLEX array, dimension (LDA,N)

On entry, the M-by-N matrix to be factored.

On exit, the factors L and U from the factorization

A = P*L*U; the unit diagonal elements of L are not stored.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*IPIV*

IPIV is INTEGER array, dimension (min(M,N))

The pivot indices; for 1 <= i <= min(M,N), row i of the

matrix was interchanged with row IPIV(i).

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, U(i,i) is exactly zero. The factorization

has been completed, but the factor U is exactly

singular, and division by zero will occur if it is used

to solve a system of equations.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dgeqrf (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer LWORK, integer INFO)¶

**DGEQRF** VARIANT: left-looking Level 3 BLAS version of the
algorithm. **Purpose:**

DGEQRF computes a QR factorization of a real M-by-N matrix A:

A = Q * R.

This is the left-looking Level 3 BLAS version of the algorithm.

**Parameters**

*M*

M is INTEGER

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

*N*

N is INTEGER

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

*A*

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,

with the array TAU, represent the orthogonal matrix Q as a

product of min(m,n) elementary reflectors (see Further

Details).

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*TAU*

TAU is DOUBLE PRECISION array, dimension (min(M,N))

The scalar factors of the elementary reflectors (see Further

Details).

*WORK*

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK is INTEGER

The dimension of the array WORK. The dimension can be divided into three parts.

1) The part for the triangular factor T. If the very last T is not bigger

than any of the rest, then this part is NB x ceiling(K/NB), otherwise,

NB x (K-NT), where K = min(M,N) and NT is the dimension of the very last T

2) The part for the very last T when T is bigger than any of the rest T.

The size of this part is NT x NT, where NT = K - ceiling ((K-NX)/NB) x NB,

where K = min(M,N), NX is calculated by

NX = MAX( 0, ILAENV( 3, 'DGEQRF', ' ', M, N, -1, -1 ) )

3) The part for dlarfb is of size max((N-M)*K, (N-M)*NB, K*NB, NB*NB)

So LWORK = part1 + part2 + part3

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further** **Details**

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v'

where tau is a real scalar, and v is a real vector with

v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),

and tau in TAU(i).

## subroutine dgetrf (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)¶

**DGETRF** VARIANT: Crout Level 3 BLAS version of the
algorithm. **DGETRF** VARIANT: iterative version of Sivan Toledo's
recursive LU algorithm

**DGETRF** VARIANT: left-looking Level 3 BLAS version of the
algorithm.

**Purpose:**

DGETRF computes an LU factorization of a general M-by-N matrix A

using partial pivoting with row interchanges.

The factorization has the form

A = P * L * U

where P is a permutation matrix, L is lower triangular with unit

diagonal elements (lower trapezoidal if m > n), and U is upper

triangular (upper trapezoidal if m < n).

This is the Crout Level 3 BLAS version of the algorithm.

**Parameters**

*M*

M is INTEGER

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

*N*

N is INTEGER

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

*A*

A is DOUBLE PRECISION array, dimension (LDA,N)

On entry, the M-by-N matrix to be factored.

On exit, the factors L and U from the factorization

A = P*L*U; the unit diagonal elements of L are not stored.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*IPIV*

IPIV is INTEGER array, dimension (min(M,N))

The pivot indices; for 1 <= i <= min(M,N), row i of the

matrix was interchanged with row IPIV(i).

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, U(i,i) is exactly zero. The factorization

has been completed, but the factor U is exactly

singular, and division by zero will occur if it is used

to solve a system of equations.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Purpose:**

DGETRF computes an LU factorization of a general M-by-N matrix A

using partial pivoting with row interchanges.

The factorization has the form

A = P * L * U

where P is a permutation matrix, L is lower triangular with unit

diagonal elements (lower trapezoidal if m > n), and U is upper

triangular (upper trapezoidal if m < n).

This is the left-looking Level 3 BLAS version of the algorithm.

**Parameters**

*M*

M is INTEGER

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

*N*

N is INTEGER

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

*A*

A is DOUBLE PRECISION array, dimension (LDA,N)

On entry, the M-by-N matrix to be factored.

On exit, the factors L and U from the factorization

A = P*L*U; the unit diagonal elements of L are not stored.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*IPIV*

IPIV is INTEGER array, dimension (min(M,N))

The pivot indices; for 1 <= i <= min(M,N), row i of the

matrix was interchanged with row IPIV(i).

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, U(i,i) is exactly zero. The factorization

has been completed, but the factor U is exactly

singular, and division by zero will occur if it is used

to solve a system of equations.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Purpose:**

DGETRF computes an LU factorization of a general M-by-N matrix A

using partial pivoting with row interchanges.

The factorization has the form

A = P * L * U

where P is a permutation matrix, L is lower triangular with unit

diagonal elements (lower trapezoidal if m > n), and U is upper

triangular (upper trapezoidal if m < n).

This code implements an iterative version of Sivan Toledo's recursive

LU algorithm[1]. For square matrices, this iterative versions should

be within a factor of two of the optimum number of memory transfers.

The pattern is as follows, with the large blocks of U being updated

in one call to DTRSM, and the dotted lines denoting sections that

have had all pending permutations applied:

1 2 3 4 5 6 7 8

+-+-+---+-------+------

| |1| | |

|.+-+ 2 | |

| | | | |

|.|.+-+-+ 4 |

| | | |1| |

| | |.+-+ |

| | | | | |

|.|.|.|.+-+-+---+ 8

| | | | | |1| |

| | | | |.+-+ 2 |

| | | | | | | |

| | | | |.|.+-+-+

| | | | | | | |1|

| | | | | | |.+-+

| | | | | | | | |

|.|.|.|.|.|.|.|.+-----

| | | | | | | | |

The 1-2-1-4-1-2-1-8-... pattern is the position of the last 1 bit in

the binary expansion of the current column. Each Schur update is

applied as soon as the necessary portion of U is available.

[1] Toledo, S. 1997. Locality of Reference in LU Decomposition with

Partial Pivoting. SIAM J. Matrix Anal. Appl. 18, 4 (Oct. 1997),

1065-1081. http://dx.doi.org/10.1137/S0895479896297744

**Parameters**

*M*

M is INTEGER

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

*N*

N is INTEGER

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

*A*

A is DOUBLE PRECISION array, dimension (LDA,N)

On entry, the M-by-N matrix to be factored.

On exit, the factors L and U from the factorization

A = P*L*U; the unit diagonal elements of L are not stored.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*IPIV*

IPIV is INTEGER array, dimension (min(M,N))

The pivot indices; for 1 <= i <= min(M,N), row i of the

matrix was interchanged with row IPIV(i).

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, U(i,i) is exactly zero. The factorization

has been completed, but the factor U is exactly

singular, and division by zero will occur if it is used

to solve a system of equations.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine sgeqrf (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)¶

**SGEQRF** VARIANT: left-looking Level 3 BLAS version of the
algorithm. **Purpose:**

SGEQRF computes a QR factorization of a real M-by-N matrix A:

A = Q * R.

This is the left-looking Level 3 BLAS version of the algorithm.

**Parameters**

*M*

M is INTEGER

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

*N*

N is INTEGER

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

*A*

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,

with the array TAU, represent the orthogonal matrix Q as a

product of min(m,n) elementary reflectors (see Further

Details).

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*TAU*

TAU is REAL array, dimension (min(M,N))

The scalar factors of the elementary reflectors (see Further

Details).

*WORK*

WORK is REAL array, dimension (MAX(1,LWORK))

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK is INTEGER

The dimension of the array WORK. The dimension can be divided into three parts.

1) The part for the triangular factor T. If the very last T is not bigger

than any of the rest, then this part is NB x ceiling(K/NB), otherwise,

NB x (K-NT), where K = min(M,N) and NT is the dimension of the very last T

2) The part for the very last T when T is bigger than any of the rest T.

The size of this part is NT x NT, where NT = K - ceiling ((K-NX)/NB) x NB,

where K = min(M,N), NX is calculated by

NX = MAX( 0, ILAENV( 3, 'SGEQRF', ' ', M, N, -1, -1 ) )

3) The part for dlarfb is of size max((N-M)*K, (N-M)*NB, K*NB, NB*NB)

So LWORK = part1 + part2 + part3

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further** **Details**

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v'

where tau is a real scalar, and v is a real vector with

v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),

and tau in TAU(i).

## subroutine sgetrf (integer M, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)¶

**SGETRF** VARIANT: Crout Level 3 BLAS version of the
algorithm. **SGETRF** VARIANT: iterative version of Sivan Toledo's
recursive LU algorithm

**SGETRF** VARIANT: left-looking Level 3 BLAS version of the
algorithm.

**Purpose:**

SGETRF computes an LU factorization of a general M-by-N matrix A

using partial pivoting with row interchanges.

The factorization has the form

A = P * L * U

where P is a permutation matrix, L is lower triangular with unit

diagonal elements (lower trapezoidal if m > n), and U is upper

triangular (upper trapezoidal if m < n).

This is the Crout Level 3 BLAS version of the algorithm.

**Parameters**

*M*

M is INTEGER

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

*N*

N is INTEGER

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

*A*

A is REAL array, dimension (LDA,N)

On entry, the M-by-N matrix to be factored.

On exit, the factors L and U from the factorization

A = P*L*U; the unit diagonal elements of L are not stored.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*IPIV*

IPIV is INTEGER array, dimension (min(M,N))

The pivot indices; for 1 <= i <= min(M,N), row i of the

matrix was interchanged with row IPIV(i).

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, U(i,i) is exactly zero. The factorization

has been completed, but the factor U is exactly

singular, and division by zero will occur if it is used

to solve a system of equations.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Purpose:**

SGETRF computes an LU factorization of a general M-by-N matrix A

using partial pivoting with row interchanges.

The factorization has the form

A = P * L * U

where P is a permutation matrix, L is lower triangular with unit

diagonal elements (lower trapezoidal if m > n), and U is upper

triangular (upper trapezoidal if m < n).

This is the left-looking Level 3 BLAS version of the algorithm.

**Parameters**

*M*

M is INTEGER

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

*N*

N is INTEGER

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

*A*

A is REAL array, dimension (LDA,N)

On entry, the M-by-N matrix to be factored.

On exit, the factors L and U from the factorization

A = P*L*U; the unit diagonal elements of L are not stored.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*IPIV*

IPIV is INTEGER array, dimension (min(M,N))

The pivot indices; for 1 <= i <= min(M,N), row i of the

matrix was interchanged with row IPIV(i).

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, U(i,i) is exactly zero. The factorization

has been completed, but the factor U is exactly

singular, and division by zero will occur if it is used

to solve a system of equations.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Purpose:**

SGETRF computes an LU factorization of a general M-by-N matrix A

using partial pivoting with row interchanges.

The factorization has the form

A = P * L * U

where P is a permutation matrix, L is lower triangular with unit

diagonal elements (lower trapezoidal if m > n), and U is upper

triangular (upper trapezoidal if m < n).

This code implements an iterative version of Sivan Toledo's recursive

LU algorithm[1]. For square matrices, this iterative versions should

be within a factor of two of the optimum number of memory transfers.

The pattern is as follows, with the large blocks of U being updated

in one call to STRSM, and the dotted lines denoting sections that

have had all pending permutations applied:

1 2 3 4 5 6 7 8

+-+-+---+-------+------

| |1| | |

|.+-+ 2 | |

| | | | |

|.|.+-+-+ 4 |

| | | |1| |

| | |.+-+ |

| | | | | |

|.|.|.|.+-+-+---+ 8

| | | | | |1| |

| | | | |.+-+ 2 |

| | | | | | | |

| | | | |.|.+-+-+

| | | | | | | |1|

| | | | | | |.+-+

| | | | | | | | |

|.|.|.|.|.|.|.|.+-----

| | | | | | | | |

The 1-2-1-4-1-2-1-8-... pattern is the position of the last 1 bit in

the binary expansion of the current column. Each Schur update is

applied as soon as the necessary portion of U is available.

[1] Toledo, S. 1997. Locality of Reference in LU Decomposition with

Partial Pivoting. SIAM J. Matrix Anal. Appl. 18, 4 (Oct. 1997),

1065-1081. http://dx.doi.org/10.1137/S0895479896297744

**Parameters**

*M*

M is INTEGER

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

*N*

N is INTEGER

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

*A*

A is REAL array, dimension (LDA,N)

On entry, the M-by-N matrix to be factored.

On exit, the factors L and U from the factorization

A = P*L*U; the unit diagonal elements of L are not stored.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*IPIV*

IPIV is INTEGER array, dimension (min(M,N))

The pivot indices; for 1 <= i <= min(M,N), row i of the

matrix was interchanged with row IPIV(i).

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, U(i,i) is exactly zero. The factorization

has been completed, but the factor U is exactly

singular, and division by zero will occur if it is used

to solve a system of equations.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine zgeqrf (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)¶

**ZGEQRF** VARIANT: left-looking Level 3 BLAS of the algorithm.
**Purpose:**

ZGEQRF computes a QR factorization of a real M-by-N matrix A:

A = Q * R.

This is the left-looking Level 3 BLAS version of the algorithm.

**Parameters**

*M*

M is INTEGER

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

*N*

N is INTEGER

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

*A*

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,

with the array TAU, represent the orthogonal matrix Q as a

product of min(m,n) elementary reflectors (see Further

Details).

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*TAU*

TAU is COMPLEX*16 array, dimension (min(M,N))

The scalar factors of the elementary reflectors (see Further

Details).

*WORK*

WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK is INTEGER

The dimension of the array WORK. The dimension can be divided into three parts.

1) The part for the triangular factor T. If the very last T is not bigger

than any of the rest, then this part is NB x ceiling(K/NB), otherwise,

NB x (K-NT), where K = min(M,N) and NT is the dimension of the very last T

2) The part for the very last T when T is bigger than any of the rest T.

The size of this part is NT x NT, where NT = K - ceiling ((K-NX)/NB) x NB,

where K = min(M,N), NX is calculated by

NX = MAX( 0, ILAENV( 3, 'ZGEQRF', ' ', M, N, -1, -1 ) )

3) The part for dlarfb is of size max((N-M)*K, (N-M)*NB, K*NB, NB*NB)

So LWORK = part1 + part2 + part3

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further** **Details**

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v'

where tau is a real scalar, and v is a real vector with

v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),

and tau in TAU(i).

## subroutine zgetrf (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)¶

**ZGETRF** VARIANT: Crout Level 3 BLAS version of the
algorithm. **ZGETRF** VARIANT: iterative version of Sivan Toledo's
recursive LU algorithm

**ZGETRF** VARIANT: left-looking Level 3 BLAS version of the
algorithm.

**Purpose:**

ZGETRF computes an LU factorization of a general M-by-N matrix A

using partial pivoting with row interchanges.

The factorization has the form

A = P * L * U

where P is a permutation matrix, L is lower triangular with unit

diagonal elements (lower trapezoidal if m > n), and U is upper

triangular (upper trapezoidal if m < n).

This is the Crout Level 3 BLAS version of the algorithm.

**Parameters**

*M*

M is INTEGER

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

*N*

N is INTEGER

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

*A*

A is COMPLEX*16 array, dimension (LDA,N)

On entry, the M-by-N matrix to be factored.

On exit, the factors L and U from the factorization

A = P*L*U; the unit diagonal elements of L are not stored.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*IPIV*

IPIV is INTEGER array, dimension (min(M,N))

The pivot indices; for 1 <= i <= min(M,N), row i of the

matrix was interchanged with row IPIV(i).

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, U(i,i) is exactly zero. The factorization

has been completed, but the factor U is exactly

singular, and division by zero will occur if it is used

to solve a system of equations.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Purpose:**

ZGETRF computes an LU factorization of a general M-by-N matrix A

using partial pivoting with row interchanges.

The factorization has the form

A = P * L * U

where P is a permutation matrix, L is lower triangular with unit

diagonal elements (lower trapezoidal if m > n), and U is upper

triangular (upper trapezoidal if m < n).

This is the left-looking Level 3 BLAS version of the algorithm.

**Parameters**

*M*

M is INTEGER

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

*N*

N is INTEGER

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

*A*

A is COMPLEX*16 array, dimension (LDA,N)

On entry, the M-by-N matrix to be factored.

On exit, the factors L and U from the factorization

A = P*L*U; the unit diagonal elements of L are not stored.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*IPIV*

IPIV is INTEGER array, dimension (min(M,N))

The pivot indices; for 1 <= i <= min(M,N), row i of the

matrix was interchanged with row IPIV(i).

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, U(i,i) is exactly zero. The factorization

has been completed, but the factor U is exactly

singular, and division by zero will occur if it is used

to solve a system of equations.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Purpose:**

ZGETRF computes an LU factorization of a general M-by-N matrix A

using partial pivoting with row interchanges.

The factorization has the form

A = P * L * U

where P is a permutation matrix, L is lower triangular with unit

diagonal elements (lower trapezoidal if m > n), and U is upper

triangular (upper trapezoidal if m < n).

This code implements an iterative version of Sivan Toledo's recursive

LU algorithm[1]. For square matrices, this iterative versions should

be within a factor of two of the optimum number of memory transfers.

The pattern is as follows, with the large blocks of U being updated

in one call to DTRSM, and the dotted lines denoting sections that

have had all pending permutations applied:

1 2 3 4 5 6 7 8

+-+-+---+-------+------

| |1| | |

|.+-+ 2 | |

| | | | |

|.|.+-+-+ 4 |

| | | |1| |

| | |.+-+ |

| | | | | |

|.|.|.|.+-+-+---+ 8

| | | | | |1| |

| | | | |.+-+ 2 |

| | | | | | | |

| | | | |.|.+-+-+

| | | | | | | |1|

| | | | | | |.+-+

| | | | | | | | |

|.|.|.|.|.|.|.|.+-----

| | | | | | | | |

The 1-2-1-4-1-2-1-8-... pattern is the position of the last 1 bit in

the binary expansion of the current column. Each Schur update is

applied as soon as the necessary portion of U is available.

[1] Toledo, S. 1997. Locality of Reference in LU Decomposition with

Partial Pivoting. SIAM J. Matrix Anal. Appl. 18, 4 (Oct. 1997),

1065-1081. http://dx.doi.org/10.1137/S0895479896297744

**Parameters**

*M*

M is INTEGER

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

*N*

N is INTEGER

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

*A*

A is COMPLEX*16 array, dimension (LDA,N)

On entry, the M-by-N matrix to be factored.

On exit, the factors L and U from the factorization

A = P*L*U; the unit diagonal elements of L are not stored.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*IPIV*

IPIV is INTEGER array, dimension (min(M,N))

The pivot indices; for 1 <= i <= min(M,N), row i of the

matrix was interchanged with row IPIV(i).

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, U(i,i) is exactly zero. The factorization

has been completed, but the factor U is exactly

singular, and division by zero will occur if it is used

to solve a system of equations.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

# Author¶

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