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SUBROUTINE ZLATTR( IMAT, UPLO, TRANS, DIAG, ISEED, N, A, LDA, B,
$ WORK, RWORK, INFO ) * * -- LAPACK test routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. CHARACTER DIAG, TRANS, UPLO INTEGER IMAT, INFO, LDA, N * .. * .. Array Arguments .. INTEGER ISEED( 4 ) DOUBLE PRECISION RWORK( * ) COMPLEX*16 A( LDA, * ), B( * ), WORK( * ) * .. * * Purpose * ======= * * ZLATTR generates a triangular test matrix in 2-dimensional storage. * IMAT and UPLO uniquely specify the properties of the test matrix, * which is returned in the array A. * * Arguments * ========= * * IMAT (input) INTEGER * An integer key describing which matrix to generate for this * path. * * UPLO (input) CHARACTER*1 * Specifies whether the matrix A will be upper or lower * triangular. * = 'U': Upper triangular * = 'L': Lower triangular * * TRANS (input) CHARACTER*1 * Specifies whether the matrix or its transpose will be used. * = 'N': No transpose * = 'T': Transpose * = 'C': Conjugate transpose * * DIAG (output) CHARACTER*1 * Specifies whether or not the matrix A is unit triangular. * = 'N': Non-unit triangular * = 'U': Unit triangular * * ISEED (input/output) INTEGER array, dimension (4) * The seed vector for the random number generator (used in * ZLATMS). Modified on exit. * * N (input) INTEGER * The order of the matrix to be generated. * * A (output) COMPLEX*16 array, dimension (LDA,N) * The triangular matrix A. If UPLO = 'U', the leading N x N * upper triangular part of the array A contains the upper * triangular matrix, and the strictly lower triangular part of * A is not referenced. If UPLO = 'L', the leading N x N lower * triangular part of the array A contains the lower triangular * matrix and the strictly upper triangular part of A is not * referenced. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * B (output) COMPLEX*16 array, dimension (N) * The right hand side vector, if IMAT > 10. * * WORK (workspace) COMPLEX*16 array, dimension (2*N) * * RWORK (workspace) DOUBLE PRECISION array, dimension (N) * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ONE, TWO, ZERO PARAMETER ( ONE = 1.0D+0, TWO = 2.0D+0, ZERO = 0.0D+0 ) * .. * .. Local Scalars .. LOGICAL UPPER CHARACTER DIST, TYPE CHARACTER*3 PATH INTEGER I, IY, J, JCOUNT, KL, KU, MODE DOUBLE PRECISION ANORM, BIGNUM, BNORM, BSCAL, C, CNDNUM, REXP, $ SFAC, SMLNUM, TEXP, TLEFT, TSCAL, ULP, UNFL, X, $ Y, Z COMPLEX*16 PLUS1, PLUS2, RA, RB, S, STAR1 * .. * .. External Functions .. LOGICAL LSAME INTEGER IZAMAX DOUBLE PRECISION DLAMCH, DLARND COMPLEX*16 ZLARND EXTERNAL LSAME, IZAMAX, DLAMCH, DLARND, ZLARND * .. * .. External Subroutines .. EXTERNAL DLABAD, DLARNV, ZCOPY, ZDSCAL, ZLARNV, ZLATB4, $ ZLATMS, ZROT, ZROTG, ZSWAP * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, DCMPLX, DCONJG, MAX, SQRT * .. * .. Executable Statements .. * PATH( 1: 1 ) = 'Zomplex precision' PATH( 2: 3 ) = 'TR' UNFL = DLAMCH( 'Safe minimum' ) ULP = DLAMCH( 'Epsilon' )*DLAMCH( 'Base' ) SMLNUM = UNFL BIGNUM = ( ONE-ULP ) / SMLNUM CALL DLABAD( SMLNUM, BIGNUM ) IF( ( IMAT.GE.7 .AND. IMAT.LE.10 ) .OR. IMAT.EQ.18 ) THEN DIAG = 'U' ELSE DIAG = 'N' END IF INFO = 0 * * Quick return if N.LE.0. * IF( N.LE.0 ) $ RETURN * * Call ZLATB4 to set parameters for CLATMS. * UPPER = LSAME( UPLO, 'U' ) IF( UPPER ) THEN CALL ZLATB4( PATH, IMAT, N, N, TYPE, KL, KU, ANORM, MODE, $ CNDNUM, DIST ) ELSE CALL ZLATB4( PATH, -IMAT, N, N, TYPE, KL, KU, ANORM, MODE, $ CNDNUM, DIST ) END IF * * IMAT <= 6: Non-unit triangular matrix * IF( IMAT.LE.6 ) THEN CALL ZLATMS( N, N, DIST, ISEED, TYPE, RWORK, MODE, CNDNUM, $ ANORM, KL, KU, 'No packing', A, LDA, WORK, INFO ) * * IMAT > 6: Unit triangular matrix * The diagonal is deliberately set to something other than 1. * * IMAT = 7: Matrix is the identity * ELSE IF( IMAT.EQ.7 ) THEN IF( UPPER ) THEN DO 20 J = 1, N DO 10 I = 1, J - 1 A( I, J ) = ZERO 10 CONTINUE A( J, J ) = J 20 CONTINUE ELSE DO 40 J = 1, N A( J, J ) = J DO 30 I = J + 1, N A( I, J ) = ZERO 30 CONTINUE 40 CONTINUE END IF * * IMAT > 7: Non-trivial unit triangular matrix * * Generate a unit triangular matrix T with condition CNDNUM by * forming a triangular matrix with known singular values and * filling in the zero entries with Givens rotations. * ELSE IF( IMAT.LE.10 ) THEN IF( UPPER ) THEN DO 60 J = 1, N DO 50 I = 1, J - 1 A( I, J ) = ZERO 50 CONTINUE A( J, J ) = J 60 CONTINUE ELSE DO 80 J = 1, N A( J, J ) = J DO 70 I = J + 1, N A( I, J ) = ZERO 70 CONTINUE 80 CONTINUE END IF * * Since the trace of a unit triangular matrix is 1, the product * of its singular values must be 1. Let s = sqrt(CNDNUM), * x = sqrt(s) - 1/sqrt(s), y = sqrt(2/(n-2))*x, and z = x**2. * The following triangular matrix has singular values s, 1, 1, * ..., 1, 1/s: * * 1 y y y ... y y z * 1 0 0 ... 0 0 y * 1 0 ... 0 0 y * . ... . . . * . . . . * 1 0 y * 1 y * 1 * * To fill in the zeros, we first multiply by a matrix with small * condition number of the form * * 1 0 0 0 0 ... * 1 + * 0 0 ... * 1 + 0 0 0 * 1 + * 0 0 * 1 + 0 0 * ... * 1 + 0 * 1 0 * 1 * * Each element marked with a '*' is formed by taking the product * of the adjacent elements marked with '+'. The '*'s can be * chosen freely, and the '+'s are chosen so that the inverse of * T will have elements of the same magnitude as T. If the *'s in * both T and inv(T) have small magnitude, T is well conditioned. * The two offdiagonals of T are stored in WORK. * * The product of these two matrices has the form * * 1 y y y y y . y y z * 1 + * 0 0 . 0 0 y * 1 + 0 0 . 0 0 y * 1 + * . . . . * 1 + . . . . * . . . . . * . . . . * 1 + y * 1 y * 1 * * Now we multiply by Givens rotations, using the fact that * * [ c s ] [ 1 w ] [ -c -s ] = [ 1 -w ] * [ -s c ] [ 0 1 ] [ s -c ] [ 0 1 ] * and * [ -c -s ] [ 1 0 ] [ c s ] = [ 1 0 ] * [ s -c ] [ w 1 ] [ -s c ] [ -w 1 ] * * where c = w / sqrt(w**2+4) and s = 2 / sqrt(w**2+4). * STAR1 = 0.25D0*ZLARND( 5, ISEED ) SFAC = 0.5D0 PLUS1 = SFAC*ZLARND( 5, ISEED ) DO 90 J = 1, N, 2 PLUS2 = STAR1 / PLUS1 WORK( J ) = PLUS1 WORK( N+J ) = STAR1 IF( J+1.LE.N ) THEN WORK( J+1 ) = PLUS2 WORK( N+J+1 ) = ZERO PLUS1 = STAR1 / PLUS2 REXP = DLARND( 2, ISEED ) IF( REXP.LT.ZERO ) THEN STAR1 = -SFAC**( ONE-REXP )*ZLARND( 5, ISEED ) ELSE STAR1 = SFAC**( ONE+REXP )*ZLARND( 5, ISEED ) END IF END IF 90 CONTINUE * X = SQRT( CNDNUM ) - 1 / SQRT( CNDNUM ) IF( N.GT.2 ) THEN Y = SQRT( 2.D0 / ( N-2 ) )*X ELSE Y = ZERO END IF Z = X*X * IF( UPPER ) THEN IF( N.GT.3 ) THEN CALL ZCOPY( N-3, WORK, 1, A( 2, 3 ), LDA+1 ) IF( N.GT.4 ) $ CALL ZCOPY( N-4, WORK( N+1 ), 1, A( 2, 4 ), LDA+1 ) END IF DO 100 J = 2, N - 1 A( 1, J ) = Y A( J, N ) = Y 100 CONTINUE A( 1, N ) = Z ELSE IF( N.GT.3 ) THEN CALL ZCOPY( N-3, WORK, 1, A( 3, 2 ), LDA+1 ) IF( N.GT.4 ) $ CALL ZCOPY( N-4, WORK( N+1 ), 1, A( 4, 2 ), LDA+1 ) END IF DO 110 J = 2, N - 1 A( J, 1 ) = Y A( N, J ) = Y 110 CONTINUE A( N, 1 ) = Z END IF * * Fill in the zeros using Givens rotations. * IF( UPPER ) THEN DO 120 J = 1, N - 1 RA = A( J, J+1 ) RB = 2.0D0 CALL ZROTG( RA, RB, C, S ) * * Multiply by [ c s; -conjg(s) c] on the left. * IF( N.GT.J+1 ) $ CALL ZROT( N-J-1, A( J, J+2 ), LDA, A( J+1, J+2 ), $ LDA, C, S ) * * Multiply by [-c -s; conjg(s) -c] on the right. * IF( J.GT.1 ) $ CALL ZROT( J-1, A( 1, J+1 ), 1, A( 1, J ), 1, -C, -S ) * * Negate A(J,J+1). * A( J, J+1 ) = -A( J, J+1 ) 120 CONTINUE ELSE DO 130 J = 1, N - 1 RA = A( J+1, J ) RB = 2.0D0 CALL ZROTG( RA, RB, C, S ) S = DCONJG( S ) * * Multiply by [ c -s; conjg(s) c] on the right. * IF( N.GT.J+1 ) $ CALL ZROT( N-J-1, A( J+2, J+1 ), 1, A( J+2, J ), 1, C, $ -S ) * * Multiply by [-c s; -conjg(s) -c] on the left. * IF( J.GT.1 ) $ CALL ZROT( J-1, A( J, 1 ), LDA, A( J+1, 1 ), LDA, -C, $ S ) * * Negate A(J+1,J). * A( J+1, J ) = -A( J+1, J ) 130 CONTINUE END IF * * IMAT > 10: Pathological test cases. These triangular matrices * are badly scaled or badly conditioned, so when used in solving a * triangular system they may cause overflow in the solution vector. * ELSE IF( IMAT.EQ.11 ) THEN * * Type 11: Generate a triangular matrix with elements between * -1 and 1. Give the diagonal norm 2 to make it well-conditioned. * Make the right hand side large so that it requires scaling. * IF( UPPER ) THEN DO 140 J = 1, N CALL ZLARNV( 4, ISEED, J-1, A( 1, J ) ) A( J, J ) = ZLARND( 5, ISEED )*TWO 140 CONTINUE ELSE DO 150 J = 1, N IF( J.LT.N ) $ CALL ZLARNV( 4, ISEED, N-J, A( J+1, J ) ) A( J, J ) = ZLARND( 5, ISEED )*TWO 150 CONTINUE END IF * * Set the right hand side so that the largest value is BIGNUM. * CALL ZLARNV( 2, ISEED, N, B ) IY = IZAMAX( N, B, 1 ) BNORM = ABS( B( IY ) ) BSCAL = BIGNUM / MAX( ONE, BNORM ) CALL ZDSCAL( N, BSCAL, B, 1 ) * ELSE IF( IMAT.EQ.12 ) THEN * * Type 12: Make the first diagonal element in the solve small to * cause immediate overflow when dividing by T(j,j). * In type 12, the offdiagonal elements are small (CNORM(j) < 1). * CALL ZLARNV( 2, ISEED, N, B ) TSCAL = ONE / MAX( ONE, DBLE( N-1 ) ) IF( UPPER ) THEN DO 160 J = 1, N CALL ZLARNV( 4, ISEED, J-1, A( 1, J ) ) CALL ZDSCAL( J-1, TSCAL, A( 1, J ), 1 ) A( J, J ) = ZLARND( 5, ISEED ) 160 CONTINUE A( N, N ) = SMLNUM*A( N, N ) ELSE DO 170 J = 1, N IF( J.LT.N ) THEN CALL ZLARNV( 4, ISEED, N-J, A( J+1, J ) ) CALL ZDSCAL( N-J, TSCAL, A( J+1, J ), 1 ) END IF A( J, J ) = ZLARND( 5, ISEED ) 170 CONTINUE A( 1, 1 ) = SMLNUM*A( 1, 1 ) END IF * ELSE IF( IMAT.EQ.13 ) THEN * * Type 13: Make the first diagonal element in the solve small to * cause immediate overflow when dividing by T(j,j). * In type 13, the offdiagonal elements are O(1) (CNORM(j) > 1). * CALL ZLARNV( 2, ISEED, N, B ) IF( UPPER ) THEN DO 180 J = 1, N CALL ZLARNV( 4, ISEED, J-1, A( 1, J ) ) A( J, J ) = ZLARND( 5, ISEED ) 180 CONTINUE A( N, N ) = SMLNUM*A( N, N ) ELSE DO 190 J = 1, N IF( J.LT.N ) $ CALL ZLARNV( 4, ISEED, N-J, A( J+1, J ) ) A( J, J ) = ZLARND( 5, ISEED ) 190 CONTINUE A( 1, 1 ) = SMLNUM*A( 1, 1 ) END IF * ELSE IF( IMAT.EQ.14 ) THEN * * Type 14: T is diagonal with small numbers on the diagonal to * make the growth factor underflow, but a small right hand side * chosen so that the solution does not overflow. * IF( UPPER ) THEN JCOUNT = 1 DO 210 J = N, 1, -1 DO 200 I = 1, J - 1 A( I, J ) = ZERO 200 CONTINUE IF( JCOUNT.LE.2 ) THEN A( J, J ) = SMLNUM*ZLARND( 5, ISEED ) ELSE A( J, J ) = ZLARND( 5, ISEED ) END IF JCOUNT = JCOUNT + 1 IF( JCOUNT.GT.4 ) $ JCOUNT = 1 210 CONTINUE ELSE JCOUNT = 1 DO 230 J = 1, N DO 220 I = J + 1, N A( I, J ) = ZERO 220 CONTINUE IF( JCOUNT.LE.2 ) THEN A( J, J ) = SMLNUM*ZLARND( 5, ISEED ) ELSE A( J, J ) = ZLARND( 5, ISEED ) END IF JCOUNT = JCOUNT + 1 IF( JCOUNT.GT.4 ) $ JCOUNT = 1 230 CONTINUE END IF * * Set the right hand side alternately zero and small. * IF( UPPER ) THEN B( 1 ) = ZERO DO 240 I = N, 2, -2 B( I ) = ZERO B( I-1 ) = SMLNUM*ZLARND( 5, ISEED ) 240 CONTINUE ELSE B( N ) = ZERO DO 250 I = 1, N - 1, 2 B( I ) = ZERO B( I+1 ) = SMLNUM*ZLARND( 5, ISEED ) 250 CONTINUE END IF * ELSE IF( IMAT.EQ.15 ) THEN * * Type 15: Make the diagonal elements small to cause gradual * overflow when dividing by T(j,j). To control the amount of * scaling needed, the matrix is bidiagonal. * TEXP = ONE / MAX( ONE, DBLE( N-1 ) ) TSCAL = SMLNUM**TEXP CALL ZLARNV( 4, ISEED, N, B ) IF( UPPER ) THEN DO 270 J = 1, N DO 260 I = 1, J - 2 A( I, J ) = 0.D0 260 CONTINUE IF( J.GT.1 ) $ A( J-1, J ) = DCMPLX( -ONE, -ONE ) A( J, J ) = TSCAL*ZLARND( 5, ISEED ) 270 CONTINUE B( N ) = DCMPLX( ONE, ONE ) ELSE DO 290 J = 1, N DO 280 I = J + 2, N A( I, J ) = 0.D0 280 CONTINUE IF( J.LT.N ) $ A( J+1, J ) = DCMPLX( -ONE, -ONE ) A( J, J ) = TSCAL*ZLARND( 5, ISEED ) 290 CONTINUE B( 1 ) = DCMPLX( ONE, ONE ) END IF * ELSE IF( IMAT.EQ.16 ) THEN * * Type 16: One zero diagonal element. * IY = N / 2 + 1 IF( UPPER ) THEN DO 300 J = 1, N CALL ZLARNV( 4, ISEED, J-1, A( 1, J ) ) IF( J.NE.IY ) THEN A( J, J ) = ZLARND( 5, ISEED )*TWO ELSE A( J, J ) = ZERO END IF 300 CONTINUE ELSE DO 310 J = 1, N IF( J.LT.N ) $ CALL ZLARNV( 4, ISEED, N-J, A( J+1, J ) ) IF( J.NE.IY ) THEN A( J, J ) = ZLARND( 5, ISEED )*TWO ELSE A( J, J ) = ZERO END IF 310 CONTINUE END IF CALL ZLARNV( 2, ISEED, N, B ) CALL ZDSCAL( N, TWO, B, 1 ) * ELSE IF( IMAT.EQ.17 ) THEN * * Type 17: Make the offdiagonal elements large to cause overflow * when adding a column of T. In the non-transposed case, the * matrix is constructed to cause overflow when adding a column in * every other step. * TSCAL = UNFL / ULP TSCAL = ( ONE-ULP ) / TSCAL DO 330 J = 1, N DO 320 I = 1, N A( I, J ) = 0.D0 320 CONTINUE 330 CONTINUE TEXP = ONE IF( UPPER ) THEN DO 340 J = N, 2, -2 A( 1, J ) = -TSCAL / DBLE( N+1 ) A( J, J ) = ONE B( J ) = TEXP*( ONE-ULP ) A( 1, J-1 ) = -( TSCAL / DBLE( N+1 ) ) / DBLE( N+2 ) A( J-1, J-1 ) = ONE B( J-1 ) = TEXP*DBLE( N*N+N-1 ) TEXP = TEXP*2.D0 340 CONTINUE B( 1 ) = ( DBLE( N+1 ) / DBLE( N+2 ) )*TSCAL ELSE DO 350 J = 1, N - 1, 2 A( N, J ) = -TSCAL / DBLE( N+1 ) A( J, J ) = ONE B( J ) = TEXP*( ONE-ULP ) A( N, J+1 ) = -( TSCAL / DBLE( N+1 ) ) / DBLE( N+2 ) A( J+1, J+1 ) = ONE B( J+1 ) = TEXP*DBLE( N*N+N-1 ) TEXP = TEXP*2.D0 350 CONTINUE B( N ) = ( DBLE( N+1 ) / DBLE( N+2 ) )*TSCAL END IF * ELSE IF( IMAT.EQ.18 ) THEN * * Type 18: Generate a unit triangular matrix with elements * between -1 and 1, and make the right hand side large so that it * requires scaling. * IF( UPPER ) THEN DO 360 J = 1, N CALL ZLARNV( 4, ISEED, J-1, A( 1, J ) ) A( J, J ) = ZERO 360 CONTINUE ELSE DO 370 J = 1, N IF( J.LT.N ) $ CALL ZLARNV( 4, ISEED, N-J, A( J+1, J ) ) A( J, J ) = ZERO 370 CONTINUE END IF * * Set the right hand side so that the largest value is BIGNUM. * CALL ZLARNV( 2, ISEED, N, B ) IY = IZAMAX( N, B, 1 ) BNORM = ABS( B( IY ) ) BSCAL = BIGNUM / MAX( ONE, BNORM ) CALL ZDSCAL( N, BSCAL, B, 1 ) * ELSE IF( IMAT.EQ.19 ) THEN * * Type 19: Generate a triangular matrix with elements between * BIGNUM/(n-1) and BIGNUM so that at least one of the column * norms will exceed BIGNUM. * 1/3/91: ZLATRS no longer can handle this case * TLEFT = BIGNUM / MAX( ONE, DBLE( N-1 ) ) TSCAL = BIGNUM*( DBLE( N-1 ) / MAX( ONE, DBLE( N ) ) ) IF( UPPER ) THEN DO 390 J = 1, N CALL ZLARNV( 5, ISEED, J, A( 1, J ) ) CALL DLARNV( 1, ISEED, J, RWORK ) DO 380 I = 1, J A( I, J ) = A( I, J )*( TLEFT+RWORK( I )*TSCAL ) 380 CONTINUE 390 CONTINUE ELSE DO 410 J = 1, N CALL ZLARNV( 5, ISEED, N-J+1, A( J, J ) ) CALL DLARNV( 1, ISEED, N-J+1, RWORK ) DO 400 I = J, N A( I, J ) = A( I, J )*( TLEFT+RWORK( I-J+1 )*TSCAL ) 400 CONTINUE 410 CONTINUE END IF CALL ZLARNV( 2, ISEED, N, B ) CALL ZDSCAL( N, TWO, B, 1 ) END IF * * Flip the matrix if the transpose will be used. * IF( .NOT.LSAME( TRANS, 'N' ) ) THEN IF( UPPER ) THEN DO 420 J = 1, N / 2 CALL ZSWAP( N-2*J+1, A( J, J ), LDA, A( J+1, N-J+1 ), $ -1 ) 420 CONTINUE ELSE DO 430 J = 1, N / 2 CALL ZSWAP( N-2*J+1, A( J, J ), 1, A( N-J+1, J+1 ), $ -LDA ) 430 CONTINUE END IF END IF * RETURN * * End of ZLATTR * END |