**Purpose**

To compute a QR factorization of an n-by-m matrix A (A = Q * R), having a p-by-min(p,m) zero triangle in the lower left-hand side corner, as shown below, for n = 8, m = 7, and p = 2: [ x x x x x x x ] [ x x x x x x x ] [ x x x x x x x ] [ x x x x x x x ] A = [ x x x x x x x ], [ x x x x x x x ] [ 0 x x x x x x ] [ 0 0 x x x x x ] and optionally apply the transformations to an n-by-l matrix B (from the left). The problem structure is exploited. This computation is useful, for instance, in combined measurement and time update of one iteration of the time-invariant Kalman filter (square root information filter).

SUBROUTINE MB04IZ( N, M, P, L, A, LDA, B, LDB, TAU, ZWORK, LZWORK, $ INFO ) C .. Scalar Arguments .. INTEGER INFO, L, LDA, LDB, LZWORK, M, N, P C .. Array Arguments .. COMPLEX*16 A(LDA,*), B(LDB,*), TAU(*), ZWORK(*)

**Input/Output Parameters**

N (input) INTEGER The number of rows of the matrix A. N >= 0. M (input) INTEGER The number of columns of the matrix A. M >= 0. P (input) INTEGER The order of the zero triagle. P >= 0. L (input) INTEGER The number of columns of the matrix B. L >= 0. A (input/output) COMPLEX*16 array, dimension (LDA,M) On entry, the leading N-by-M part of this array must contain the matrix A. The elements corresponding to the zero P-by-MIN(P,M) lower trapezoidal/triangular part (if P > 0) are not referenced. On exit, the elements on and above the diagonal of this array contain the MIN(N,M)-by-M upper trapezoidal matrix R (R is upper triangular, if N >= M) of the QR factorization, and the relevant elements below the diagonal contain 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) COMPLEX*16 array, dimension (LDB,L) On entry, the leading N-by-L part of this array must contain the matrix B. On exit, the leading N-by-L part of this array contains the updated matrix B. If L = 0, this array is not referenced. LDB INTEGER The leading dimension of array B. LDB >= MAX(1,N) if L > 0; LDB >= 1 if L = 0. TAU (output) COMPLEX*16 array, dimension MIN(N,M) The scalar factors of the elementary reflectors used.

ZWORK COMPLEX*16 array, dimension (LZWORK) On exit, if INFO = 0, ZWORK(1) returns the optimal value of LZWORK. LZWORK The length of the array ZWORK. LZWORK >= MAX(1,M-1,M-P,L). For optimum performance LZWORK should be larger. If LZWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the ZWORK array, returns this value as the first entry of the ZWORK array, and no error message related to LZWORK is issued by XERBLA.

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

The routine uses min(N,M) Householder transformations exploiting the zero pattern of the 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 (N-P+I-2)-vector. The components of v are stored i i in the i-th column of A, beginning from the location i+1, and tau is stored in TAU(i). i

The algorithm is backward stable.

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**Program Text**

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