NAME

SYNOPSIS

  SUBROUTINE CGELSX( M, N, NRHS, A, LDA, B, LDB, JPIVOT, RCOND, IRANK, 
 *      WORK, WORK2, INFO)
  COMPLEX A(LDA,*), B(LDB,*), WORK(*)
  INTEGER M, N, NRHS, LDA, LDB, IRANK, INFO
  INTEGER JPIVOT(*)
  REAL RCOND
  REAL WORK2(*)
 
  SUBROUTINE CGELSX_64( M, N, NRHS, A, LDA, B, LDB, JPIVOT, RCOND, 
 *      IRANK, WORK, WORK2, INFO)
  COMPLEX A(LDA,*), B(LDB,*), WORK(*)
  INTEGER*8 M, N, NRHS, LDA, LDB, IRANK, INFO
  INTEGER*8 JPIVOT(*)
  REAL RCOND
  REAL WORK2(*)
 

F95 INTERFACE

  SUBROUTINE GELSX( [M], [N], [NRHS], A, [LDA], B, [LDB], JPIVOT, 
 *       RCOND, IRANK, [WORK], [WORK2], [INFO])
  COMPLEX, DIMENSION(:) :: WORK
  COMPLEX, DIMENSION(:,:) :: A, B
  INTEGER :: M, N, NRHS, LDA, LDB, IRANK, INFO
  INTEGER, DIMENSION(:) :: JPIVOT
  REAL :: RCOND
  REAL, DIMENSION(:) :: WORK2
 
  SUBROUTINE GELSX_64( [M], [N], [NRHS], A, [LDA], B, [LDB], JPIVOT, 
 *       RCOND, IRANK, [WORK], [WORK2], [INFO])
  COMPLEX, DIMENSION(:) :: WORK
  COMPLEX, DIMENSION(:,:) :: A, B
  INTEGER(8) :: M, N, NRHS, LDA, LDB, IRANK, INFO
  INTEGER(8), DIMENSION(:) :: JPIVOT
  REAL :: RCOND
  REAL, DIMENSION(:) :: WORK2
 

C INTERFACE

void cgelsx(int m, int n, int nrhs, complex *a, int lda, complex *b, int ldb, int *jpivot, float rcond, int *irank, int *info);

void cgelsx_64(long m, long n, long nrhs, complex *a, long lda, complex *b, long ldb, long *jpivot, float rcond, long *irank, long *info);

PURPOSE

CGELSX 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' [ inv(T11)*Q1'*B ]
              [        0       ]

where Q1 consists of the first RANK columns of Q.

ARGUMENTS

* M (input)

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

* N (input)

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

* NRHS (input)

The number of right hand sides, i.e., the number of columns of matrices B and X. NRHS >= 0.

* A (input/output)

On entry, the M-by-N matrix A. On exit, A has been overwritten by details of its complete orthogonal factorization.

* LDA (input)

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

* B (input/output)

On entry, the M-by-NRHS right hand side matrix B. On exit, the N-by-NRHS solution matrix X. If m >= n and IRANK = 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 (input)

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

* JPIVOT (input)

On entry, if JPIVOT(i) .ne. 0, the i-th column of A is an initial column, otherwise it is a free column. Before the QR factorization of A, all initial columns are permuted to the leading positions; only the remaining free columns are moved as a result of column pivoting during the factorization. On exit, if JPIVOT(i) = k, then the i-th column of A*P was the k-th column of A.

* RCOND (input)

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.

* IRANK (output)

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 (workspace)

(min(M,N) + max( N, 2*min(M,N)+NRHS )),

* WORK2 (workspace)

* INFO (output)