CGELS solves overdetermined or underdetermined complex linear systems


 involving an M-by-N matrix A, or its conjugate-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 = 'C' and m >= n:  find the minimum norm solution of


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


 4. If TRANS = 'C' and m < n:  find the least squares solution of


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


                 minimize || B - A**H * 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.

TRANS

          TRANS is CHARACTER*1


          = 'N': the linear system involves A;


          = 'C': the linear system involves A**H.

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 COMPLEX array, dimension (LDA,N)


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


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


                       factorization as returned by CGEQRF;


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


                       factorization as returned by CGELQF.

LDA

          LDA is INTEGER


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

B

          B is COMPLEX 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 = 'C'.


          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 the


          modulus 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 = 'C' and m >= n, rows 1 to M of B contain the


          minimum norm solution vectors;


          if TRANS = 'C' 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 the modulus 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 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.


          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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 CGELSD 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.

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 COMPLEX 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 COMPLEX 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 the modulus of elements n+1:m in that column.

LDB

          LDB is INTEGER


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

S

          S is REAL 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 REAL


          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 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. 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


              2 * N + N * NRHS


          if M is greater than or equal to N or


              2 * M + M * NRHS


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


          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 array WORK and the


          minimum sizes of the arrays RWORK and IWORK, and returns


          these values as the first entries of the WORK, RWORK and


          IWORK arrays, and no error message related to LWORK is issued


          by XERBLA.

RWORK

          RWORK is REAL array, dimension (MAX(1,LRWORK))


          LRWORK >=


             10*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +


             MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )


          if M is greater than or equal to N or


             10*M + 2*M*SMLSIZ + 8*M*NLVL + 3*SMLSIZ*NRHS +


             MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )


          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 )


          On exit, if INFO = 0, RWORK(1) returns the minimum LRWORK.

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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA

 CGELSS computes the minimum norm solution to a complex 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.

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 COMPLEX 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 COMPLEX 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 the modulus of elements n+1:m in that column.

LDB

          LDB is INTEGER


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

S

          S is REAL 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 REAL


          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 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. LWORK >= 1, and also:


          LWORK >=  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.

RWORK

          RWORK is REAL array, dimension (5*min(M,N))

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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 CGELST solves overdetermined or underdetermined real linear systems


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


 or LQ factorization of A with compact WY representation of Q.


 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 = 'C' and m >= n:  find the minimum norm solution of


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


 4. If TRANS = 'C' 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.

TRANS

          TRANS is CHARACTER*1


          = 'N': the linear system involves A;


          = 'C': the linear system involves A**H.

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 COMPLEX 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 CGEQRT;


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


                       factorization as returned by CGELQT.

LDA

          LDA is INTEGER


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

B

          B is COMPLEX 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 = 'C'.


          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


          modulus 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 = 'C' and m >= n, rows 1 to M of B contain the


          minimum norm solution vectors;


          if TRANS = 'C' 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 the modulus 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 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.


          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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

  November 2022,  Igor Kozachenko,


                  Computer Science Division,


                  University of California, Berkeley

 CGELSY computes the minimum-norm solution to a complex 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 unitary 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**H [ inv(T11)*Q1**H*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 permutation of matrix B (the right hand side) is faster and


     more simple.


   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.

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 COMPLEX 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 COMPLEX 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 A*P


          was the k-th column of A.

RCOND

          RCOND is REAL


          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 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 unblocked strategy requires that:


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


          where MN = min(M,N).


          The block algorithm requires that:


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


          where NB is an upper bound on the blocksize returned


          by ILAENV for the routines CGEQP3, CTZRZF, CTZRQF, CUNMQR,


          and CUNMRZ.


          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.

RWORK

          RWORK is REAL array, dimension (2*N)

INFO

          INFO is INTEGER


          = 0: successful exit


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

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain

 CGESV computes the solution to a complex 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.

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 COMPLEX 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 COMPLEX 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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 CGESVX uses the LU factorization to compute the solution to a complex


 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.

 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.

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)

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 COMPLEX 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 COMPLEX 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 CGETRF.  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 CGETRF; 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 REAL 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 REAL 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 COMPLEX 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 COMPLEX 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 REAL


          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 REAL 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 REAL 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 COMPLEX array, dimension (2*N)

RWORK

          RWORK is REAL array, dimension (2*N)


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


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


          used. If RWORK(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


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


          leading INFO columns of A.

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.

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    CGESVXX uses the LU factorization to compute the solution to a


    complex 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. CGESVXX 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.


    CGESVXX 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


    CGESVXX 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 CGESVXX would itself produce.

    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 (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.

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)

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 COMPLEX 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 COMPLEX 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 CGETRF.  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 CGETRF; 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 REAL 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 REAL 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 COMPLEX 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 COMPLEX 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 REAL


     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 REAL


     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 CGESVX, this quantity is


     returned in WORK(1).

BERR

          BERR is REAL 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 REAL 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) * slamch('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) * slamch('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) * slamch('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 REAL 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) * slamch('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) * slamch('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) * slamch('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 REAL 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: 1.0


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


                    computed.


            = 1.0:  Use the double-precision refinement algorithm,


                    possibly with doubled-single computations if the


                    compilation environment does not support DOUBLE


                    PRECISION.


              (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 COMPLEX array, dimension (2*N)

RWORK

          RWORK is REAL array, dimension (2*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.

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NAG Ltd.

 CGETSLS solves overdetermined or underdetermined complex 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 = 'C' and m >= n:  find the minimum norm solution of


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


 4. If TRANS = 'C' 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.

TRANS

          TRANS is CHARACTER*1


          = 'N': the linear system involves A;


          = 'C': the linear system involves A**H.

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 COMPLEX 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 CGEQR or CGELQ.

LDA

          LDA is INTEGER


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

B

          B is COMPLEX 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 = 'C'.


          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 = 'C' and m >= n, rows 1 to M of B contain the


          minimum norm solution vectors;


          if TRANS = 'C' 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) COMPLEX array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) contains optimal (or either minimal


          or optimal, if query was assumed) LWORK.


          See LWORK for details.

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.

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