Charles M. Rader and Allan O. Steinhardt, Hyperbolic Householder transformations, IEEE Trans. Acoust., Speech, Signal Processing, ASSP-34 (1986), pp. 1589--1602.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and there is no particular strategy to minimize the condition numbers of the individual rotations. 4.2. Reduction by unified Householder reflectors. Unified Householder reflectors [4] include standard orthogonal Householder transformations [11] together with hyperbolic Householder reflectors [20], 21] Given a signature matrix J = diag(# i ) a unified Householder matrix has the form H = H(J, k, v) P 2vv , v 0, 4.12) where P is a permutation matrix in the (1, k) plane. For any vector x such that x 0, the unified Householder vector v can be chosen so that H ....

Charles M. Rader and Allan O. Steinhardt, Hyperbolic Householder transformations, IEEE Trans. Acoust., Speech, Signal Processing, ASSP-34 (1986), pp. 1589--1602.

....G i Theta i . It is also clear that the nonzero entry could have been placed in any of the last q positions of the prearray. The rotation matrix Theta i can be implemented in a variety of ways: by using a sequence of elementary Givens and hyperbolic rotations [94] Householder transformations [161, 27], as well as square root and division free versions of the elementary rotations (see, e.g. 84, 103] 4.4.1. Elementary Circular and Hyperbolic Rotations. An elementary 2 Theta 2 unitary rotation Theta (also known as Givens or circular rotation) takes a row vector x = Theta a b and ....

C. M. Rader and A. O. Steinhardt, Hyperbolic Householder transformations, IEEE Transactions on Acoustics, Speech and Signal Processing, 34 (1986), pp. 1589--1602.

....i (t) We say that G i (t) is a proper generator of R i (t) and Theta i (t) is a J(t) Gammaunitary rotation that transforms G i (t) to proper form. This can be achieved in a variety of ways: by using a sequence of elementary Givens and hyperbolic rotations [18] Householder transformations [26 28], etc. Referring to (14) we see that the effect of Theta i (t) is the following h f i (t)d 1=2 i (t Gamma 1) g i (t) i 1 0 0 Theta i (t) h f i (t)d 1=2 i (t Gamma 1) ffi i (t) 0 i ; 16) where we still need to annihilate the nonzero entry ffi i (t) This can be achieved by ....

C. M. Rader and A. O. Steinhardt, Hyperbolic Householder transformations. IEEE Transactions on Acoustics, Speech and Signal Processing, vol. ASSP-34, no. 6, pp. 1589--1602, December 1986.

....0 x (j) i 0 : 0 i ; 4.2) with a nonzero entry in a single (convenient) column, say the j th column. So assume we use this choice for Theta i , which can be implemented in a variety of ways : we may use elementary rotations such as Givens or hyperbolic [12] Householder transformations [12, 30], etc. Using the above choice leads to the following algorithm in the strongly regular case. BLOCK SCHUR ALGORITHM 9 Algorithm 4.1 (Strongly Regular Step) The generator recursion for a strongly regular step is given by 0 G i 1 = G i Theta i 2 4 I j 0 0 0 0 0 0 0 I r Gammaj Gamma1 ....

C. M. Rader and A. O. Steinhardt, Hyperbolic Householder transformations, IEEE Transactions on Acoustics, Speech and Signal Processing, 34 (1986), pp. 1589--1602.

....= G T WmpG = G T U T WmpUG = h R T 0 i 2 6 4 I mp 0 0 GammaI mp 3 7 5 2 6 4 R 0 3 7 5 = R T R; 2. 24) which gives us the Cholesky factorization of T [22] The transformation matrix U that satisfies the property U T WmpU = Wmp is called a hyperbolic Householder transformation [52]. The basic properties of hyperbolic Householder reflectors are discussed in Section 2.2.2. Since the matrix G consists of two upper triangular block Toeplitz matrices, we show in Section 2.2.4 that considerable computational savings can be obtained by working with a generator matrix defined using ....

....to the new signature matrix Wmp ) to reduce G to an upper triangular matrix. A detailed discussion of the Schur algorithm for indefinite Toeplitz matrices is presented in Chapter 4. 2.2. 2 Hyperbolic Householder transformations In [22] Cybenko and Berry use hyperbolic Householder transformations [52] to reduce the generator matrix G of a scalar Toeplitz matrix to an upper triangular matrix. We extend their idea to block hyperbolic Householder transformations (required in the block Schur algorithm) using representations very similar to those proposed in [53] and [54] Let W be a diagonal ....

C. M. Rader and A. O. Steinhardt, "Hyperbolic Householder transformations," IEEE Trans. Acoust. Speech Signal Process., vol. 34, pp. 1589--1602, 1986.

....0. To construct a transformation U such that Ua = b is then to determine the vector y and the scalar # from a and b. Let # be a nonsingular signature matrix, namely, # is a diagonal matrix with 1 diagonal elements. # A matrix U is unitary with respect to #, or simply # unitary, if U H #U = # [8]. Unitary matrices are the special case of # = I. Hyperbolic rotations are well known 2 2 # unitary transformations with # = diag(1, 1) they have been used to e#ciently implement down dating of various matrix factorizations [9,10,11,12,13] In comparison, hyperbolic Householder ....

....[9,10,11,12,13] In comparison, hyperbolic Householder transformations were first used in linear algebra algorithms by Bunse Gerstner [14] for a theoretical interest in generalizing the QR algorithm for eigenvalue problems. They were further studied and proposed by Rader and Steinhardt [8,15] to use in practical computing such as in solving signal processing problems with least squares based adaptive methods. The hyperbolic Householder matrices used are of the Hermitian form H = # 2yy H , y H #y = 1, cf. for instance, 14] and [8] As for elementary unitary transformations, the ....

[Article contains additional citation context not shown here]

C. M. Rader and A. O. Steinhardt. Hyperbolic Householder transformations. IEEE Tras. on A.S.S.P., 34:1589--1602, 1986.

....T R; R upper triangular; in the case when the matrix A is given in terms of matrices Z 1 and Z 2 as A = Z T 1 Z 1 Gamma Z T 2 Z 2 ; Z 1 square: 6.1) Obviously, A is positive definite, if and only if Z 1 is non singular and kZ 2 Z Gamma1 1 k 1: 6. 2) The Cholesky factor R is obtained [9, 10] by premultiplying the matrix Z = Z 1 Z 2 by a sequence of J unitaries with J = I 0 0 GammaI : Here the partition is that of Z above, and we have QZ = R 0 ; Q J unitary: We have Theorem 6.1 Let A, Z 1 and Z 2 be as in (6.1,6.2) Then all J unitaries performing QZ = ....

....T 1 Z 1 ) Delta 1 kZ 2 Z Gamma1 1 k 2 1 Gamma kZ 2 Z Gamma1 1 k 2 (A) Z T 1 Z 1 ) 1 kZ 2 Z Gamma1 1 k 2 1 Gamma kZ 2 Z Gamma1 1 k 2 ; 6.5) which shows (mind the squares ) why it is often better to compute R directly from Z than by forming A first. In practice ([9, 10]) the matrix Q is obtained as a product of a sequence of elementary transformations of Givens or Householder type. While it is pleasant to know that all such Q s have the same condition number (6.4) independently of the algorithm used, there is of course, no guarantee that intermediate condition ....

Rader, C. M., Steinhardt, Hyperbolic Householder transformations, IEEE Trans. ASSP, 34:1589-1602 (1986).

....positive definite matrices of displacement rank k 2 (satisfying additional conditions similar to those listed in Definition 4. 2) see [18] For matrices of displacement rank k the factorization algorithm uses elementary rank k downdating via hyperbolic Householder or mixed Householder reflections [8, 20]. We conclude by noting that the Bareiss algorithms guarantee small residual errors in the sense of Definition 7.2, but the Levinson algorithm can yield residuals at least five orders of magnitude larger than those expected for a backward stable method. This result suggests that, if the Levinson ....

C. M. Rader and A. O. Steinhardt, Hyperbolic Householder transformations, IEEE Trans. Acoustics, Speech and Signal Processing, ASSP--34 (1986), pp. 1584--1602.

....are R and Z. The updating and downdating for least squares problems with k = 1 have been extensively studied in literature. Downdating for k = 1 has been studied in [1, 2, 5, 6, 11, 14] Downdating a block, i.e. when k 1, with variations of the Householder transformations has been studied in [3, 12]. Generalizations of the LINPACK downdating algorithm [14] to block downdating were first given in [4] The single row updating and block updating of the Cholesky factorization is equivalent to an intermediate step of the QR decomposition, of which the sensitivity has been well understood [17] ....

....T 1 Q 1 ) 2. Determine an orthogonal matrix, W , such that I k Z 0 R : W T Q 1 R Gamma 0 : Other algorithms for block downdating, which produce more accurate results than Algorithm BDLIN, are given in [4] There also exist block algorithms based on hyperbolic transformations [3, 12]. We emphasize that the perturbation analysis of this paper is for the problem of downdating a block of rows in the Cholesky factorization, and, equivalently in a QR decomposition, when the upper triangular factor R and the block of rows to be removed, Z, are known, but the Q factor is not known. ....

Rader, C., Steinhardt, A. (1986): Hyperbolic Householder transformations. IEEE Trans. Acoust., Speech, Signal Processing 34, 1589--1602

.... then we have T = G T WmpG = G T U T WmpUG = h R T 0 i I mp 0 0 GammaI mp # R 0 # = R T R; 4) which gives us the Cholesky factorization of T [8] The transformation matrix U which satisfies the property U T WmpU = Wmp , is called a hyperbolic Householder transformation [26]. The basic properties of hyperbolic Householder reflectors are discussed in Section 2.2. Since the matrix G comprises of two upper triangular block Toeplitz matrix, we show in Section 2.4 that considerable computational savings can be obtained by working with a generator matrix defined using the ....

....signature matrix Wmp ) to reduce G to an upper triangular matrix. A detailed discussion of the Schur algorithm for indefinite Toeplitz matrices is presented in Section 3. 2. 2 Hyperbolic Householder transformations In their paper [8] Cybenko and Berry use hyperbolic Householder transformations [26] to reduce the generator matrix G of a scalar Toeplitz matrix to an upper triangular matrix. We extend their idea to block hyperbolic Householder transformations (required in the block Schur algorithm) using representations very similar to those proposed in [2] and [29] Let W be a diagonal ....

C. M. Rader and A. O. Steinhardt, Hyperbolic Householder transformations, IEEE Trans. Acoust. Speech Signal Process., 34 (1986), pp. 1589--1602.

.... 1 j j 2 ; s = sign(x 2 ) p jffij; c = s; end if ffi 0 oe 1 = oe 1 ; oe 2 = oe 2 ; else oe 1 = Gammaoe 1 ; oe 2 = Gammaoe 2 ; end The unified Householder transformation unifies and generalizes unitary Householder transformation [5] and hyperbolic Householder transformation [7, 8]. Definition 2 (Householder Transformation) Given a signature matrix S = diag(oe i ) and a complex vector u such that u H Su 6= 0 and it is scaled so that ju H Suj = 2, a unified Householder transformation is of the form PJ where P = S Gamma sign(u H Su)uu H (2) and J is the matrix ....

....singular values are: oe Gamma1 min = oe max = u H u ju H Suj s u H u u H Su 2 Gamma 1: When S = SigmaI , then oe min = oe max = 1, J = I, and PJ is the unitary Householder matrix. In the case when S 6= SigmaI and J = I and P is the hyperbolic Householder defined in [7, 8], the above result on the condition number coincides with [2] For a general signature matrix S, the ratio (u H u) ju H Suj can be arbitrarily large. In other words, cond(PJ) can be arbitrarily large. In the following, analogous to [11] we propose a pivoting scheme. 2.2 Pivoting As shown ....

C. M. Rader and A. O. Steinhardt. Hyperbolic Householder transformations. SIAM Journal on Matrix Analysis and Applications, 9:269--290, 1988.

.... 1 j j 2 ; s = sign(x 2 ) p jffij; c = s; end if ffi 0 oe 1 = oe 1 ; oe 2 = oe 2 ; else oe 1 = Gammaoe 1 ; oe 2 = Gammaoe 2 ; end The unified Householder transformation unifies and generalizes unitary Householder transformation [5] and hyperbolic Householder transformation [7, 8]. Definition 2 (Householder Transformation) Given a signature matrix S = diag(oe i ) and a complex vector u such that u H Su 6= 0 and it is scaled so that ju H Suj = 2, a unified Householder transformation is of the form PJ where P = S Gamma sign(u H Su)uu H (2) and J is the matrix ....

....singular values are: oe Gamma1 min = oe max = u H u ju H Suj s u H u u H Su 2 Gamma 1: When S = SigmaI , then oe min = oe max = 1, J = I, and PJ is the unitary Householder matrix. In the case when S 6= SigmaI and J = I and P is the hyperbolic Householder defined in [7, 8], the above result on the condition number coincides with [2] For a general signature matrix S, the ratio (u H u) ju H Suj can be arbitrarily large. In other words, cond(PJ) can be arbitrarily large. In the following, analogous to [11] we propose a pivoting scheme. 2.2 Pivoting As shown ....

C. M. Rader and A. O. Steinhardt. Hyperbolic Householder transformations. IEEE Transactions on Acoustics, Speech and Signal Processing , 34:1589--1602, 1986.

.... then we have T = G T WmpG = G T U T WmpUG = Theta R T 0 I mp 0 0 GammaI mp R 0 = R T R; 4) which gives us the Cholesky factorization of T [8] The transformation matrix U which satisfies the property U T WmpU = Wmp , is called a hyperbolic Householder transformation [26]. The basic properties of hyperbolic Householder reflectors are discussed in Section 2.2. Since the matrix G comprises of two upper triangular block Toeplitz matrix, we show in Section 2.4 that considerable computational savings can be obtained by working with a generator matrix defined using the ....

....matrix Wmp ) to reduce G to an upper triangular matrix. A detailed discussion of the Schur algorithm for indefinite Toeplitz matrices is presented in Section 3. 5 2.2. Hyperbolic Householder transformations In their paper [8] Cybenko and Berry use hyperbolic Householder transformations [26] to reduce the generator matrix G of a scalar Toeplitz matrix to an upper triangular matrix. We extend their idea to block hyperbolic Householder transformations (required in the block Schur algorithm) using representations very similar to those proposed in [2] and [29] Let W be a diagonal ....

C. M. Rader and A. O. Steinhardt. Hyperbolic Householder transformations. IEEE Trans. Acoust. Speech Signal Process., 34:1589--1602,

....P r x = r 0 and P H 1 0 0 GammaI P = 1 0 0 GammaI where r represents a diagonal element of the upper triangular R, x is a column in X in (2) and r gives a diagonal element of R. Hyperbolic rotations [2] or hyperbolic Householder transformations (reflectors) [6, 7] are this kind of transformations. In [2] a real hyperbolic rotation is defined by Q = c Gammas Gammas c ; c 2 Gamma s 2 = 1: The hyperbolic rotation Q satisfies Q T SQ = S; where S = diag(1; Gamma1) and the above property is called S orthogonality. A hyperbolic rotation can ....

....the above algorithm fails to satisfy the equation c 2 Gamma s 2 = 1 and, consequently, the hyperbolic rotation Q fails to satisfy the S orthogonality. We also note that it is not obvious how to generalize the above algorithm to the complex case. Given a complex vector x, Rader and Steinhardt [6, 7] proposed a hyperbolic Householder transformation P RS = S Gamma ffi Gamma1 RS uu H (3) where S = diag(oe i ) oe i = Sigma1; called signature matrix, and u = Sx ffe 1 (ff = p x H Sx) and ffi RS = u H Su) 2: The hyperbolic Householder transformation defined above is ....

[Article contains additional citation context not shown here]

C. M. Rader and A. O. Steinhardt. Hyperbolic Householder transformations. SIAM Journal on Matrix Analysis and Applications, 9:269--290, 1988.

....P r x = r 0 and P H 1 0 0 GammaI P = 1 0 0 GammaI where r represents a diagonal element of the upper triangular R, x is a column in X in (2) and r gives a diagonal element of R. Hyperbolic rotations [2] or hyperbolic Householder transformations (reflectors) [6, 7] are this kind of transformations. In [2] a real hyperbolic rotation is defined by Q = c Gammas Gammas c ; c 2 Gamma s 2 = 1: The hyperbolic rotation Q satisfies Q T SQ = S; where S = diag(1; Gamma1) and the above property is called S orthogonality. A hyperbolic rotation can ....

....the above algorithm fails to satisfy the equation c 2 Gamma s 2 = 1 and, consequently, the hyperbolic rotation Q fails to satisfy the S orthogonality. We also note that it is not obvious how to generalize the above algorithm to the complex case. Given a complex vector x, Rader and Steinhardt [6, 7] proposed a hyperbolic Householder transformation P RS = S Gamma ffi Gamma1 RS uu H (3) where S = diag(oe i ) oe i = Sigma1; called signature matrix, and u = Sx ffe 1 (ff = p x H Sx) and ffi RS = u H Su) 2: The hyperbolic Householder transformation defined above is ....

[Article contains additional citation context not shown here]

C. M. Rader and A. O. Steinhardt. Hyperbolic Householder transformations. IEEE Transactions on Acoustics, Speech and Signal Processing , 34:1589--1602, 1986.

....positive definite matrices of displacement rank k 2 (satisfying additional conditions similar to those listed in Definition 4. 2) see [18] For matrices of displacement rank k the factorization algorithm uses elementary rank k downdating via hyperbolic Householder or mixed Householder reflections [8, 20]. We conclude by noting that the Bariess algorithms guarantee small residual errors in the sense of Definition 7.2, but the Levinson algorithm can yield residuals at least five orders of magnitude larger than those expected for a backward stable method. This result suggests that, if the Levinson ....

C.M. Rader and A.O. Steinhardt, "Hyperbolic Householder Transformations", IEEE Transaction on Acoustics, Speech and Signal Processing, vol ASSP-34, 1986, pp 1584-1602.

....U such that Ua = b is then to determine the vector y and the scalar fl from a and b. Let Phi be a nonsingular signature matrix, namely, Phi is a diagonal matrix with Sigma1 diagonal elements. A matrix U is unitary with respect to Phi, or simply Phi unitary, if U H PhiU = Phi [8]. Unitary matrices are the special case of Phi = SigmaI . Hyperbolic rotations are well known 2 Theta 2 Phi unitary transformations with Phi = Sigmadiag(1; Gamma1) they have been used to efficiently implement down dating of various matrix factorizations [9,10,11,12,13] In comparison, ....

....[9,10,11,12,13] In comparison, hyperbolic Householder transformations were first used in linear algebra algorithms by Bunse Gerstner [14] for a theoretical interest in generalizing the QR algorithm for eigenvalue problems. They were further studied and proposed by Rader and Steinhardt [8,15] to use in practical computing such as in solving signal processing problems with least squares based adaptive methods. The hyperbolic Householder matrices used are of the Hermitian form H = Phi Gamma 2yy H , y H Phiy = 1, cf. for instance, 14] and [8] As for elementary unitary ....

[Article contains additional citation context not shown here]

C. M. Rader and A. O. Steinhardt. Hyperbolic Householder transformations. IEEE Tras. on A.S.S.P., 34:1589--1602, 1986.

....the above equation, we obtain oe Gamma1 min = oe max = u H u ju H Suj s u H u u H Su 2 Gamma 1: When S = SigmaI , then oe min = oe max = 1, J = I, and PJ is a unitary Householder matrix. In the case when S 6= SigmaI and J = I, P is the hyperbolic Householder defined in [8, 9], the above result on the condition number coincides with [3] For a general signature matrix S, the ratio (u H u) ju H Suj can be arbitrarily large. Thus cond(PJ) can also be arbitrarily large. However, as shown in Algorithm 1, when a Householder transformation is constructed to introduce ....

C. M. Rader and A. O. Steinhardt. Hyperbolic Householder transformations. SIAM Journal on Matrix Analysis and Applications, 9:269--290, 1988.

....the above equation, we obtain oe Gamma1 min = oe max = u H u ju H Suj s u H u u H Su 2 Gamma 1: When S = SigmaI , then oe min = oe max = 1, J = I, and PJ is a unitary Householder matrix. In the case when S 6= SigmaI and J = I, P is the hyperbolic Householder defined in [8, 9], the above result on the condition number coincides with [3] For a general signature matrix S, the ratio (u H u) ju H Suj can be arbitrarily large. Thus cond(PJ) can also be arbitrarily large. However, as shown in Algorithm 1, when a Householder transformation is constructed to introduce ....

C. M. Rader and A. O. Steinhardt. Hyperbolic Householder transformations. IEEE Transactions on Acoustics, Speech and Signal Processing , 34:1589--1602, 1986.

.... 1 : Solving for in the above equation, we obtain 1 min = max = u H u ju H Suj s u H u u H Su 2 1: When S = I , then min = max = 1, J = I , and PJ is a unitary Householder matrix. In the case when S 6= I and J = I , P is the hyperbolic Householder de ned in [9, 10], the above result on the condition number coincides with [3] For a general signature matrix S, the ratio (u H u) ju H Suj can be arbitrarily large. Thus cond(PJ) can also be arbitrarily large. However, as shown in Algorithm 1, when a Householder transformation is constructed to introduce ....

C. M. Rader and A.O. Steinhardt. Hyperbolic Householder transformations. SIAM Journal on Matrix Analysis and Applications , 9:269-290, 1988.

.... 1 : Solving for in the above equation, we obtain 1 min = max = u H u ju H Suj s u H u u H Su 2 1: When S = I , then min = max = 1, J = I , and PJ is a unitary Householder matrix. In the case when S 6= I and J = I , P is the hyperbolic Householder de ned in [9, 10], the above result on the condition number coincides with [3] For a general signature matrix S, the ratio (u H u) ju H Suj can be arbitrarily large. Thus cond(PJ) can also be arbitrarily large. However, as shown in Algorithm 1, when a Householder transformation is constructed to introduce ....

C. M. Rader and A.O. Steinhardt. Hyperbolic Householder transformations. IEEE Transactions on Acoustics, Speech and Signal Processing , 34:1589-1602, 1986.

No context found.

Charles M. Rader and Allan O. Steinhardt, Hyperbolic Householder transformations, IEEE Trans. Acoust., Speech, Signal Processing, ASSP-34 (1986), pp. 1589--1602.

No context found.

Charles M. Rader and Allan O. Steinhardt. Hyperbolic Householder transformations. IEEE Trans. Acoust., Speech, Signal Processing, ASSP-34(6):1589--1602, 1986.

No context found.

C. M. Rader and A. O. Steinhardt, Hyperbolic Householder transformations, IEEE Transactions on Acoustics, Speech and Signal Processing, vol. ASSP-34, no. 6, pp. 1589--1602, December 1986.

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C.M. RADER AND A.O. STEINHARDT, Hyperbolic Householder transformations, IEEE Trans. Acoust. Speech, Signal Proc., 34 (1986), pp. 1589--1602.

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