## table of contents

doubleGEsolve(3) | LAPACK | doubleGEsolve(3) |

# NAME¶

doubleGEsolve - double

# SYNOPSIS¶

## Functions¶

subroutine **dgels** (TRANS, M, N, NRHS, A, LDA, B, LDB, WORK,
LWORK, INFO)

** DGELS solves overdetermined or underdetermined systems for GE matrices**
subroutine **dgelsd** (M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK,
LWORK, IWORK, INFO)

** DGELSD computes the minimum-norm solution to a linear least squares
problem for GE matrices** subroutine **dgelss** (M, N, NRHS, A, LDA, B,
LDB, S, RCOND, RANK, WORK, LWORK, INFO)

** DGELSS solves overdetermined or underdetermined systems for GE
matrices** subroutine **dgelsy** (M, N, NRHS, A, LDA, B, LDB, JPVT,
RCOND, RANK, WORK, LWORK, INFO)

** DGELSY solves overdetermined or underdetermined systems for GE
matrices** subroutine **dgesv** (N, NRHS, A, LDA, IPIV, B, LDB, INFO)

** DGESV computes the solution to system of linear equations A * X = B for GE
matrices** subroutine **dgesvx** (FACT, TRANS, N, NRHS, A, LDA, AF,
LDAF, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK,
INFO)

** DGESVX computes the solution to system of linear equations A * X = B for
GE matrices** subroutine **dgesvxx** (FACT, TRANS, N, NRHS, A, LDA, AF,
LDAF, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS,
ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO)

** DGESVXX computes the solution to system of linear equations A * X = B for
GE matrices** subroutine **dgetsls** (TRANS, M, N, NRHS, A, LDA, B,
LDB, WORK, LWORK, INFO)

**DGETSLS** subroutine **dsgesv** (N, NRHS, A, LDA, IPIV, B, LDB, X,
LDX, WORK, SWORK, ITER, INFO)

** DSGESV computes the solution to system of linear equations A * X = B for
GE matrices** (mixed precision with iterative refinement)

# Detailed Description¶

This is the group of double solve driver functions for GE matrices

# Function Documentation¶

## subroutine dgels (character TRANS, integer M, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( * ) WORK, integer LWORK, integer INFO)¶

** DGELS solves overdetermined or underdetermined systems for GE
matrices**

**Purpose:**

DGELS solves overdetermined or underdetermined real linear systems

involving an M-by-N matrix A, or its transpose, using a QR or LQ

factorization of A. It is assumed that A has full rank.

The following options are provided:

1. If TRANS = 'N' and m >= n: find the least squares solution of

an overdetermined system, i.e., solve the least squares problem

minimize || B - A*X ||.

2. If TRANS = 'N' and m < n: find the minimum norm solution of

an underdetermined system A * X = B.

3. If TRANS = 'T' and m >= n: find the minimum norm solution of

an underdetermined system A**T * X = B.

4. If TRANS = 'T' and m < n: find the least squares solution of

an overdetermined system, i.e., solve the least squares problem

minimize || B - A**T * X ||.

Several right hand side vectors b and solution vectors x can be

handled in a single call; they are stored as the columns of the

M-by-NRHS right hand side matrix B and the N-by-NRHS solution

matrix X.

**Parameters**

*TRANS*

TRANS is CHARACTER*1

= 'N': the linear system involves A;

= 'T': the linear system involves A**T.

*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.

*NRHS*

NRHS is INTEGER

The number of right hand sides, i.e., the number of

columns of the matrices B and X. NRHS >=0.

*A*

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

On entry, the M-by-N matrix A.

On exit,

if M >= N, A is overwritten by details of its QR

factorization as returned by DGEQRF;

if M < N, A is overwritten by details of its LQ

factorization as returned by DGELQF.

*LDA*

LDA is INTEGER

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

*B*

B is DOUBLE PRECISION array, dimension (LDB,NRHS)

On entry, the matrix B of right hand side vectors, stored

columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS

if TRANS = 'T'.

On exit, if INFO = 0, B is overwritten by the solution

vectors, stored columnwise:

if TRANS = 'N' and m >= n, rows 1 to n of B contain the least

squares solution vectors; the residual sum of squares for the

solution in each column is given by the sum of squares of

elements N+1 to M in that column;

if TRANS = 'N' and m < n, rows 1 to N of B contain the

minimum norm solution vectors;

if TRANS = 'T' and m >= n, rows 1 to M of B contain the

minimum norm solution vectors;

if TRANS = 'T' and m < n, rows 1 to M of B contain the

least squares solution vectors; the residual sum of squares

for the solution in each column is given by the sum of

squares of elements M+1 to N in that column.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= MAX(1,M,N).

*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.

LWORK >= max( 1, MN + max( MN, NRHS ) ).

For optimal performance,

LWORK >= max( 1, MN + max( MN, NRHS )*NB ).

where MN = min(M,N) and NB is the optimum block size.

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

> 0: if INFO = i, the i-th diagonal element of the

triangular factor of A is zero, so that A does not have

full rank; the least squares solution could not be

computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine dgelsd (integer M, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( * ) S, double precision RCOND, integer RANK, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO)¶

** DGELSD computes the minimum-norm solution to a linear least
squares problem for GE matrices**

**Purpose:**

DGELSD computes the minimum-norm solution to a real linear least

squares problem:

minimize 2-norm(| b - A*x |)

using the singular value decomposition (SVD) of A. A is an M-by-N

matrix which may be rank-deficient.

Several right hand side vectors b and solution vectors x can be

handled in a single call; they are stored as the columns of the

M-by-NRHS right hand side matrix B and the N-by-NRHS solution

matrix X.

The problem is solved in three steps:

(1) Reduce the coefficient matrix A to bidiagonal form with

Householder transformations, reducing the original problem

into a "bidiagonal least squares problem" (BLS)

(2) Solve the BLS using a divide and conquer approach.

(3) Apply back all the Householder transformations to solve

the original least squares problem.

The effective rank of A is determined by treating as zero those

singular values which are less than RCOND times the largest singular

value.

The divide and conquer algorithm makes very mild assumptions about

floating point arithmetic. It will work on machines with a guard

digit in add/subtract, or on those binary machines without guard

digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or

Cray-2. It could conceivably fail on hexadecimal or decimal machines

without guard digits, but we know of none.

**Parameters**

*M*

M is INTEGER

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

*N*

N is INTEGER

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

*NRHS*

NRHS is INTEGER

The number of right hand sides, i.e., the number of columns

of the matrices B and X. NRHS >= 0.

*A*

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

On entry, the M-by-N matrix A.

On exit, A has been destroyed.

*LDA*

LDA is INTEGER

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

*B*

B is DOUBLE PRECISION array, dimension (LDB,NRHS)

On entry, the M-by-NRHS right hand side matrix B.

On exit, B is overwritten by the N-by-NRHS solution

matrix X. If m >= n and RANK = n, the residual

sum-of-squares for the solution in the i-th column is given

by the sum of squares of elements n+1:m in that column.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,max(M,N)).

*S*

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

The singular values of A in decreasing order.

The condition number of A in the 2-norm = S(1)/S(min(m,n)).

*RCOND*

RCOND is DOUBLE PRECISION

RCOND is used to determine the effective rank of A.

Singular values S(i) <= RCOND*S(1) are treated as zero.

If RCOND < 0, machine precision is used instead.

*RANK*

RANK is INTEGER

The effective rank of A, i.e., the number of singular values

which are greater than RCOND*S(1).

*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. LWORK must be at least 1.

The exact minimum amount of workspace needed depends on M,

N and NRHS. As long as LWORK is at least

12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2,

if M is greater than or equal to N or

12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2,

if M is less than N, the code will execute correctly.

SMLSIZ is returned by ILAENV and is equal to the maximum

size of the subproblems at the bottom of the computation

tree (usually about 25), and

NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )

For good performance, LWORK should generally be larger.

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.

*IWORK*

IWORK is INTEGER array, dimension (MAX(1,LIWORK))

LIWORK >= max(1, 3 * MINMN * NLVL + 11 * MINMN),

where MINMN = MIN( M,N ).

On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

*INFO*

INFO is INTEGER

= 0: successful exit

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

> 0: the algorithm for computing the SVD failed to converge;

if INFO = i, i off-diagonal elements of an intermediate

bidiagonal form did not converge to zero.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

Osni Marques, LBNL/NERSC, USA

## subroutine dgelss (integer M, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( * ) S, double precision RCOND, integer RANK, double precision, dimension( * ) WORK, integer LWORK, integer INFO)¶

** DGELSS solves overdetermined or underdetermined systems for GE
matrices**

**Purpose:**

DGELSS computes the minimum norm solution to a real linear least

squares problem:

Minimize 2-norm(| b - A*x |).

using the singular value decomposition (SVD) of A. A is an M-by-N

matrix which may be rank-deficient.

Several right hand side vectors b and solution vectors x can be

handled in a single call; they are stored as the columns of the

M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix

X.

The effective rank of A is determined by treating as zero those

singular values which are less than RCOND times the largest singular

value.

**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.

*NRHS*

NRHS is INTEGER

The number of right hand sides, i.e., the number of columns

of the matrices B and X. NRHS >= 0.

*A*

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

On entry, the M-by-N matrix A.

On exit, the first min(m,n) rows of A are overwritten with

its right singular vectors, stored rowwise.

*LDA*

LDA is INTEGER

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

*B*

B is DOUBLE PRECISION array, dimension (LDB,NRHS)

On entry, the M-by-NRHS right hand side matrix B.

On exit, B is overwritten by the N-by-NRHS solution

matrix X. If m >= n and RANK = n, the residual

sum-of-squares for the solution in the i-th column is given

by the sum of squares of elements n+1:m in that column.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,max(M,N)).

*S*

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

The singular values of A in decreasing order.

The condition number of A in the 2-norm = S(1)/S(min(m,n)).

*RCOND*

RCOND is DOUBLE PRECISION

RCOND is used to determine the effective rank of A.

Singular values S(i) <= RCOND*S(1) are treated as zero.

If RCOND < 0, machine precision is used instead.

*RANK*

RANK is INTEGER

The effective rank of A, i.e., the number of singular values

which are greater than RCOND*S(1).

*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. LWORK >= 1, and also:

LWORK >= 3*min(M,N) + max( 2*min(M,N), max(M,N), NRHS )

For good performance, LWORK should generally be larger.

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.

> 0: the algorithm for computing the SVD failed to converge;

if INFO = i, i off-diagonal elements of an intermediate

bidiagonal form did not converge to zero.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine dgelsy (integer M, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, integer, dimension( * ) JPVT, double precision RCOND, integer RANK, double precision, dimension( * ) WORK, integer LWORK, integer INFO)¶

** DGELSY solves overdetermined or underdetermined systems for GE
matrices**

**Purpose:**

DGELSY computes the minimum-norm solution to a real linear least

squares problem:

minimize || A * X - B ||

using a complete orthogonal factorization of A. A is an M-by-N

matrix which may be rank-deficient.

Several right hand side vectors b and solution vectors x can be

handled in a single call; they are stored as the columns of the

M-by-NRHS right hand side matrix B and the N-by-NRHS solution

matrix X.

The routine first computes a QR factorization with column pivoting:

A * P = Q * [ R11 R12 ]

[ 0 R22 ]

with R11 defined as the largest leading submatrix whose estimated

condition number is less than 1/RCOND. The order of R11, RANK,

is the effective rank of A.

Then, R22 is considered to be negligible, and R12 is annihilated

by orthogonal transformations from the right, arriving at the

complete orthogonal factorization:

A * P = Q * [ T11 0 ] * Z

[ 0 0 ]

The minimum-norm solution is then

X = P * Z**T [ inv(T11)*Q1**T*B ]

[ 0 ]

where Q1 consists of the first RANK columns of Q.

This routine is basically identical to the original xGELSX except

three differences:

o The call to the subroutine xGEQPF has been substituted by the

the call to the subroutine xGEQP3. This subroutine is a Blas-3

version of the QR factorization with column pivoting.

o Matrix B (the right hand side) is updated with Blas-3.

o The permutation of matrix B (the right hand side) is faster and

more simple.

**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.

*NRHS*

NRHS is INTEGER

The number of right hand sides, i.e., the number of

columns of matrices B and X. NRHS >= 0.

*A*

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

On entry, the M-by-N matrix A.

On exit, A has been overwritten by details of its

complete orthogonal factorization.

*LDA*

LDA is INTEGER

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

*B*

B is DOUBLE PRECISION array, dimension (LDB,NRHS)

On entry, the M-by-NRHS right hand side matrix B.

On exit, the N-by-NRHS solution matrix X.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,M,N).

*JPVT*

JPVT is INTEGER array, dimension (N)

On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted

to the front of AP, otherwise column i is a free column.

On exit, if JPVT(i) = k, then the i-th column of AP

was the k-th column of A.

*RCOND*

RCOND is DOUBLE PRECISION

RCOND is used to determine the effective rank of A, which

is defined as the order of the largest leading triangular

submatrix R11 in the QR factorization with pivoting of A,

whose estimated condition number < 1/RCOND.

*RANK*

RANK is INTEGER

The effective rank of A, i.e., the order of the submatrix

R11. This is the same as the order of the submatrix T11

in the complete orthogonal factorization of A.

*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 unblocked strategy requires that:

LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ),

where MN = min( M, N ).

The block algorithm requires that:

LWORK >= MAX( MN+2*N+NB*(N+1), 2*MN+NB*NRHS ),

where NB is an upper bound on the blocksize returned

by ILAENV for the routines DGEQP3, DTZRZF, STZRQF, DORMQR,

and DORMRZ.

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.

**Contributors:**

E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain

G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain

## subroutine dgesv (integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision, dimension( ldb, * ) B, integer LDB, integer INFO)¶

** DGESV computes the solution to system of linear equations A *
X = B for GE matrices**

**Purpose:**

DGESV computes the solution to a real system of linear equations

A * X = B,

where A is an N-by-N matrix and X and B are N-by-NRHS matrices.

The LU decomposition with partial pivoting and row interchanges is

used to factor A as

A = P * L * U,

where P is a permutation matrix, L is unit lower triangular, and U is

upper triangular. The factored form of A is then used to solve the

system of equations A * X = B.

**Parameters**

*N*

N is INTEGER

The number of linear equations, i.e., the order of the

matrix A. N >= 0.

*NRHS*

NRHS is INTEGER

The number of right hand sides, i.e., the number of columns

of the matrix B. NRHS >= 0.

*A*

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

On entry, the N-by-N coefficient matrix A.

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,N).

*IPIV*

IPIV is INTEGER array, dimension (N)

The pivot indices that define the permutation matrix P;

row i of the matrix was interchanged with row IPIV(i).

*B*

B is DOUBLE PRECISION array, dimension (LDB,NRHS)

On entry, the N-by-NRHS matrix of right hand side matrix B.

On exit, if INFO = 0, the N-by-NRHS solution matrix X.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

*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, so the solution could not be computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine dgesvx (character FACT, character TRANS, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, character EQUED, double precision, dimension( * ) R, double precision, dimension( * ) C, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldx, * ) X, integer LDX, double precision RCOND, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶

** DGESVX computes the solution to system of linear equations A *
X = B for GE matrices**

**Purpose:**

DGESVX uses the LU factorization to compute the solution to a real

system of linear equations

A * X = B,

where A is an N-by-N matrix and X and B are N-by-NRHS matrices.

Error bounds on the solution and a condition estimate are also

provided.

**Description:**

The following steps are performed:

1. If FACT = 'E', real scaling factors are computed to equilibrate

the system:

TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B

TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B

TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B

Whether or not the system will be equilibrated depends on the

scaling of the matrix A, but if equilibration is used, A is

overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')

or diag(C)*B (if TRANS = 'T' or 'C').

2. If FACT = 'N' or 'E', the LU decomposition is used to factor the

matrix A (after equilibration if FACT = 'E') as

A = P * L * U,

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

matrix, and U is upper triangular.

3. If some U(i,i)=0, so that U is exactly singular, then the routine

returns with INFO = i. Otherwise, the factored form of A is used

to estimate the condition number of the matrix A. If the

reciprocal of the condition number is less than machine precision,

INFO = N+1 is returned as a warning, but the routine still goes on

to solve for X and compute error bounds as described below.

4. The system of equations is solved for X using the factored form

of A.

5. Iterative refinement is applied to improve the computed solution

matrix and calculate error bounds and backward error estimates

for it.

6. If equilibration was used, the matrix X is premultiplied by

diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so

that it solves the original system before equilibration.

**Parameters**

*FACT*

FACT is CHARACTER*1

Specifies whether or not the factored form of the matrix A is

supplied on entry, and if not, whether the matrix A should be

equilibrated before it is factored.

= 'F': On entry, AF and IPIV contain the factored form of A.

If EQUED is not 'N', the matrix A has been

equilibrated with scaling factors given by R and C.

A, AF, and IPIV are not modified.

= 'N': The matrix A will be copied to AF and factored.

= 'E': The matrix A will be equilibrated if necessary, then

copied to AF and factored.

*TRANS*

TRANS is CHARACTER*1

Specifies the form of the system of equations:

= 'N': A * X = B (No transpose)

= 'T': A**T * X = B (Transpose)

= 'C': A**H * X = B (Transpose)

*N*

N is INTEGER

The number of linear equations, i.e., the order of the

matrix A. N >= 0.

*NRHS*

NRHS is INTEGER

The number of right hand sides, i.e., the number of columns

of the matrices B and X. NRHS >= 0.

*A*

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

On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is

not 'N', then A must have been equilibrated by the scaling

factors in R and/or C. A is not modified if FACT = 'F' or

'N', or if FACT = 'E' and EQUED = 'N' on exit.

On exit, if EQUED .ne. 'N', A is scaled as follows:

EQUED = 'R': A := diag(R) * A

EQUED = 'C': A := A * diag(C)

EQUED = 'B': A := diag(R) * A * diag(C).

*LDA*

LDA is INTEGER

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

*AF*

AF is DOUBLE PRECISION array, dimension (LDAF,N)

If FACT = 'F', then AF is an input argument and on entry

contains the factors L and U from the factorization

A = P*L*U as computed by DGETRF. If EQUED .ne. 'N', then

AF is the factored form of the equilibrated matrix A.

If FACT = 'N', then AF is an output argument and on exit

returns the factors L and U from the factorization A = P*L*U

of the original matrix A.

If FACT = 'E', then AF is an output argument and on exit

returns the factors L and U from the factorization A = P*L*U

of the equilibrated matrix A (see the description of A for

the form of the equilibrated matrix).

*LDAF*

LDAF is INTEGER

The leading dimension of the array AF. LDAF >= max(1,N).

*IPIV*

IPIV is INTEGER array, dimension (N)

If FACT = 'F', then IPIV is an input argument and on entry

contains the pivot indices from the factorization A = P*L*U

as computed by DGETRF; row i of the matrix was interchanged

with row IPIV(i).

If FACT = 'N', then IPIV is an output argument and on exit

contains the pivot indices from the factorization A = P*L*U

of the original matrix A.

If FACT = 'E', then IPIV is an output argument and on exit

contains the pivot indices from the factorization A = P*L*U

of the equilibrated matrix A.

*EQUED*

EQUED is CHARACTER*1

Specifies the form of equilibration that was done.

= 'N': No equilibration (always true if FACT = 'N').

= 'R': Row equilibration, i.e., A has been premultiplied by

diag(R).

= 'C': Column equilibration, i.e., A has been postmultiplied

by diag(C).

= 'B': Both row and column equilibration, i.e., A has been

replaced by diag(R) * A * diag(C).

EQUED is an input argument if FACT = 'F'; otherwise, it is an

output argument.

*R*

R is DOUBLE PRECISION array, dimension (N)

The row scale factors for A. If EQUED = 'R' or 'B', A is

multiplied on the left by diag(R); if EQUED = 'N' or 'C', R

is not accessed. R is an input argument if FACT = 'F';

otherwise, R is an output argument. If FACT = 'F' and

EQUED = 'R' or 'B', each element of R must be positive.

*C*

C is DOUBLE PRECISION array, dimension (N)

The column scale factors for A. If EQUED = 'C' or 'B', A is

multiplied on the right by diag(C); if EQUED = 'N' or 'R', C

is not accessed. C is an input argument if FACT = 'F';

otherwise, C is an output argument. If FACT = 'F' and

EQUED = 'C' or 'B', each element of C must be positive.

*B*

B is DOUBLE PRECISION array, dimension (LDB,NRHS)

On entry, the N-by-NRHS right hand side matrix B.

On exit,

if EQUED = 'N', B is not modified;

if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by

diag(R)*B;

if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is

overwritten by diag(C)*B.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

*X*

X is DOUBLE PRECISION array, dimension (LDX,NRHS)

If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X

to the original system of equations. Note that A and B are

modified on exit if EQUED .ne. 'N', and the solution to the

equilibrated system is inv(diag(C))*X if TRANS = 'N' and

EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'

and EQUED = 'R' or 'B'.

*LDX*

LDX is INTEGER

The leading dimension of the array X. LDX >= max(1,N).

*RCOND*

RCOND is DOUBLE PRECISION

The estimate of the reciprocal condition number of the matrix

A after equilibration (if done). If RCOND is less than the

machine precision (in particular, if RCOND = 0), the matrix

is singular to working precision. This condition is

indicated by a return code of INFO > 0.

*FERR*

FERR is DOUBLE PRECISION array, dimension (NRHS)

The estimated forward error bound for each solution vector

X(j) (the j-th column of the solution matrix X).

If XTRUE is the true solution corresponding to X(j), FERR(j)

is an estimated upper bound for the magnitude of the largest

element in (X(j) - XTRUE) divided by the magnitude of the

largest element in X(j). The estimate is as reliable as

the estimate for RCOND, and is almost always a slight

overestimate of the true error.

*BERR*

BERR is DOUBLE PRECISION array, dimension (NRHS)

The componentwise relative backward error of each solution

vector X(j) (i.e., the smallest relative change in

any element of A or B that makes X(j) an exact solution).

*WORK*

WORK is DOUBLE PRECISION array, dimension (4*N)

On exit, WORK(1) contains the reciprocal pivot growth

factor norm(A)/norm(U). The "max absolute element" norm is

used. If WORK(1) is much less than 1, then the stability

of the LU factorization of the (equilibrated) matrix A

could be poor. This also means that the solution X, condition

estimator RCOND, and forward error bound FERR could be

unreliable. If factorization fails with 0<INFO<=N, then

WORK(1) contains the reciprocal pivot growth factor for the

leading INFO columns of A.

*IWORK*

IWORK is INTEGER array, dimension (N)

*INFO*

INFO is INTEGER

= 0: successful exit

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

> 0: if INFO = i, and i is

<= N: U(i,i) is exactly zero. The factorization has

been completed, but the factor U is exactly

singular, so the solution and error bounds

could not be computed. RCOND = 0 is returned.

= N+1: U is nonsingular, but RCOND is less than machine

precision, meaning that the matrix is singular

to working precision. Nevertheless, the

solution and error bounds are computed because

there are a number of situations where the

computed solution can be more accurate than the

value of RCOND would suggest.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine dgesvxx (character FACT, character TRANS, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, character EQUED, double precision, dimension( * ) R, double precision, dimension( * ) C, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldx , * ) X, integer LDX, double precision RCOND, double precision RPVGRW, double precision, dimension( * ) BERR, integer N_ERR_BNDS, double precision, dimension( nrhs, * ) ERR_BNDS_NORM, double precision, dimension( nrhs, * ) ERR_BNDS_COMP, integer NPARAMS, double precision, dimension( * ) PARAMS, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶

** DGESVXX computes the solution to system of linear equations A
* X = B for GE matrices**

**Purpose:**

DGESVXX uses the LU factorization to compute the solution to a

double precision system of linear equations A * X = B, where A is an

N-by-N matrix and X and B are N-by-NRHS matrices.

If requested, both normwise and maximum componentwise error bounds

are returned. DGESVXX will return a solution with a tiny

guaranteed error (O(eps) where eps is the working machine

precision) unless the matrix is very ill-conditioned, in which

case a warning is returned. Relevant condition numbers also are

calculated and returned.

DGESVXX accepts user-provided factorizations and equilibration

factors; see the definitions of the FACT and EQUED options.

Solving with refinement and using a factorization from a previous

DGESVXX call will also produce a solution with either O(eps)

errors or warnings, but we cannot make that claim for general

user-provided factorizations and equilibration factors if they

differ from what DGESVXX would itself produce.

**Description:**

The following steps are performed:

1. If FACT = 'E', double precision scaling factors are computed to equilibrate

the system:

TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B

TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B

TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B

Whether or not the system will be equilibrated depends on the

scaling of the matrix A, but if equilibration is used, A is

overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')

or diag(C)*B (if TRANS = 'T' or 'C').

2. If FACT = 'N' or 'E', the LU decomposition is used to factor

the matrix A (after equilibration if FACT = 'E') as

A = P * L * U,

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

matrix, and U is upper triangular.

3. If some U(i,i)=0, so that U is exactly singular, then the

routine returns with INFO = i. Otherwise, the factored form of A

is used to estimate the condition number of the matrix A (see

argument RCOND). If the reciprocal of the condition number is less

than machine precision, the routine still goes on to solve for X

and compute error bounds as described below.

4. The system of equations is solved for X using the factored form

of A.

5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),

the routine will use iterative refinement to try to get a small

error and error bounds. Refinement calculates the residual to at

least twice the working precision.

6. If equilibration was used, the matrix X is premultiplied by

diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so

that it solves the original system before equilibration.

Some optional parameters are bundled in the PARAMS array. These

settings determine how refinement is performed, but often the

defaults are acceptable. If the defaults are acceptable, users

can pass NPARAMS = 0 which prevents the source code from accessing

the PARAMS argument.

**Parameters**

*FACT*

FACT is CHARACTER*1

Specifies whether or not the factored form of the matrix A is

supplied on entry, and if not, whether the matrix A should be

equilibrated before it is factored.

= 'F': On entry, AF and IPIV contain the factored form of A.

If EQUED is not 'N', the matrix A has been

equilibrated with scaling factors given by R and C.

A, AF, and IPIV are not modified.

= 'N': The matrix A will be copied to AF and factored.

= 'E': The matrix A will be equilibrated if necessary, then

copied to AF and factored.

*TRANS*

TRANS is CHARACTER*1

Specifies the form of the system of equations:

= 'N': A * X = B (No transpose)

= 'T': A**T * X = B (Transpose)

= 'C': A**H * X = B (Conjugate Transpose = Transpose)

*N*

N is INTEGER

The number of linear equations, i.e., the order of the

matrix A. N >= 0.

*NRHS*

NRHS is INTEGER

The number of right hand sides, i.e., the number of columns

of the matrices B and X. NRHS >= 0.

*A*

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

On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is

not 'N', then A must have been equilibrated by the scaling

factors in R and/or C. A is not modified if FACT = 'F' or

'N', or if FACT = 'E' and EQUED = 'N' on exit.

On exit, if EQUED .ne. 'N', A is scaled as follows:

EQUED = 'R': A := diag(R) * A

EQUED = 'C': A := A * diag(C)

EQUED = 'B': A := diag(R) * A * diag(C).

*LDA*

LDA is INTEGER

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

*AF*

AF is DOUBLE PRECISION array, dimension (LDAF,N)

If FACT = 'F', then AF is an input argument and on entry

contains the factors L and U from the factorization

A = P*L*U as computed by DGETRF. If EQUED .ne. 'N', then

AF is the factored form of the equilibrated matrix A.

If FACT = 'N', then AF is an output argument and on exit

returns the factors L and U from the factorization A = P*L*U

of the original matrix A.

If FACT = 'E', then AF is an output argument and on exit

returns the factors L and U from the factorization A = P*L*U

of the equilibrated matrix A (see the description of A for

the form of the equilibrated matrix).

*LDAF*

LDAF is INTEGER

The leading dimension of the array AF. LDAF >= max(1,N).

*IPIV*

IPIV is INTEGER array, dimension (N)

If FACT = 'F', then IPIV is an input argument and on entry

contains the pivot indices from the factorization A = P*L*U

as computed by DGETRF; row i of the matrix was interchanged

with row IPIV(i).

If FACT = 'N', then IPIV is an output argument and on exit

contains the pivot indices from the factorization A = P*L*U

of the original matrix A.

If FACT = 'E', then IPIV is an output argument and on exit

contains the pivot indices from the factorization A = P*L*U

of the equilibrated matrix A.

*EQUED*

EQUED is CHARACTER*1

Specifies the form of equilibration that was done.

= 'N': No equilibration (always true if FACT = 'N').

= 'R': Row equilibration, i.e., A has been premultiplied by

diag(R).

= 'C': Column equilibration, i.e., A has been postmultiplied

by diag(C).

= 'B': Both row and column equilibration, i.e., A has been

replaced by diag(R) * A * diag(C).

EQUED is an input argument if FACT = 'F'; otherwise, it is an

output argument.

*R*

R is DOUBLE PRECISION array, dimension (N)

The row scale factors for A. If EQUED = 'R' or 'B', A is

multiplied on the left by diag(R); if EQUED = 'N' or 'C', R

is not accessed. R is an input argument if FACT = 'F';

otherwise, R is an output argument. If FACT = 'F' and

EQUED = 'R' or 'B', each element of R must be positive.

If R is output, each element of R is a power of the radix.

If R is input, each element of R should be a power of the radix

to ensure a reliable solution and error estimates. Scaling by

powers of the radix does not cause rounding errors unless the

result underflows or overflows. Rounding errors during scaling

lead to refining with a matrix that is not equivalent to the

input matrix, producing error estimates that may not be

reliable.

*C*

C is DOUBLE PRECISION array, dimension (N)

The column scale factors for A. If EQUED = 'C' or 'B', A is

multiplied on the right by diag(C); if EQUED = 'N' or 'R', C

is not accessed. C is an input argument if FACT = 'F';

otherwise, C is an output argument. If FACT = 'F' and

EQUED = 'C' or 'B', each element of C must be positive.

If C is output, each element of C is a power of the radix.

If C is input, each element of C should be a power of the radix

to ensure a reliable solution and error estimates. Scaling by

powers of the radix does not cause rounding errors unless the

result underflows or overflows. Rounding errors during scaling

lead to refining with a matrix that is not equivalent to the

input matrix, producing error estimates that may not be

reliable.

*B*

B is DOUBLE PRECISION array, dimension (LDB,NRHS)

On entry, the N-by-NRHS right hand side matrix B.

On exit,

if EQUED = 'N', B is not modified;

if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by

diag(R)*B;

if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is

overwritten by diag(C)*B.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

*X*

X is DOUBLE PRECISION array, dimension (LDX,NRHS)

If INFO = 0, the N-by-NRHS solution matrix X to the original

system of equations. Note that A and B are modified on exit

if EQUED .ne. 'N', and the solution to the equilibrated system is

inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or

inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'.

*LDX*

LDX is INTEGER

The leading dimension of the array X. LDX >= max(1,N).

*RCOND*

RCOND is DOUBLE PRECISION

Reciprocal scaled condition number. This is an estimate of the

reciprocal Skeel condition number of the matrix A after

equilibration (if done). If this is less than the machine

precision (in particular, if it is zero), the matrix is singular

to working precision. Note that the error may still be small even

if this number is very small and the matrix appears ill-

conditioned.

*RPVGRW*

RPVGRW is DOUBLE PRECISION

Reciprocal pivot growth. On exit, this contains the reciprocal

pivot growth factor norm(A)/norm(U). The "max absolute element"

norm is used. If this is much less than 1, then the stability of

the LU factorization of the (equilibrated) matrix A could be poor.

This also means that the solution X, estimated condition numbers,

and error bounds could be unreliable. If factorization fails with

0<INFO<=N, then this contains the reciprocal pivot growth factor

for the leading INFO columns of A. In DGESVX, this quantity is

returned in WORK(1).

*BERR*

BERR is DOUBLE PRECISION array, dimension (NRHS)

Componentwise relative backward error. This is the

componentwise relative backward error of each solution vector X(j)

(i.e., the smallest relative change in any element of A or B that

makes X(j) an exact solution).

*N_ERR_BNDS*

N_ERR_BNDS is INTEGER

Number of error bounds to return for each right hand side

and each type (normwise or componentwise). See ERR_BNDS_NORM and

ERR_BNDS_COMP below.

*ERR_BNDS_NORM*

ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)

For each right-hand side, this array contains information about

various error bounds and condition numbers corresponding to the

normwise relative error, which is defined as follows:

Normwise relative error in the ith solution vector:

max_j (abs(XTRUE(j,i) - X(j,i)))

------------------------------

max_j abs(X(j,i))

The array is indexed by the type of error information as described

below. There currently are up to three pieces of information

returned.

The first index in ERR_BNDS_NORM(i,:) corresponds to the ith

right-hand side.

The second index in ERR_BNDS_NORM(:,err) contains the following

three fields:

err = 1 "Trust/don't trust" boolean. Trust the answer if the

reciprocal condition number is less than the threshold

sqrt(n) * dlamch('Epsilon').

err = 2 "Guaranteed" error bound: The estimated forward error,

almost certainly within a factor of 10 of the true error

so long as the next entry is greater than the threshold

sqrt(n) * dlamch('Epsilon'). This error bound should only

be trusted if the previous boolean is true.

err = 3 Reciprocal condition number: Estimated normwise

reciprocal condition number. Compared with the threshold

sqrt(n) * dlamch('Epsilon') to determine if the error

estimate is "guaranteed". These reciprocal condition

numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some

appropriately scaled matrix Z.

Let Z = S*A, where S scales each row by a power of the

radix so all absolute row sums of Z are approximately 1.

See Lapack Working Note 165 for further details and extra

cautions.

*ERR_BNDS_COMP*

ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)

For each right-hand side, this array contains information about

various error bounds and condition numbers corresponding to the

componentwise relative error, which is defined as follows:

Componentwise relative error in the ith solution vector:

abs(XTRUE(j,i) - X(j,i))

max_j ----------------------

abs(X(j,i))

The array is indexed by the right-hand side i (on which the

componentwise relative error depends), and the type of error

information as described below. There currently are up to three

pieces of information returned for each right-hand side. If

componentwise accuracy is not requested (PARAMS(3) = 0.0), then

ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most

the first (:,N_ERR_BNDS) entries are returned.

The first index in ERR_BNDS_COMP(i,:) corresponds to the ith

right-hand side.

The second index in ERR_BNDS_COMP(:,err) contains the following

three fields:

err = 1 "Trust/don't trust" boolean. Trust the answer if the

reciprocal condition number is less than the threshold

sqrt(n) * dlamch('Epsilon').

err = 2 "Guaranteed" error bound: The estimated forward error,

almost certainly within a factor of 10 of the true error

so long as the next entry is greater than the threshold

sqrt(n) * dlamch('Epsilon'). This error bound should only

be trusted if the previous boolean is true.

err = 3 Reciprocal condition number: Estimated componentwise

reciprocal condition number. Compared with the threshold

sqrt(n) * dlamch('Epsilon') to determine if the error

estimate is "guaranteed". These reciprocal condition

numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some

appropriately scaled matrix Z.

Let Z = S*(A*diag(x)), where x is the solution for the

current right-hand side and S scales each row of

A*diag(x) by a power of the radix so all absolute row

sums of Z are approximately 1.

See Lapack Working Note 165 for further details and extra

cautions.

*NPARAMS*

NPARAMS is INTEGER

Specifies the number of parameters set in PARAMS. If <= 0, the

PARAMS array is never referenced and default values are used.

*PARAMS*

PARAMS is DOUBLE PRECISION array, dimension (NPARAMS)

Specifies algorithm parameters. If an entry is < 0.0, then

that entry will be filled with default value used for that

parameter. Only positions up to NPARAMS are accessed; defaults

are used for higher-numbered parameters.

PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative

refinement or not.

Default: 1.0D+0

= 0.0: No refinement is performed, and no error bounds are

computed.

= 1.0: Use the extra-precise refinement algorithm.

(other values are reserved for future use)

PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual

computations allowed for refinement.

Default: 10

Aggressive: Set to 100 to permit convergence using approximate

factorizations or factorizations other than LU. If

the factorization uses a technique other than

Gaussian elimination, the guarantees in

err_bnds_norm and err_bnds_comp may no longer be

trustworthy.

PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code

will attempt to find a solution with small componentwise

relative error in the double-precision algorithm. Positive

is true, 0.0 is false.

Default: 1.0 (attempt componentwise convergence)

*WORK*

WORK is DOUBLE PRECISION array, dimension (4*N)

*IWORK*

IWORK is INTEGER array, dimension (N)

*INFO*

INFO is INTEGER

= 0: Successful exit. The solution to every right-hand side is

guaranteed.

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

> 0 and <= N: U(INFO,INFO) is exactly zero. The factorization

has been completed, but the factor U is exactly singular, so

the solution and error bounds could not be computed. RCOND = 0

is returned.

= N+J: The solution corresponding to the Jth right-hand side is

not guaranteed. The solutions corresponding to other right-

hand sides K with K > J may not be guaranteed as well, but

only the first such right-hand side is reported. If a small

componentwise error is not requested (PARAMS(3) = 0.0) then

the Jth right-hand side is the first with a normwise error

bound that is not guaranteed (the smallest J such

that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)

the Jth right-hand side is the first with either a normwise or

componentwise error bound that is not guaranteed (the smallest

J such that either ERR_BNDS_NORM(J,1) = 0.0 or

ERR_BNDS_COMP(J,1) = 0.0). See the definition of

ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information

about all of the right-hand sides check ERR_BNDS_NORM or

ERR_BNDS_COMP.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine dgetsls (character TRANS, integer M, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( * ) WORK, integer LWORK, integer INFO)¶

**DGETSLS**

**Purpose:**

DGETSLS solves overdetermined or underdetermined real linear systems

involving an M-by-N matrix A, using a tall skinny QR or short wide LQ

factorization of A. It is assumed that A has full rank.

The following options are provided:

1. If TRANS = 'N' and m >= n: find the least squares solution of

an overdetermined system, i.e., solve the least squares problem

minimize || B - A*X ||.

2. If TRANS = 'N' and m < n: find the minimum norm solution of

an underdetermined system A * X = B.

3. If TRANS = 'T' and m >= n: find the minimum norm solution of

an undetermined system A**T * X = B.

4. If TRANS = 'T' and m < n: find the least squares solution of

an overdetermined system, i.e., solve the least squares problem

minimize || B - A**T * X ||.

Several right hand side vectors b and solution vectors x can be

handled in a single call; they are stored as the columns of the

M-by-NRHS right hand side matrix B and the N-by-NRHS solution

matrix X.

**Parameters**

*TRANS*

TRANS is CHARACTER*1

= 'N': the linear system involves A;

= 'T': the linear system involves A**T.

*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.

*NRHS*

NRHS is INTEGER

The number of right hand sides, i.e., the number of

columns of the matrices B and X. NRHS >=0.

*A*

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

On entry, the M-by-N matrix A.

On exit,

A is overwritten by details of its QR or LQ

factorization as returned by DGEQR or DGELQ.

*LDA*

LDA is INTEGER

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

*B*

B is DOUBLE PRECISION array, dimension (LDB,NRHS)

On entry, the matrix B of right hand side vectors, stored

columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS

if TRANS = 'T'.

On exit, if INFO = 0, B is overwritten by the solution

vectors, stored columnwise:

if TRANS = 'N' and m >= n, rows 1 to n of B contain the least

squares solution vectors.

if TRANS = 'N' and m < n, rows 1 to N of B contain the

minimum norm solution vectors;

if TRANS = 'T' and m >= n, rows 1 to M of B contain the

minimum norm solution vectors;

if TRANS = 'T' and m < n, rows 1 to M of B contain the

least squares solution vectors.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= MAX(1,M,N).

*WORK*

(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.

*LWORK*

LWORK is INTEGER

The dimension of the array WORK.

If LWORK = -1 or -2, then a workspace query is assumed.

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).

*INFO*

INFO is INTEGER

= 0: successful exit

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

> 0: if INFO = i, the i-th diagonal element of the

triangular factor of A is zero, so that A does not have

full rank; the least squares solution could not be

computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine dsgesv (integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldx, * ) X, integer LDX, double precision, dimension( n, * ) WORK, real, dimension( * ) SWORK, integer ITER, integer INFO)¶

** DSGESV computes the solution to system of linear equations A *
X = B for GE matrices** (mixed precision with iterative refinement)

**Purpose:**

DSGESV computes the solution to a real system of linear equations

A * X = B,

where A is an N-by-N matrix and X and B are N-by-NRHS matrices.

DSGESV first attempts to factorize the matrix in SINGLE PRECISION

and use this factorization within an iterative refinement procedure

to produce a solution with DOUBLE PRECISION normwise backward error

quality (see below). If the approach fails the method switches to a

DOUBLE PRECISION factorization and solve.

The iterative refinement is not going to be a winning strategy if

the ratio SINGLE PRECISION performance over DOUBLE PRECISION

performance is too small. A reasonable strategy should take the

number of right-hand sides and the size of the matrix into account.

This might be done with a call to ILAENV in the future. Up to now, we

always try iterative refinement.

The iterative refinement process is stopped if

ITER > ITERMAX

or for all the RHS we have:

RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX

where

o ITER is the number of the current iteration in the iterative

refinement process

o RNRM is the infinity-norm of the residual

o XNRM is the infinity-norm of the solution

o ANRM is the infinity-operator-norm of the matrix A

o EPS is the machine epsilon returned by DLAMCH('Epsilon')

The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00

respectively.

**Parameters**

*N*

N is INTEGER

The number of linear equations, i.e., the order of the

matrix A. N >= 0.

*NRHS*

NRHS is INTEGER

The number of right hand sides, i.e., the number of columns

of the matrix B. NRHS >= 0.

*A*

A is DOUBLE PRECISION array,

dimension (LDA,N)

On entry, the N-by-N coefficient matrix A.

On exit, if iterative refinement has been successfully used

(INFO = 0 and ITER >= 0, see description below), then A is

unchanged, if double precision factorization has been used

(INFO = 0 and ITER < 0, see description below), then the

array A contains 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,N).

*IPIV*

IPIV is INTEGER array, dimension (N)

The pivot indices that define the permutation matrix P;

row i of the matrix was interchanged with row IPIV(i).

Corresponds either to the single precision factorization

(if INFO = 0 and ITER >= 0) or the double precision

factorization (if INFO = 0 and ITER < 0).

*B*

B is DOUBLE PRECISION array, dimension (LDB,NRHS)

The N-by-NRHS right hand side matrix B.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

*X*

X is DOUBLE PRECISION array, dimension (LDX,NRHS)

If INFO = 0, the N-by-NRHS solution matrix X.

*LDX*

LDX is INTEGER

The leading dimension of the array X. LDX >= max(1,N).

*WORK*

WORK is DOUBLE PRECISION array, dimension (N,NRHS)

This array is used to hold the residual vectors.

*SWORK*

SWORK is REAL array, dimension (N*(N+NRHS))

This array is used to use the single precision matrix and the

right-hand sides or solutions in single precision.

*ITER*

ITER is INTEGER

< 0: iterative refinement has failed, double precision

factorization has been performed

-1 : the routine fell back to full precision for

implementation- or machine-specific reasons

-2 : narrowing the precision induced an overflow,

the routine fell back to full precision

-3 : failure of SGETRF

-31: stop the iterative refinement after the 30th

iterations

> 0: iterative refinement has been successfully used.

Returns the number of iterations

*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) computed in DOUBLE PRECISION is

exactly zero. The factorization has been completed,

but the factor U is exactly singular, so the solution

could not be computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

# Author¶

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