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Effects of extreme rainfall on phreatic eruptions: A case study of Mt. Ontake in Japan

  • Sometimes, natural disasters caused by volcanic eruptions have tragic consequences. Phreatic eruptions are large explosions of steam rocks and hot water caused by the sudden evaporation of water to steam. The September 2014 eruption of Mt. Ontake in Japan was the deadliest in recorded history. Numerous studies have analyzed the occurrence of phreatic eruptions of Mt. Ontake. However, at present, although it is explained that the magma did not move, studies on the cause of the eruption and the elucidation of the process are limited. This study investigates the role of external water of meteoric origin and determines its role in the eruption process. According to a survey of rainfall records by the Japan Meteorological Agency, heavy rain that broke historical records occurred immediately before the phreatic eruption of Mt. Ontake. It was hypothesized that extreme rainfall was the source of the external water supply that caused the phreatic eruption without the magma moving. Various studies on eruptions have confirmed the consistency of this hypothesis. Regarding the eruptive process, extreme rainfall collided with the hot rocks outside the magma chamber, triggering frequent occurrences of vaporization associated with boiling, leading to large explosions in sealed rocks above the zone of water infiltration. This research can contribute to disaster prevention in the future. In order to achieve this, it is necessary to install rainfall measuring instruments on all volcanoes and perform a comparative, multidisciplinary approach to all the monitored parameters.

    Citation: Nobuo Uchida. Effects of extreme rainfall on phreatic eruptions: A case study of Mt. Ontake in Japan[J]. AIMS Geosciences, 2024, 10(2): 208-227. doi: 10.3934/geosci.2024012

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  • Sometimes, natural disasters caused by volcanic eruptions have tragic consequences. Phreatic eruptions are large explosions of steam rocks and hot water caused by the sudden evaporation of water to steam. The September 2014 eruption of Mt. Ontake in Japan was the deadliest in recorded history. Numerous studies have analyzed the occurrence of phreatic eruptions of Mt. Ontake. However, at present, although it is explained that the magma did not move, studies on the cause of the eruption and the elucidation of the process are limited. This study investigates the role of external water of meteoric origin and determines its role in the eruption process. According to a survey of rainfall records by the Japan Meteorological Agency, heavy rain that broke historical records occurred immediately before the phreatic eruption of Mt. Ontake. It was hypothesized that extreme rainfall was the source of the external water supply that caused the phreatic eruption without the magma moving. Various studies on eruptions have confirmed the consistency of this hypothesis. Regarding the eruptive process, extreme rainfall collided with the hot rocks outside the magma chamber, triggering frequent occurrences of vaporization associated with boiling, leading to large explosions in sealed rocks above the zone of water infiltration. This research can contribute to disaster prevention in the future. In order to achieve this, it is necessary to install rainfall measuring instruments on all volcanoes and perform a comparative, multidisciplinary approach to all the monitored parameters.



    Throughout this paper, the complex m×n matrix space is denoted by Cm×n. The conjugate transpose, the Moore-Penrose inverse, the range space and the null space of a complex matrix ACm×n are denoted by AH, A+, R(A) and N(A), respectively. In denotes the n×n identity matrix. PL stands for the orthogonal projector on the subspace LCn. Furthermore, for a matrix ACm×n, EA and FA stand for two orthogonal projectors: EA=ImAA+=PN(AH), FA=InA+A=PN(A).

    A number of papers have been published for solving linear matrix equations. For example, Chen et al. [1] proposed LSQR iterative method to solve common symmetric solutions of matrix equations AXB=E and CXD=F. Zak and Toutounian [2] studied the matrix equation of AXB=C with nonsymmetric coefficient matrices by using nested splitting conjugate gradient (NSCG) iteration method. By applying a Hermitian and skew-Hermitian splitting (HSS) iteration method, Wang et al. [3] computed the solution of the matrix equation AXB=C. Tian et al. [4] obtained the solution of the matrix equation AXB=C by applying the Jacobi and Gauss-Seidel-type iteration methods. Liu et al. [5] solved the matrix equation AXB=C by employing stationary splitting iterative methods. In addition, some scholars studied matrix equations by direct methods. Yuan and Dai [6] obtained generalized reflexive solutions of the matrix equation AXB=D and the optimal approximation solution by using the generalized singular value decomposition. Zhang et al. [7] provided the explicit expression of the minimal norm least squares Hermitian solution of the complex matrix equation AXB+CXD=E by using the structure of the real representations of complex matrices and the Moore-Penrose inverse. By means of the definitions of the rank and inertias of matrices, Song and Yu [8] obtained the existence conditions and expressions of the nonnegative (positive) definite and the Re-nonnegative (Re-positive) definite solutions to the matrix equations AXAH=C and BXBH=D.

    In this paper, we will focus on the restricted solutions to the following well-known linear matrix equation

    AXB=C, (1)

    where ACm×n,BCn×p and CCm×p. We observe that the structured matrix, such as Hermitian matrix, skew-Hermitian matrix, Re-nonnegative definite matrix and Re-positive definite matrix, is of important applications in structural dynamics, numerical analysis, stability and robust stability analysis of control theory and so on [9,10,11,12,13,14]. Conditions for the existence of Hermitian solutions to Eq (1) were studied in [15,16,17]. A solvability criterion for the existence of Re-nonnegative definite solutions of Eq (1) by using generalized inner inverses was investigated by Cvetković-Ilić [19]. Recently, a direct method for solving Eq (1) by using the generalized inverses of matrices and orthogonal projectors was proposed by Yuan and Zuo [21]. In addition, the Re-nonnegative definite and Re-positive definite solutions to some special cases of Eq (1) were considered by Wu [22], Wu and Cain [23] and Groß [24]. In [25], necessary and sufficient conditions for the existence of common Re-nonnegative definite and Re-positive definite solutions to the matrix equations AX=C,XB=D were discussed by virtue of the extremal ranks of matrix polynomials.

    In this paper, necessary and sufficient conditions for the existence of Hermitian (skew-Hermitian), Re-nonnegative (Re-positive) definite, and Re-nonnegative (Re-positive) definite least-rank solutions to Eq (1) are deduced by using the Moore-Penrose inverse of matrices, and the explicit representations of the general solutions are given when the solvability conditions are satisfied. Compared with the approaches proposed in [18,19,20,21], the coefficient matrices of Eq (1) have no any constraints and the method in this paper is straightforward and easy to implement.

    Definition 1. A matrix ACn×n is said to be Re-nonnegative definite (Re-nnd) if H(A):=12(A+AH) is Hermitian nonnegative definite (i.e., H(A)0), and A is said to be Re-positive definite (Re-pd) if H(A) is Hermitian positive definite (i.e., H(A)>0). The set of all Re-nnd (Re-pd) matrices in Cn×n is denoted by RNDn×n (RPDn×n).

    Lemma 1. [26] Let ACm×n,BCn×p and CCm×p. Then the matrix equation AXB=C is solvable if and only if AA+CB+B=C. In this case, the general solution can be written in the following parametric form

    X=A+CB++FAL1+L2EB,

    where L1,L2Cn×n are arbitrary matrices.

    Lemma 2. [27,28] Let B1Cl×q,D1Cl×l. Then the matrix equation

    YBH1±B1YH=D1,

    has a solution YCl×q if and only if

    D1=±DH1, EB1D1EB1=0.

    In which case, the general solution is

    Y=12D1(B+1)H+12EB1D1(B+1)H+2VVB+1B1B1VH(B+1)HEB1VB+1B1,

    where VCl×q is an arbitrary matrix.

    Lemma 3. [29] Let ACm×n and BCm×p. Then the Moore-Penrose inverse of the matrix [A,B] is

    [A,B]+=[(I+TTH)1(A+A+BC+)C++TH(I+TTH)1(A+A+BC+)],

    where C=(IAA+)B and T=A+B(IC+C).

    Lemma 4. [30] Suppose that a Hermitian matrix M is partitioned as

    M=[M11M12MH12M22],

    where M11, M22 are square. Then

    (i). M is Hermitian nonnegative definite if and only if

    M110, M11M+11M12=M12, M22MH12M+11M12H20.

    In which case, M can be expressed as

    M=[M11M11H1HH1M11H2+HH1M11H1],

    where H1 is an arbitrary matrix and H2 is an arbitrary Hermitian nonnegative definite matrix.

    (ii). M is Hermitian positive definite if and only if

    M11>0, M22MH12M111M12H3>0.

    In the case, M can be expressed as

    M=[M11M12MH12H3+MH12M111M12],

    where H3 is an arbitrary Hermitian positive definite matrix.

    Lemma 5. [31] Let

    M=[CAB0], N=[0In], S=[0In], N1=NFM, S1=EMS.

    Then the general least-rank solution to Eq.(1) can be written as

    X=NM+S+N1R1+R2S1,

    where R1C(p+n)×n,R2Cn×(m+n) are arbitrary matrices.

    Theorem 1. Eq (1) has a Hermitian solution X if and only if

    AA+CB+B=C,PT(A+CB+(A+CB+)H)PT=0, (2)

    where T=R(AH)R(B). In which case, the general Hermitian solution of Eq (1) is

    X=A+CB++FAL1+L2EB, (3)

    where

    L1=P1D1+12P1D1WH1+2VH12FAZH1+P1(V1FAV2EB)+FAZH1WH1, (4)
    L2=D1QH112W1D1QH1+2V2+2Z1EB+(FAVH1EBVH2)QH1W1Z1EB, (5)
    D1=A+CB+(A+CB+)H, C1=(IFAF+A)EB, T1=F+AEB(IC+1C1),P1=(I+T1TH1)1(F+A+F+AEBC+1), Q1=C+1+TH1P1,W1=FAP1EBQ1, Z1=V1P1+V2Q1,

    and V1,V2Cn×n are arbitrary matrices.

    Proof. By Lemma 1, if the first condition of (2) holds, then the general solution of Eq (1) is given by (3). Now, we will find L1 and L2 such that AXB=C has a Hermitian solution, that is,

    A+CB++FAL1+L2EB=(A+CB+)H+LH1FA+EBLH2. (6)

    Clearly, Eq (6) can be equivalently written as

    X1AH1A1XH1=D1, (7)

    where A1=[FA,EB], X1=[LH1,L2], D1=A+CB+(A+CB+)H.

    By Lemma 2, Eq (7) has a solution X1 if and only if

    D1=DH1, EA1D1EA1=0. (8)

    The first condition of (8) is obviously satisfied. And note that

    EA1=PN(AH1)=PN(FA)N(EB)=PR(AH)R(B).

    Thus, the second condition of (8) is equivalent to

    PTD1PT=0,

    where T=R(AH)R(B), which is the second condition of (2). In which case, the general solution of Eq (7) is

    X1=D1(A+1)H12A1A+1D1(A+1)H+2V2VA+1A1+A1VH(A+1)H+A1A+1VA+1A1, (9)

    where V=[V1,V2] is an arbitrary matrix. By Lemma 3, we have

    [FA,EB]+=[(I+T1TH1)1(F+A+F+AEBC+1)C+1+TH1(I+T1TH1)1(F+A+F+AEBC+1)], (10)

    where C1=(IFAF+A)EB and T1=F+AEB(IC+1C1). Upon substituting (10) into (9), we can get (4) and (5).

    Corollary 1. Eq (1) has a skew-Hermitian solution X if and only if

    AA+CB+B=C,PT(A+CB++(A+CB+)H)PT=0, (11)

    where T=R(AH)R(B). In which case, the general skew-Hermitian solution of Eq (1) is

    X=A+CB++FAL3+L4EB, (12)

    where

    L3=P2D212P2D2WH2+2VH32FAZH2P2(V3FA+V4EB)+FAZH2WH2, (13)
    L4=D2QH212W2D2QH2+2V42Z2EB(FAVH3+EBVH4)QH2+W2Z2EB, (14)
    D2=A+CB+(A+CB+)H, C2=(IFAF+A)EB, T2=F+AEB(IC+2C2),P2=(I+T2TH2)1(F+AF+AEBC+2), Q2=C+2+TH2P2,W2=FAP2+EBQ2, Z2=V3P2+V4Q2,

    and V3, V4Cn×n are arbitrary matrices.

    Theorem 2. Let ACm×n, BCn×p, CCm×p and T=R(AH)R(B). Assume that the spectral decomposition of PT is

    PT=U[Is000]UH, (15)

    where U=[U1,U2]Cn×n is a unitary matrix and s=dim(T). Then

    (a). Eq (1) has a Re-nnd solution if and only if

    AA+CB+B=C, UH1A+CB+U1RNDs×s. (16)

    In which case, the general Re-nnd solution of (1) is

    X=A+CB++FAJ1+J2EB, (17)

    where

    J1=P3D312P3D3WH3+2VH52FAZH3P3(V5FA+V6EB)+FAZH3WH3, (18)
    J2=D3QH312W3D3QH3+2V62Z3EB(FAVH5+EBVH6)QH3+W3Z3EB, (19)
    D3=KA+CB+(A+CB+)H, C3=(IFAF+A)EB,T3=F+AEB(IC+3C3), P3=(I+T3TH3)1(F+AF+AEBC+3), Q3=C+3+TH3P3,W3=FAP3+EBQ3, Z3=V5P3+V6Q3, K11=UH1(A+CB++(A+CB+)H)U1,K=U[K11K11H1HH1K11H2+HH1K11H1]UH,

    V5,V6Cn×n,H1Cs×(ns) are arbitrary matrices, and H2C(ns)×(ns) is an arbitrary Hermitian nonnegative definite matrix.

    (b). Eq (1) has a Re-pd solution if and only if

    AA+CB+B=C, UH1A+CB+U1RPDs×s. (20)

    In which case, the general Re-pd solution of (1) is

    X=A+CB++FAJ1+J2EB, (21)

    where

    K=U[K11K12KH12H3+KH12K111K12]UH,

    J1,J2,D3,C3,T3,P3,Q3,W3,Z3 and K11 are given by (18) and (19), K12Cs×(ns) is an arbitrary matrix and H3C(ns)×(ns) is an arbitrary Hermitian positive definite matrix.

    Proof. By Lemma 1, if the first condition of (16) holds, then the general solution of Eq (1) is given by (17). Now, we will find J1 and J2 such that AXB=C has a Re-nnd (Re-pd) solution, that is, we will choose suitable matrices J1 and J2 such that

    A+CB++(A+CB+)H+FAJ1+JH1FA+J2EB+EBJH2K0 (K>0). (22)

    Clearly, Eq (22) can be equivalently written as

    X3AH3+A3XH3=D3, (23)

    where A3=[FA,EB], X3=[JH1,J2], D3=KA+CB+(A+CB+)H.

    By Lemma 2, Eq (23) has a solution X1 if and only if

    D3=DH3, EA3D3EA3=0. (24)

    The first condition of (24) is obviously satisfied. And note that

    EA3=PN(AH3)=PN(FA)N(EB)=PR(AH)R(B).

    Then the second condition of (24) is equivalent to

    PTD3PT=0,

    where T=R(AH)R(B). By (15), we can obtain

    [Is000]UHKU[Is000]=[Is000]UH(A+CB++(A+CB+)H)U[Is000], (25)

    Let

    UHKU=[K11K12KH12K22], (26)

    where U=[U1,U2]. By (25), we can obtain

    K11=UH1(A+CB++(A+CB+)H)U1, (27)

    it is easily known that K0 (K>0) if and only if UHKU0 (UHKU>0). And X is Re-nnd (Re-pd) if and only if K0 (K>0). Thus, by (26) and (27), we can get

    K0K11=UH1(A+CB++(A+CB+)H)U10,K>0K11=UH1(A+CB++(A+CB+)H)U1>0,

    equivalently,

    K0UH1A+CB+U1RNDs×s,K>0UH1A+CB+U1RPDs×s,

    which are the second conditions of (16) and (20). In which case, by Lemma 4,

    K0K=U[K11K11H1HH1K11H2+HH1K11H1]UH,
    K>0K=U[K11K12KH12H3+KH12K111K12]UH,

    where H1Cs×(ns) is an arbitrary matrix, H2C(ns)×(ns) is an arbitrary Hermitian nonnegative definite matrix and H3C(ns)×(ns) is an arbitrary Hermitian positive definite matrix. And the general solution of Eq (23) is

    X3=D3(A+3)H12A3A+3D3(A+3)H+2V2VA+3A3A3VH(A+3)H+A3A+3VA+3A3, (28)

    where V=[V5,V6] is an arbitrary matrix. By Lemma 3, we have

    [FA,EB]+=[(I+T3TH3)1(F+AF+AEBC+3)C+3+TH3(I+T3TH3)1(F+AF+AEBC+3)], (29)

    where C3=(IFAF+A)EB and T3=F+AEB(IC+3C3). Upon substituting (29) into (28), we can get (18) and (19).

    Theorem 3. Let ACm×n,BCn×p, CCm×p and ˜T=N(NH1)N(S1). Assume that the spectral decomposition of P˜T is

    P˜T=˜U[Ik000]˜UH, (30)

    where ˜U=[˜U1,˜U2]Cn×n is a unitary matrix and k=dim(˜T), and M,N,S,N1,S1 are given by Lemma 5. Then

    (a). Eq (1) has a Re-nnd least-rank solution if and only if

    AA+CB+B=C, ˜UH1(NM+S)˜U1RNDk×k. (31)

    In which case, the general Re-nnd least-rank solution of Eq (1) is

    X=NM+S+N1R1+R2S1, (32)

    where

    R1=P4D412P4D4WH4+2VH72NH1ZH4P4(V7NH1+V8S1)+NH1ZH4WH4, (33)
    R2=D4QH412W4D4QH4+2V82Z4SH1(N1VH7+SH1VH8)QH4+W4Z4SH1, (34)
    D4=˜K+NM+S+(NM+S)H, C4=(IN1N+1)SH1, T4=N+1SH1(IC+4C4),P4=(I+T4TH4)1(N+1N+1SH1C+4), Q4=C+4+TH4P4,W4=N1P4+SH1Q4, Z4=V7P4+V8Q4, ˜K11=˜UH1(NM+S(NM+S)H)˜U1,˜K=˜U[˜K11˜K11˜H1˜HH1˜K11˜H2+˜HH1˜K11˜H1]˜UH,

    V7Cn×(p+n), V8Cn×(m+n), ˜H1Ck×(nk) are arbitrary matrices, and ˜H2C(nk)×(nk) is an arbitrary Hermitian nonnegative definite matrix.

    (b). Eq (1) has a Re-pd least-rank solution if and only if

    AA+CB+B=C, ˜UH1(NM+S)˜U1RPDk×k. (35)

    In which case, the general Re-nnd least-rank solution of Eq (1) is

    X=NM+S+N1R1+R2S1, (36)

    where

    ˜K=˜U[˜K11˜K12˜KH12˜H3+˜KH12˜K111˜K12]˜UH,

    R1,R2,D4,C4,T4,P4,Q4,W4,Z4 and ˜K11 are given by (33) and (34), ˜K12Ck×(nk) is an arbitrary matrix and ˜H3C(nk)×(nk) is an arbitrary Hermitian positive definite matrix.

    Proof. By Lemmas 1 and 5, if the first condition of (31) holds, then the least-rank solution of Eq (1) is given by (32). Now, we will find R1 and R2 such that AXB=C has a Re-nnd (Re-pd) least-rank solution, that is, we will choose suitable matrices R1 and R2 such that

    NM+S(NM+S)H+N1R1+RH1NH1+R2S1+SH1RH2˜K0 (˜K>0). (37)

    Clearly, Eq (37) can be equivalently written as

    X4AH4+A4XH4=D4, (38)

    where A4=[N1,SH1], X4=[RH1,R2], D4=˜K+NM+S+(NM+S)H.

    By Lemma 2, Eq (38) has a solution X4 if and only if

    D4=DH4, EA4D4EA4=0. (39)

    The first condition of (39) is obviously satisfied. And note that

    EA4=PN(AH4)=PN(NH1)N(S1).

    Thus, the second condition of (39) is equivalent to

    P˜TD4P˜T=0,

    where ˜T=N(NH1)N(S1). By (30), we can obtain

    [Ik000]˜UH˜K˜U[Ik000]=[Ik000]˜UH(NM+S(NM+S)H)˜U[Ik000], (40)

    Let

    ˜UH˜K˜U=[˜K11˜K12˜KH12˜K22], (41)

    where ˜U=[˜U1,˜U2]. By (40), we can obtain

    ˜K11=˜UH1(NM+S(NM+S)H)˜U1, (42)

    it is easily known that ˜K0 (˜K>0) if and only if ˜UH˜K˜U0 (˜UH˜K˜U>0). And X is Re-nnd (Re-pd) least-rank solution if and only if ˜K0 (˜K>0). Thus, by (41) and (42), we can get

    ˜K0˜K11=˜UH1(NM+S(NM+S)H)˜U10,˜K>0˜K11=˜UH1(NM+S(NM+S)H)˜U1>0,

    equivalently,

    ˜K0˜UH1(NM+S)˜U1RNDk×k,˜K>0˜UH1(NM+S)˜U1RPDk×k,

    which are the second conditions of (31) and (35). In which case, by Lemma 4,

    ˜K0˜K=˜U[˜K11˜K11˜H1˜HH1˜K11˜H2+HH1˜K11˜H1]˜UH,
    ˜K>0˜K=˜U[˜K11˜K12˜KH12˜H3+˜KH12˜K111˜K12]˜UH,

    where ˜H1Ck×(nk) is an arbitrary matrix, ˜H2C(nk)×(nk) is an arbitrary Hermitian nonnegative definite matrix and ˜H3C(nk)×(nk) is an arbitrary Hermitian positive definite matrix. And the general solution of Eq (38) is

    X4=D4(A+4)H12A4A+4D4(A+4)H+2V2VA+4A4A4VH(A+4)H+A4A+4VA+4A4, (43)

    where V=[V7,V8] is an arbitrary matrix. By Lemma 3, we have

    [N1,SH1]+=[(I+T4TH4)1(N+1N+1SH1C+4)C+4+TH4(I+T4TH4)1(N+1N+1SH1C+4)], (44)

    where C4=(IN1N+1)SH1 and T4=N+1SH1(IC+4C4). Upon substituting (44) into (43), we can get (33) and (34).

    The following example comes from [9].

    Example 1. Consider a 7-DOF system modelled analytically with the first three measured modal data given by

    Λ=diag(3.5498, 101.1533, 392.8443), X=[0.55850.47510.42410.08410.23530.28380.30940.17170.25120.08000.16460.08520.09960.35620.05080.05530.04040.21050.00840.17880.4113].

    and the corrected symmetric mass matrix M and symmetric stiffness matrix K should satisfy the orthogonality conditions, that is,

    XMX=I3,XKX=Λ.

    Since

    X(X)+X+XI3=1.5442×1015,PT((X)+X+((X)+X+))PT=0,

    which means that the conditions of (2) are satisfied. Choose V1=0,V2=0. Then, by the equation of (3), we can obtain a corrected mass matrix given by

    M=[1.19680.10730.82010.16780.19770.20790.14390.10730.30570.62920.09980.29530.25470.20950.82010.62922.27480.13471.13980.72900.32490.16780.09980.13470.09980.34270.01770.28040.19770.29531.13980.34271.87750.03701.48780.20790.25470.72900.01770.03700.37420.64610.14390.20950.32490.28041.48780.64612.1999],

    and

    XMXI3=1.5016×1015,

    which implies that M is a symmetric solution of XMX=I3.

    Since

    X(X)+ΛX+XΛ=4.0942×1013,PT((X)+ΛX+((X)+ΛX+))PT=1.5051×1014,

    which means that the conditions of (2) are satisfied. Choose V1=0,V2=0. Then, by the equation of (3), we obtain a corrected stiffness matrix given by

    K=[50.036447.836966.10521.462121.655862.090444.891547.836993.9748169.400741.942349.965378.9100244.131566.1052169.4007176.99573.6625103.7349173.8782169.81701.462141.94233.662525.75282.6730189.498698.311021.655849.9653103.73492.673095.347351.6395446.806262.090478.9100173.8782189.498651.639556.3200448.690044.8915244.1315169.817098.3110446.8062448.6900394.1690],

    and

    XKXΛ=3.5473×1013,

    which implies that K is a symmetric solution of XKX=Λ.

    Example 2. Given matrices

    A=[7.94829.79751.36526.61445.82792.25959.56842.71450.11762.84414.23505.79815.22592.52338.93904.69225.15517.60378.80148.75741.99140.64783.33955.29821.72967.37312.98729.88334.32916.4053],B=[1.93433.78378.21633.41193.70416.82228.60016.44915.34087.02743.02768.53668.17977.27115.46575.41675.93566.60233.09294.44881.50874.96553.41978.38506.94576.97908.99772.89735.68076.2131],C=[745.63171194.55431060.3913995.63791010.8200535.5044831.5304791.3676684.6897711.7632845.43241338.10651123.73801077.06151096.2601629.07681006.4762928.6566804.5134824.7220868.41581299.0559995.07861012.70461054.9264].

    Since AA+CB+BC=9.3907×1012, and the eigenvalues of K11 are

    Λ1=diag{0.3561,11.0230,6.8922,5,3274},

    which means that the conditions of (16) are satisfied. Choose V,H1 and H2 as

    V=[I6,I6], H1=[0.52110.67910.23160.39550.48890.36740.62410.9880], H2=[6000].

    Then, by the equation of (17), we get a solution of Eq (1):

    X=[7.93051.85151.13640.56150.54111.28251.72282.34252.30291.46770.49750.49200.85821.68172.52851.83251.08950.80953.05140.82500.79223.37411.52491.69186.17873.85492.52942.69452.45680.69230.19681.06000.32140.63830.31253.2860]

    with corresponding residual

    AXBC=6.4110×1012.

    Furthermore, it can be computed that the eigenvalues of X+XH are

    Λ2=diag{0, 0.3555, 3.9363, 7.0892, 11.1802, 21.2758},

    which implies that X is a Re-nnd solution of Eq (1).

    Example 3. Let the matrices A and B be the same as those in Example 2 and the matrix C be given by

    C=[1841.23232726.84152555.29262172.76882365.39391281.07881957.59321843.72521614.62111699.39851826.13253077.61412600.81682539.01812467.35581583.35432422.31692153.66251910.01562035.21232065.87612950.77832362.00602197.71292389.1173].

    Since AA+CB+BC=1.7944×1011, and the eigenvalues of K11 are

    Λ1=diag{25.3998, 20.3854, 19.4050, 19.0160},

    which means that the conditions of (20) are satisfied. If select V,K12 and H3 as

    V=[I6,I6], K12=[0.52110.67910.23160.39550.48890.36740.62410.9880], H3=[6008].

    Then, by the equation of (21), we can achieve a Re-pd solution of Eq (1):

    X=[7.07011.25000.86771.76852.77351.46891.01529.90320.15350.43830.05711.22970.80630.487410.08780.59562.02351.14262.16620.01360.83648.29371.87212.58871.66270.72791.28871.21985.66820.49251.60281.17371.80042.67180.27298.1489],

    and the eigenvalues of X+XH are

    Λ2=diag{5.9780, 7.9395, 19.1387, 19.4286, 20.4585, 25.4004}.

    Furthermore, it can be computed that

    AXBC=1.7808×1011.

    In this paper, we mainly consider some special solutions of Eq (1). By imposing some constraints on the expression X=A+CB++FAL1+L2EB, we succeed in obtaining a set of necessary and sufficient conditions for the existence of the Hermitian, skew-Hermitian, Re-nonnegative definite, Re-positive definite, Re-nonnegative definite least-rank and Re-positive definite least-rank solutions of Eq (1), respectively. Moreover, we give the explicit expressions for these special solutions, when the consistent conditions are satisfied.

    The authors wish to give special thanks to the editor and the anonymous reviewers for their helpful comments and suggestions which have improved the presentation of the paper.

    The authors declare no conflict of interest.



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