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Mise à jour de lapack vers la version 3.3.0.
1: SUBROUTINE DLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY ) 2: * 3: * -- LAPACK auxiliary routine (version 3.2) -- 4: * -- LAPACK is a software package provided by Univ. of Tennessee, -- 5: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 6: * November 2006 7: * 8: * .. Scalar Arguments .. 9: INTEGER K, LDA, LDT, LDY, N, NB 10: * .. 11: * .. Array Arguments .. 12: DOUBLE PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ), 13: $ Y( LDY, NB ) 14: * .. 15: * 16: * Purpose 17: * ======= 18: * 19: * DLAHRD reduces the first NB columns of a real general n-by-(n-k+1) 20: * matrix A so that elements below the k-th subdiagonal are zero. The 21: * reduction is performed by an orthogonal similarity transformation 22: * Q' * A * Q. The routine returns the matrices V and T which determine 23: * Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T. 24: * 25: * This is an OBSOLETE auxiliary routine. 26: * This routine will be 'deprecated' in a future release. 27: * Please use the new routine DLAHR2 instead. 28: * 29: * Arguments 30: * ========= 31: * 32: * N (input) INTEGER 33: * The order of the matrix A. 34: * 35: * K (input) INTEGER 36: * The offset for the reduction. Elements below the k-th 37: * subdiagonal in the first NB columns are reduced to zero. 38: * 39: * NB (input) INTEGER 40: * The number of columns to be reduced. 41: * 42: * A (input/output) DOUBLE PRECISION array, dimension (LDA,N-K+1) 43: * On entry, the n-by-(n-k+1) general matrix A. 44: * On exit, the elements on and above the k-th subdiagonal in 45: * the first NB columns are overwritten with the corresponding 46: * elements of the reduced matrix; the elements below the k-th 47: * subdiagonal, with the array TAU, represent the matrix Q as a 48: * product of elementary reflectors. The other columns of A are 49: * unchanged. See Further Details. 50: * 51: * LDA (input) INTEGER 52: * The leading dimension of the array A. LDA >= max(1,N). 53: * 54: * TAU (output) DOUBLE PRECISION array, dimension (NB) 55: * The scalar factors of the elementary reflectors. See Further 56: * Details. 57: * 58: * T (output) DOUBLE PRECISION array, dimension (LDT,NB) 59: * The upper triangular matrix T. 60: * 61: * LDT (input) INTEGER 62: * The leading dimension of the array T. LDT >= NB. 63: * 64: * Y (output) DOUBLE PRECISION array, dimension (LDY,NB) 65: * The n-by-nb matrix Y. 66: * 67: * LDY (input) INTEGER 68: * The leading dimension of the array Y. LDY >= N. 69: * 70: * Further Details 71: * =============== 72: * 73: * The matrix Q is represented as a product of nb elementary reflectors 74: * 75: * Q = H(1) H(2) . . . H(nb). 76: * 77: * Each H(i) has the form 78: * 79: * H(i) = I - tau * v * v' 80: * 81: * where tau is a real scalar, and v is a real vector with 82: * v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in 83: * A(i+k+1:n,i), and tau in TAU(i). 84: * 85: * The elements of the vectors v together form the (n-k+1)-by-nb matrix 86: * V which is needed, with T and Y, to apply the transformation to the 87: * unreduced part of the matrix, using an update of the form: 88: * A := (I - V*T*V') * (A - Y*V'). 89: * 90: * The contents of A on exit are illustrated by the following example 91: * with n = 7, k = 3 and nb = 2: 92: * 93: * ( a h a a a ) 94: * ( a h a a a ) 95: * ( a h a a a ) 96: * ( h h a a a ) 97: * ( v1 h a a a ) 98: * ( v1 v2 a a a ) 99: * ( v1 v2 a a a ) 100: * 101: * where a denotes an element of the original matrix A, h denotes a 102: * modified element of the upper Hessenberg matrix H, and vi denotes an 103: * element of the vector defining H(i). 104: * 105: * ===================================================================== 106: * 107: * .. Parameters .. 108: DOUBLE PRECISION ZERO, ONE 109: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 110: * .. 111: * .. Local Scalars .. 112: INTEGER I 113: DOUBLE PRECISION EI 114: * .. 115: * .. External Subroutines .. 116: EXTERNAL DAXPY, DCOPY, DGEMV, DLARFG, DSCAL, DTRMV 117: * .. 118: * .. Intrinsic Functions .. 119: INTRINSIC MIN 120: * .. 121: * .. Executable Statements .. 122: * 123: * Quick return if possible 124: * 125: IF( N.LE.1 ) 126: $ RETURN 127: * 128: DO 10 I = 1, NB 129: IF( I.GT.1 ) THEN 130: * 131: * Update A(1:n,i) 132: * 133: * Compute i-th column of A - Y * V' 134: * 135: CALL DGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, 136: $ A( K+I-1, 1 ), LDA, ONE, A( 1, I ), 1 ) 137: * 138: * Apply I - V * T' * V' to this column (call it b) from the 139: * left, using the last column of T as workspace 140: * 141: * Let V = ( V1 ) and b = ( b1 ) (first I-1 rows) 142: * ( V2 ) ( b2 ) 143: * 144: * where V1 is unit lower triangular 145: * 146: * w := V1' * b1 147: * 148: CALL DCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 ) 149: CALL DTRMV( 'Lower', 'Transpose', 'Unit', I-1, A( K+1, 1 ), 150: $ LDA, T( 1, NB ), 1 ) 151: * 152: * w := w + V2'*b2 153: * 154: CALL DGEMV( 'Transpose', N-K-I+1, I-1, ONE, A( K+I, 1 ), 155: $ LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 ) 156: * 157: * w := T'*w 158: * 159: CALL DTRMV( 'Upper', 'Transpose', 'Non-unit', I-1, T, LDT, 160: $ T( 1, NB ), 1 ) 161: * 162: * b2 := b2 - V2*w 163: * 164: CALL DGEMV( 'No transpose', N-K-I+1, I-1, -ONE, A( K+I, 1 ), 165: $ LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 ) 166: * 167: * b1 := b1 - V1*w 168: * 169: CALL DTRMV( 'Lower', 'No transpose', 'Unit', I-1, 170: $ A( K+1, 1 ), LDA, T( 1, NB ), 1 ) 171: CALL DAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 ) 172: * 173: A( K+I-1, I-1 ) = EI 174: END IF 175: * 176: * Generate the elementary reflector H(i) to annihilate 177: * A(k+i+1:n,i) 178: * 179: CALL DLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ), 1, 180: $ TAU( I ) ) 181: EI = A( K+I, I ) 182: A( K+I, I ) = ONE 183: * 184: * Compute Y(1:n,i) 185: * 186: CALL DGEMV( 'No transpose', N, N-K-I+1, ONE, A( 1, I+1 ), LDA, 187: $ A( K+I, I ), 1, ZERO, Y( 1, I ), 1 ) 188: CALL DGEMV( 'Transpose', N-K-I+1, I-1, ONE, A( K+I, 1 ), LDA, 189: $ A( K+I, I ), 1, ZERO, T( 1, I ), 1 ) 190: CALL DGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, T( 1, I ), 1, 191: $ ONE, Y( 1, I ), 1 ) 192: CALL DSCAL( N, TAU( I ), Y( 1, I ), 1 ) 193: * 194: * Compute T(1:i,i) 195: * 196: CALL DSCAL( I-1, -TAU( I ), T( 1, I ), 1 ) 197: CALL DTRMV( 'Upper', 'No transpose', 'Non-unit', I-1, T, LDT, 198: $ T( 1, I ), 1 ) 199: T( I, I ) = TAU( I ) 200: * 201: 10 CONTINUE 202: A( K+NB, NB ) = EI 203: * 204: RETURN 205: * 206: * End of DLAHRD 207: * 208: END