**Purpose**

To calculate an LQ factorization of the first block row and apply the orthogonal transformations (from the right) also to the second block row of a structured matrix, as follows _ [ L A ] [ L 0 ] [ ]*Q = [ ] [ 0 B ] [ C D ] _ where L and L are lower triangular. The matrix A can be full or lower trapezoidal/triangular. The problem structure is exploited. This computation is useful, for instance, in combined measurement and time update of one iteration of the Kalman filter (square root covariance filter).

SUBROUTINE MB04LD( UPLO, N, M, P, L, LDL, A, LDA, B, LDB, C, LDC, $ TAU, DWORK ) C .. Scalar Arguments .. CHARACTER UPLO INTEGER LDA, LDB, LDC, LDL, M, N, P C .. Array Arguments .. DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), $ L(LDL,*), TAU(*)

**Mode Parameters**

UPLO CHARACTER*1 Indicates if the matrix A is or not triangular as follows: = 'L': Matrix A is lower trapezoidal/triangular; = 'F': Matrix A is full.

N (input) INTEGER _ The order of the matrices L and L. N >= 0. M (input) INTEGER The number of columns of the matrices A, B and D. M >= 0. P (input) INTEGER The number of rows of the matrices B, C and D. P >= 0. L (input/output) DOUBLE PRECISION array, dimension (LDL,N) On entry, the leading N-by-N lower triangular part of this array must contain the lower triangular matrix L. On exit, the leading N-by-N lower triangular part of this _ array contains the lower triangular matrix L. The strict upper triangular part of this array is not referenced. LDL INTEGER The leading dimension of array L. LDL >= MAX(1,N). A (input/output) DOUBLE PRECISION array, dimension (LDA,M) On entry, if UPLO = 'F', the leading N-by-M part of this array must contain the matrix A. If UPLO = 'L', the leading N-by-MIN(N,M) part of this array must contain the lower trapezoidal (lower triangular if N <= M) matrix A, and the elements above the diagonal are not referenced. On exit, the leading N-by-M part (lower trapezoidal or triangular, if UPLO = 'L') of this array contains the trailing components (the vectors v, see Method) of the elementary reflectors used in the factorization. LDA INTEGER The leading dimension of array A. LDA >= MAX(1,N). B (input/output) DOUBLE PRECISION array, dimension (LDB,M) On entry, the leading P-by-M part of this array must contain the matrix B. On exit, the leading P-by-M part of this array contains the computed matrix D. LDB INTEGER The leading dimension of array B. LDB >= MAX(1,P). C (output) DOUBLE PRECISION array, dimension (LDC,N) The leading P-by-N part of this array contains the computed matrix C. LDC INTEGER The leading dimension of array C. LDC >= MAX(1,P). TAU (output) DOUBLE PRECISION array, dimension (N) The scalar factors of the elementary reflectors used.

DWORK DOUBLE PRECISION array, dimension (N)

The routine uses N Householder transformations exploiting the zero pattern of the block matrix. A Householder matrix has the form ( 1 ), H = I - tau *u *u', u = ( v ) i i i i i ( i) where v is an M-vector, if UPLO = 'F', or an min(i,M)-vector, if i UPLO = 'L'. The components of v are stored in the i-th row of A, i and tau is stored in TAU(i). i

The algorithm is backward stable.

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**Program Text**

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