Real Tunneling and Black Hole Creation

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Real Tunneling and Black Hole Creation Wu Zhong Chao Dept.of Physics Beijing Normal University Beijing,100875,China Ref:Int.J.Modern Phys.D 7,111(1998)Abstract We discuss the Hawking theory of quantum cosmology with regard to approximation at the lowest order of the Planck constant.At this level,the quantum scenario will be reduced to its classical evolutions in real and imaginary times.We restrict our attention to the so-called real tunneling case.It can be shown that,even at this level,there still exist some quantum e?ects,the classical ?eld equation may not hold at the transition surface.One can introduce the concept of constrained gravitational instanton.It may play some important role in the scenario of black hole creation in the in?ationary background at the Planckian era of the universe.From the constrained gravitational instanton,the real tunneling can occur through di?erent ways.Consequently,it will

lead to the creation of di?erent parts of the black hole spacetime in the de Sitter background.The global aspects of the black hole creation are discussed.

PACS number(s):98.80.Hw,98.80.Bp,04.60.Kz,04.70.Dy

e-mail:wu@axp3g9.icra.it

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I.Introduction

The Hawking theory of the No-Boundary Universe is the?rst success in obtaining a self-contained cosmology.Now,in principle,one can predict everything in the universe solely from the theory.

According to the no-boundary proposal of Hartle and Hawking,the quantum state of the uni-verse is in the ground state,which is de?ned by a path integral over compact Euclidean metrics [1].Conceptually,this proposal is very appealing.Technically,like any other theory of quantum cosmology,it encounters enormous di?culty in calculations due to the lack of a complete theory of quantum gravity.The task of this article is to deal with the situation at the modest level,i.e,the lowest order in the Planck constantˉh.It turns out that the problem is not really as trivial as it looks at the?rst glance.

In quantum?eld theory,it is well known that quantum tunneling can be studied by using the instanton theory.Instanton is a sort of Euclidean solution of the?eld equation.Quantum pene-trating can be described by an analytic continuation from the Euclidean solution to its Lorentzian counterpart.In?at spacetime background,one can readily realize this by simply changing the time value from imaginary to real.However,except for some very special cases,a complex solution of the Einstein?eld equation does not typically have both purely Euclidean and Lorentzian sectors. Therefore,the instanton theory cannot be used here without modi?cation.

In some simple models,the creation of the universe can be considered as a quantum tunneling from an Euclidean spacetime to a Lorentzian one.If the?eld equation is not only satis?ed in the Euclidean and Lorentzian spacetimes,respectively,but also at the location of the transition,then the instanton theory can be used as it is.Unfortunately,very few models in quantum cosmology share such luck[2].The de Sitter model is an exception.

The de Sitter model is the?rst nontrivial model in quantum cosmology.Since the4-sphere and the de Sitter spacetime are the Euclidean and Lorentzian sectors of a complex solution to the vacuum Einstein equation with a cosmological constant,the instanton theory does apply here.The main reason that the creation of the de Sitter universe can be considered as tunneling from the4-sphere is that it is a model with only one degree of freedom,the scale of the3-metric.Indeed,the instanton theory can be used to all models with only one degree of freedom.

However,if one discusses a model with more than one degrees of freedom,then the situation be-

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comes more complicated.For more realistic models such as the Hawking massive scalar or primordial black hole creation,the instanton theory has to be modi?ed and generalized.

In the Hawking model a massive scalarφwith mass m is coupled to an isotropic and homogeneous universe[3].It can be shown that there does not exist any instanton solution in this model.In fact, there does not exist any compact sector of a complex solution,as the4-sphere solution to the de Sitter model.

Section II will review the Wheeler-DeWitt equation at the lowest order.It is known that,in general,the wave packet represents two ensembles of classical evolutions,one is in real time and other in imaginary.They satisfy classical equations with modi?cation due to their mutual interactions. These two ensembles of trajectories are mutually orthogonal with respect to the supermetric of the con?guration space.

Section III will deal with the modi?cation of the instanton theory for the so-called real tunneling case.That is,there is no interaction between Euclidean and Lorentzian evolutions,or the two ensembles decouple.To classically interpret the quantum tunneling,one has to begin with the action from?rst principles.Then we can introduce the concept of a constrained gravitational instanton [4],which is an Euclidean stationary action solution under some constraints.It satis?es the Einstein equation with the possible exception at the transition surface where the constraints are imposed.

A very interesting application of the theory will be the primordial black hole creation in quantum cosmology[5].Although some attempts were made one decade ago on the Schwarzschild-de Sitter black hole creation in quantum cosmology[6][7],its conclusive solution was obtained only very recently.This will be the content of Section IV.The problem of a black hole of the whole Kerr-Newman family in the de Sitter background has been completely resolved.Its probability,at the W K

B level,is the exponential of a quarter of the sum of the black hole and cosmological horizons. It turns out that the de Sitter evolution is the most probable trajectory in the Planckian era of the universe.

Section V shows that there are alternative ways of real tunneling in the black hole creation with di?erent probabilities.They will lead to the Lorentzian evolutions of parts of interior of the black hole horizon or exterior of the cosmological horizon.Section VI is devoted to the global aspects of

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black hole creation.Section VII is a summary.

II.The wave function at the lowest approximation

In the No-Boundary Universe the wave function of the universe is given by[1]

Ψ(h ij,φ)= C d[gμν]d[φ]exp(?ˉI([gμν,φ]),(1) where the path integral is over the class C of compact Euclidean4-metrics and matter?eld con?g-urations,which agree with the given3-metrics h ij of the only boundary and matter con?gurationφon it.HereˉI means the Euclidean action.

The Euclidean action for the gravitational part for a smooth spacetime manifold M with bound-ary?M is

ˉI=?1

8π ?M K,(2) whereΛis the cosmological constant,R is the scalar curvature and K is the trace of the second fundamental form of the boundary.

The dominant contribution to the path integral comes from some stationary action manifolds with matter?elds on them,which are the saddle points of the path integral.In general,the wave function takes a superposition form of wave packets

Ψ≈C exp(?S/ˉh),(3) where we have writtenˉh explicitly;C is a slowly varying prefactor;and S≡S r+iS i is a complex phase.

Since the wave packets of form(3)are not independent in the decomposition of the wave function, one more restriction should be imposed.That is,the wave packets itself should obey the Wheeler-DeWitt equation.Classically,it means that the evolutions represented by the wave packet should satisfy the Einstein equation with some quantum corrections,as it will be shown below.

The Wheeler-DeWitt equation takes the following form module to some operator ordering am-biguities: ?1

where△is the Laplacian in the supermetric of the con?guration space and V is considered as the potential term.Classically,it means that the evolutions represented by the wave packet should satisfy the Einstein equation with some quantum corrections.We have implicitly assumed the universe to be closed.

Inserting the wave packet form into the Wheeler-DeWitt equation,one obtains:

?12△S+▽S·▽ Cˉh?1

2(▽S r)2+

1

2

△S r+▽S r·▽ Cˉh?1

2

C△S i+▽S i·▽C ˉh=0.(7)

If we ignore the quantum e?ects represented by the terms associated with powers ofˉh in these equations,then Eqs.(6)and(7)become

?1

2

(▽S i)2+V=0,(8)

and

▽S r·▽S i=0.(9) Eq.(8)is the Lorentzian(or Euclidean)Hamilton-Jacobi equation,with S i(or S r)and▽S i (or▽S r)identi?ed as the classical action and the canonical momenta,respectively.One can de?ne Lorentzian(or Euclidean)orbits along integral curves with?

?τ

≡▽S r·▽).The wave function represents an ensemble of classical trajectories.The Lorentzian(or Euclidean)trajectories

will trace out orbits in the presence of the potential V?1

2

(▽S i)2).

Here,two kinds of quantum corrections are involved:one due to the higherˉh terms of Eq.(6) and the other due to the nonzero value of▽S r(or▽S i).For the Lorentzian case,if the modi?cation is not negligible,then the evolutions should deviate quite dramatically from classical dynamics.This deviation,which has little e?ect on the short range behavior,may be crucial to the global properties of the universe.At the lowest approximation the?rst kind of corrections are neglected.

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The quantityΨ?Ψor the factor exp(?2S r)in the wave packet can be interpreted as the relative probability of the Lorentzian trajectories in the region of the con?guration space with varying S i. From Eq.(9)we know that the Lorentzian and Euclidean trajectories are mutually perpendicular, or S r remains constant along the orbits at the lowest order inˉh.Theˉh term of Eq.(7)represents the probability creation rates during the Lorentzian evolutions,and the probabilities are conserved if and only if the?rst term vanishes.Theˉh term of Eq.(6)represents the dynamic e?ects of probability creation due to the Euclidean evolution.Brie?y speaking,the evolution in real time is causal,while the evolution in imaginary time is stochastic.

While imaginary time becomes a commonly accepted notion in quantum cosmology,it has seemed not accepted yet by workers in other?elds.Many people on quantum optics are talking about light propagation at a speed higher than c when transpassing a classically prohibited region.They notice that the light takes zero real time lapse for the tunneling.If one agrees that the time lapse is imaginary within this region and the only observational e?ect of imaginary time is an exponential decay of the signal,then there would not be any puzzle left[8].

Eqs.(8)(9)represents the Wheeler-DeWitt equation for quantum cosmology at the lowest order approximation.The second order partial functional di?erential equation has been degraded into the ?rst order one.One of the consequences of this approximation is that the propagation property associated with the wavelike equation of the exact equation has been ignored at this level.We shall restrict our following arguments to the lowest order approximation unless otherwise stated.

In some models there exists a so-called Euclidean regime in con?guration space with a purely real phase in the wave packet.At the boundary of this region a transition from Euclidean evolution to Lorentzian evolution occurs through a3-geometryΣ.This transition is called real tunneling by Gibbons and Hartle[2].

During real tunneling,the Euclidean spacetime is connected to the Lorentzian spacetime with common boundaryΣ.Gibbons and Hartle argue that if the Einstein equation holds at the both sides ofΣ,then the second fundamental form K ij has to vanish from the both sides[2],

K ij=0.(10) There is no such restriction on the normal derivative of the matter?eld there.SinceΣhas a vanishing second fundamental form,one can construct the vacuum instanton manifold by joining

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the Euclidean manifold with its orientation reversal acrossΣ.

In the next section we shall show that the presumption made by Gibbons and Hartle at the tran-sition surface is too restrictive,it would exclude many interesting phenomena in quantum cosmology, in particular the scenario of black hole creation at the birth of the universe[5].

From the above argument,we learn that at the lowest level,for the real tunneling,the Lorentzian or Euclidean classical equations are satis?ed along the trajectories in the interiors of the Euclidean or Lorentzian regimes.

One may wonder whether at the lowest approximation the quantum theory can be fully reduced into the classical theory.We shall argue that this is not always the case.So we have to look closely at the transition surface of the manifold,or the boundary of the two regimes in the con?guration space.At some cases the classical equation will not be recovered there,and consequently condition (10)may not always hold there.

We shall deal with real tunneling cases in the rest of the paper unless stated otherwise.

III.The constrained gravitational instanton

The probability of the Lorentzian trajectory emanating from the3-surfaceΣwith the matter ?eldφon it can be written as

P=Ψ?Ψ= C d[gμν]d[φ]exp(?ˉI([gμν,φ]),(11) where the class C is composed by all no-boundary compact Euclidean4-metrics and matter?eld con?gurations which agree with the given3-metric h ij and the matter?eldφonΣ.

Here,we do not restrict the class C to contain regular metrics only,since the derivation from Eq.

(1)to Eq.(11)has already led to some jump discontinuities in the extrinsic curvature atΣ.This point is crucial for the wave function of the universe,otherwise it would be impossible to factorize expression(11)into the ground state de?nition(1).

The main contribution to the path integral in Eq.(11)is due to the stationary action4-metric, which meets all requirements on the3-surfaceΣ.At the W KB level,the exponential of the negative of the stationary action is the probability of the corresponding Lorentzian trajectory.

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From the above viewpoint,an extension of the class C to include metrics with some mild sin-gularities is essential.Indeed,it is recognized that,in some sense,the set of all regular metrics is not complete.For many cases,under the usual regularity conditions and the requirements at the equatorΣ,there may not exist any stationary action metric,i.e.a gravitational instanton.It is not clear,how large the class C should be.A necessary condition for a metric to be a member is that its scalar curvature should be well-de?ned mathematically.It is reasonable to include met-rics with jump discontinuities of extrinsic curvature and their degenerate cases,that is,the conical or pancake singularities.For this kind of singularity,the quantity g1/2R can be interpreted as a distribution-valued density[9].

If we lift the requirement on the3-metric of the equator,then the stationary action solution becomes the regular gravitational instanton,as it satis?es the Einstein equation everywhere.Then the Gibbons-Hartle condition(10)should hold at the equator.The probability of the corresponding trajectory takes stationary value;it may be maximum,minimum or neither[10].

The wave packet represents an ensemble of Lorentzian trajectories.If the regular gravitational instanton has minimum action,then the3-metric from which the Lorentzian evolution supposes to emanate is determined,and the most probable trajectory is therefore singled out.As a result, quantum cosmology fully realizes its prediction power:there is no degree of freedom left,with the exception of physical time[10].If one does not use the instanton theory,the degree of freedom is reduced to half by the ground state proposal,roughly speaking,due to the regularity condition at the south pole of the Euclidean manifolds in the path integral.

In general,the regularity conditions on the4-metrics and the requirements from the equatorΣsometimes are so strong that no gravitational instanton exists.The reason is that one cannot require the regularity condition and the given3-metric at the equator simultaneously in the variational calculation.Therefore,hopefully,one can only?nd a nonregular gravitational instanton with some mild singularities within the class C.If this is the case,then Eq.(10)will no longer hold at the singularities.The probability of the real tunneling will not be stationary when lifting the3-metric requirements.

It has been proven[9]that a stationary action regular solution keeps its status under the extension of the class C.However,if a stationary action regular solution cannot be found,then it can probably

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be expected with some singularities at its equator among the class C.

One can rephrase this by saying that the solution obeys the generalized Einstein equation in the whole manifold.Since this result is derived from?rst principles,and if one believes that Nature is quantum,then one should not feel upset about this situation.

IV.The creation of a black hole

In this section,we shall apply the constrained gravitational instanton theory to the problem of primordial black hole.The creation of a black hole in the whole Kerr-Newman family has been resolved[5].

In the Hawking model,the universe at the Euclidean and in?ation stage can be approximated by a S4space and the de Sitter space with an e?ective cosmological constantΛ≡3m2φ20,whereφ0 is the initial value of the scalar?eld.This is the motivation to discuss the black hole creation in the de Sitter background.A chargeless and nonrotating black hole sitting in the de Sitter background can be described by the Schwarzschild-de Sitter spacetime.It is the unique spherically symmetric vacuum solution to the Einstein equation with a cosmological constantΛ.The S2×S2Nariai spacetime is its degenerate case.

Its Euclidean metric can be written as[11]

ds2= 1?2m3 dτ2+ 1?2m3 ?1dr2+r2d?22.(12) For convenience one can factorize the potential[11]

?=1?2m

3

=?

Λ

1

3 ,r3=2

Λcos α?π1

3

arccos(3mΛ1/2),(15) where r2,r3are the black hole and cosmological horizons,and r0is the horizon for the negative r. We are interested in the Euclidean sector r2≤r≤r3for0≤m≤m c=Λ?1/2/3.For the extreme case m=m c the sector degenerates into the S2×S2space.

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The black hole and cosmological surface gravitiesκ2andκ3are[11]

Λ

κ2=

(r3?r2)(r3?r0).(17)

6r3

Now we are making a constrained gravitational instanton.In the(τ?r)plane r=r2is an axis of symmetry,the imaginary time coordinateτis identi?ed with periodβ2=2πκ?12,andβ?12 is the Hawking temperature.This makes the Euclidean manifold regular at the black hole horizon. One can also apply this procedure to the cosmological horizon with periodβ3=2πκ?13,andβ?13is the Gibbons-Hawking temperature.For the S2×S2case these two horizons are identical,thus one obtains a regular instanton.Except for the S2×S2spacetime,one cannot simultaneously regularize at both horizons.In fact,there is no way to avoid singularity in compacting the Euclidean spacetime because of the inequalityβ?12>β?13.

To form a constrained gravitational instanton[5],one can have two cuts atτ=consts.between r=r2and r=r3and then glue them.Then the f2-fold cover turns the(τ?r)plane into a cone with a de?cit angle2π(1?f2)at the black hole horizon.In a similar way one can have an f3-fold cover at the cosmological horizon.Both f2and f3can take any pair of real numbers with the relation

f2β2=f3β3(18) for a fairly symmetric Euclidean manifold.

If f2or f3is di?erent from1(at least one should be),then the cone at the black hole or cosmological horizon will have an extra contribution to the action of the manifold.We shall see that after the transition to Lorentzian spacetime,the conical singularities will only a?ect the real part of the phase of the wave function,i.e.the probability of the black hole creation.The black hole creation can be described by an analytic continuation from imaginary time to real time of the constrained gravitational instanton at the equator which is two jointτsectors,sayτ=±f2β2/4 through the two horizons.

Since the integral of K with respect to the3-area in the boundary term of the action(2)is the area increase rate along its normal,then the extra contribution due to the conical singularities can

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be considered as the degenerate form shown below

ˉI

2,deficit=?

1

8π

·4πr23·2π(1?f3).(20) The volume term of the action for the manifold can be calculated

ˉI vol =?

Λ

Λ (24)

and the second is the Nariai model,or pair black hole creation,with m=m c,

P c≈exp 2π

For the case m?m c,we have

ˉI

instanton

≈?π 33

Λ

(1+2α2)(28) and

P m≈exp 2π

Λ .(29) The probability is an exponentially decreasing function in terms of the mass parameter.The de Sitter case has the maximum probability and the Nariai case has the minimum probability.

The topology of the3-metric of the equator is S2×S1,the con?guration space has two degrees of freedom,one being the size of the universe(or the scale of S2),the other being the mass parameter. This situation is very fortunate in that the quantum creation of a black hole can be realized by a real tunneling.

If one includes an electromagnetic?eld into the model,one would be able to carry out a similar calculation.One simply replaces the potential by

?=1?2m

r2

?

Λr2

3r2

(r?r0)(r?r1)(r?r2)(r?r3),(30)

where Q is the charge parameter of the black hole.

For the magnetically charged black hole case,the con?guration of the wave function is the3-metric and magnetic charge.However,the con?guration for the wave function of an electrically charged black hole is not well de?ned[5][13][14][15],if one naively uses the folding and gluing techniques described above.For the electric case the con?guration of the wave function is the3-metric and the canonical momentum conjugate to the charge.In order to get the wave function for

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the charge,one has to appeal to a Fourier transformation,by which the duality between electric and magnetic black holes is recovered.

The quantum creation scenario of a Schwarzschild-de Sitter black hole and the Reissner-Nordstr¨o m-de Sitter black hole can be clearly depicted by using the so-called synchronous coordinates[15]:

ds2=?dη2+1

?ξ η

2dξ2+r2(η,ξ)(dθ2+sin2θdφ2).(31)

The Einstein constraint implies that

˙r2+1?2m

r2

?

Λr2

(E2??)1/2

,(33)

t= Edr

r +

Q2

3

.(36)

Its classical evolution is equivalent to a coherent motion of particles along a congruence of timelike geodesics,labeled by(ξ,θ,φ),in a potential hill?.One can release a particle from the potential hill between r2and r3.Except for the case with the initial position at the top of the potential, the particle will approach in?nity or hit the singularity r=0for the chargeless black hole case. Therefore,the synchronous coordinates cover the whole spacetime manifold.For the charged case, the potential will blow up near r=0,and the particle will transpass the inner horizon r=r1and is bounced back by the singularity at r=0.

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By using this coordinate system one can obtain the wave function for the Schwarzschild-de Sitter and the Reissner-Nordstr¨o m-de Sitter spacetimes[15]for the spacelike3-geometry covered by the coordinates.

The whole scenario of the chargeless black hole creation is shown in Fig.1.The S2space(θ?φ) is represented by a S1space around the vertical axis.The radius of S2is r.The bottom part shows the instanton,the upper part shows the black hole created.The inner edge of the donut collapses from the black horizon to the singularity r=0.The outer edge expands from the cosmological horizon to r=∞.The conical singularities are not shown in this very sketchy picture.The scenario of the charged black hole creation is similar except that the motion of the infalling trajectories are bounced by the singularity.

For the rotating and charged black hole case,the spacetime metric takes the Kerr-Newman form dθ2)+ρ?2Ξ?2?θsin2θ(adt?(r2+a2)dφ)2?ρ?2Ξ?2?r(dt?a sin2θdφ)2, ds2=ρ2(??1r dr2+??1

θ

(37) where

ρ2=r2+a2cos2θ,(38)

?r=(r2+a2)(1?Λr23?1)?2mr+Q2+P2,(39)

?θ=1+Λa23?1cos2θ,(40)

Ξ=1+Λa23?1(41) and m,a,Q and P are constants,m and ma represent mass and angular momentum.Q and P are electric and magnetic charges.

One can factorize?r as follows:

Λ

?r=?

If one were to naively factorize the probability from Eq.(11),he would get the wave function for the3-metric and the di?erential rotation of two horizons only.So one has to use another Fourier transformation to obtain the wave function for angular momentum[5].If the hole is electrically charged,one has to appeal again to the Fourier transformation,as we did for the nonrotating case. Here,two Fourier transformations are involved.

At any case,the probability of a black creation in the de Sitter background,at the W KB level, is the exponential of a quarter of the sum of the black hole and cosmological horizons,a quarter of the sum is the total entropy of the universe.By the no-hair theorem,the problem of a single black hole creation in quantum cosmology has been resolved completely.

The probability is an exponentially decreasing function of mass,charge magnitude and angular momentum.The de Sitter evolution is the most probable one at the Planckian era[5].

V.The alternative tunnelings

Now we are going to discuss a black hole creation from an alternative route.We begin with the vacuum Kantowski-Sachs model[16]with the positive cosmological constant.The3-surface is homogeneous and has topology S1×S2.The treatment in this section is suitable for the Kaluza-Klein S1×S n case[10],but we shall study the n=2case only below for simplicity.In Ref.[16] a black hole will be formed when the massive scalar?eld rolls down the potential hill and starts to oscillate.The e?ect of the massive scalar?eld is approximated by a cosmological constant during the imaginary time stage and the in?ationary stage.Here,we investigate the case of a black hole creation at the exact moment of the birth of the universe.

The Euclidean metric of the Kantowski-Sachs model takes the form:

ds2=dτ2+a2(τ)dω2+b2(τ)d?22,(43)

whereωis identi?ed with a period2π.

The Euclidean?eld equation is

b¨b+˙b2

2

+

Λb2

and

b˙b˙a+a˙b2

2

+

Λab2

Λ/3is the Hubble constant.

After the scale of S2reaches maximum,then one can make an analytical continuation along real time direction to get its Lorentzian counterpart

ds2=?dt2+a20sinh2Htdω2+H?2cosh2Htd?22.(48)

The Lorentzian metric is a part of the de Sitter space with a cone singularity at the birth of the universe due to the identi?cation of the circle S1with the?nite period.The Einstein equation does not hold at the transition asτ=π/2H and t=0unless at the Euclidean side as a0=H?1. The solution has a scale invariance associated with the circle.Thus there essentially exists only one trajectory.This mild singularity is acceptable.The fact that there are so many beautiful representations for the de Sitter spacetime is a manifestation of its versatility.

The gravitational instanton with the boundary condition(ii)is the Schwarzschild-de Sitter man-ifold of the last section,in which r is identi?ed as b,the size of S2,?is identi?ed as a2,and??1dr2 becomes dτ2here.

If one regularizes the black hole horizon by setting f2=1as in the last section,then one can consider the horizon as the south pole.It is noted that the expansion rate of S2space vanishes at both horizons.The spacetime becomes Lorentzian as one enters the exterior of the cosmological horizon r>r3.Here,the r coordinate becomes timelike.At the transition there is a conical singularity, while the scale of S1shrinks to zero.The Lorentzian evolution represents a part of the exterior of cosmological horizon with S1of extension?t=β2.The constrained gravitational instanton is formed by joining the above manifold with its orientation reversal across the cosmological horizon.

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Therefore,the action should be twice as much as the action of the constrained instanton discussed in the last section.The probability of the creation,at the W KB level,is

P3≈exp(2π(r22+r23)).(49)

On the other hand,if one lets f3=1,then the cosmological horizon can be considered as the south pole and quantum transition will occur at the black horizon.The Lorentzian evolution describes an interior part of the black hole.One just exchanges the two horizons in the statement of the preceding paragraph.One will obtain the same creation probability at the W KB level

P2≈exp(2π(r22+r23)).(50)

The manifolds created by the alternative tunnelings are shown as regions OGH and EMN in Fig. 2,respectively.The curves OG and OH(curves EM and EN)are identi?ed due to the periodicity condition required by Euclidean compacti?cation at the cosmological(black hole)horizon.

To apply the above arguments on the alternative tunnelings to the black holes of the whole Kerr-Newman family is straightforward.

VI.Global aspects of the black hole creation

Now we discuss the global property of the black hole creation.

The synchronous coordinates cover the whole manifold of the Schwarzschild-de Sitter spacetime. This can be shown clearly by the Penrose-Carter diagrams[11]in Fig.2.There is an in?nite sequence of diamond shape regions,singularities r=0and spacelike in?nities r=∞.Therefore,the3-geometry is not closed and not suitable for No-Boundary Universe.However,in the scenario of a black hole creation the whole manifold created can be obtained by an identi?cation,for instance, with lines ABC and DEF.The equator in the instanton is identi?ed as the closed line BE here.

It is worth mentioning that t coordinate is future(past)directed in the right(left)triangle above line BE.This is consistent with the analytic continuation of the imaginary time at the two cuts τ=consts for the south hemisphere of the constrained gravitational instanton,or the two ends of

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the imaginary time lapse.The same argument applies to the other members of the Kerr-Newman family.In the Schwarzschild space,r3becomes r=+∞,and no identi?cation is necessary.

Fig.3shows the Penrose-Carter diagram for Reissner-Nordstr¨o m-de Sitter black hole creation. The lines ABC and DEF are identi?ed.The line BE is the quantum transition surface,from which the region of HABOEDLG is created.The synchronous coordinates cover the whole manifold minus neighborhoods of the singularity.One may travel to other universes by passing through the“wormholes”made by the charge.In particular,if one follows an infalling trajectory in the synchronous coordinates,he will be bounced by the singularity and enter the another universe.But this does not bother us right now,since the relevant3-metric will no longer be spacelike,and the wave function is not well de?ned under this circumstance.In the Reissner-Nordstr¨o m space,r3 becomes r=+∞,and no identi?cation is necessary.

Fig.4shows the Penrose-Carter diagram of the symmetry axis of the Kerr-Newman-de Sitter black hole creation.The in?nities r=+∞and r=?∞are not joined together.The open circles mark where the ring singularity occurs,although it is not on the symmetry axis.The lines ABC and DEF are identi?ed.The line BE is the quantum transition surface,from which the region of HABOEDLG is created.In the Kerr-Newman space,r3becomes r=+∞,no identi?cation is necessary.

The alternative tunnelings are shown by the shaded bands in Figs.2and3with the two bound-aries identi?ed.It should be similar for the Kerr-Newman case.

VII.Summary

The complex tunnelings are common phenomena in Nature.We investigate the real tunneling problem in this paper.However the conventional concept of real tunneling is too narrow.If one insists on this,then many interesting phenomena,such as creation of a single black hole in quantum cosmology,would be excluded from studying,.On the other hand,if one begins with the?rst principles in quantum framework,it is naturally leads to the constrained gravitational instanton. The?eld equation holds in the instanton except for the location where the constraint is imposed; here is where one will?nd the equator of the instanton.If one believes that the nature is quantum,

18

then the?eld equation becomes secondary,and one should welcome this kind of generalization.

As an example,we have shown that the real tunnelings occur at the quantum creation of a black hole in the de Sitter background.This is a quite rare case in nature,taking into consideration the fact that the model has more than one degree of freedom.

By the analysis of this article,one learns that real tunnelings of quantum transition originates from the same Schwarzschild-de Sitter instanton in several ways.No matter what kind analytic continuation is made,one always obtain the whole or parts of the same Lorentzian spacetime,but with di?erent probabilities.If we are working with the de Sitter model,the situation is not so transparent,since it is of the maximum symmetry.All equators of the4-sphere are identical.

It is also interesting to examine how the Euclidean manifold is joined to the Lorentzian counter-part in the black hole creation.The Lorentzian spacetimes have to be compacti?ed by a periodic identi?cation,then the global aspects of the whole scenario is clari?ed.

Acknowledgment:

I would like to thank S.W.Hawking of Cambridge University and K.Sato of Tokyo University for their hospitality.I am grateful to G.W.Gibbons for discussions.

References:

1.J.B.Hartle and S.W.Hawking,Phys.Rev.D28

,2458(1990).

3.S.W.Hawking,Nucl.Phys.B239

,199(1997).

6.L.Z.Fang and M.Li,Phys.Lett.B169

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10.X.M.Hu and Z.C.Wu,Phys.Lett.B149

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19

12.R.Bousso and S.W.Hawking,Phys.Rev.D52

,2254(1995).

14.S.F.Ross and S.W.Hawking,Phys.Rev.D52

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20

数值积分在MATLAB中的应用

数值积分在MATLAB 中的应用 摘 要:介绍了数值积分法的几种计算公式及相应的MATLAB 命令,并给出了用MATLAB 编程求数值积分的实例.牛顿—莱布尼兹公式在计算积分的方法和解决实际问题中期了很大作用，但在某些领域遇到一些复杂情况，用牛顿—莱布尼兹公式则无法求解。这时可以“数值积分”的方法求定积分。“数值积分”法中常用的方法有“矩形公式”，“梯形公式”和“辛普森公式”等。MATLAB 中求数值积分的命令有：矩形公式命令 sum ；梯形公式命令 trapz ；辛普森公式命令 quad 。使用这些命令可以快速计算一些数值积分问题。 关键词:MATLAB ;数值积分;矩形公式;梯形公式;辛普森公式 Numerical integration in MATLAB Applications Abstract : Introduced several numerical integration formula and the corresponding MATLAB commands, and gives the Numerical Integration with MATLAB programming examples. Newton - Leibniz formula in calculating the integral method to solve practical problems and a significant role in the medium-term However, the complexities encountered in some areas, with Newton - Leibniz formula can not be solved. Then you can "numerical integration" method seeking the definite integral. "Numerical integration" method commonly used method in the "rectangular formula", "trapezoidal rule" and "Simpson formula," and so on. Numerical Integration in MATLAB commands are: rectangle formula order sum ; trapezoidal formula order trapz ; Simpson formula command quad . Use these commands to quickly calculate some numerical integration problems. Key words: MATLAB ; Numerical integration; Rectangular formula; Trapezoid formula; Simpson formula 引言 MATLAB 是一个包含大量计算算法的集合。 其拥有600多个工程中要用到的数学运算函数，可以方便的实现用户所需的各种计算功能。函数中所使用的算法都是科研和工程计算中的最新研究成果，而前经过了各种优化和容错处理。在通常情况下，可以用它来代替底层编程语言，如C 和++C 。在计算要求相同的情况下，使用MATLAB 的编程工作量会大大减少。MATLAB 的这些函数集包括从最简单最基本的函数到诸如距阵，特征向量、快速傅立叶变换的复杂函数。函数所能解决的问题其大致包括矩阵运算和线性方程组的求解、微分方程及偏微分方程的组的求解、符号运算、傅立叶变换和数据的统计分析、工程中的优化问题、稀疏矩阵运算、复数的各种运算、三角函数和其他初等数学运算、多维数组操作以及建模动态仿真等。 数值积分在众多方面都有着重用的作用，但其计算太过庞大和复杂。而MATLAB 拥有庞大的数学运算函数，使数值积分在MATLAB 中的计算变得简单，所以我们要了解数值积

MATLAB及其应用1

一、选择题 1. 设A=[2 4 3; 5 3 1; 3 6 7]，则sum(A)，length(A)和size(A)的结果( D ) A. [10 13 11] 9 [3 3] B. [9 9 16] 3 [3 3] C. [9 9 16] 9 [3 3] D. [10 13 11] 3 [3 3] 2. 下列关于脚本文件和函数文件的描述中不正确的是( B ) A. 去掉函数文件第一行的定义行可转变成脚本文件; B. 函数文件可以在命令窗口直接运行; C. 脚本文件可以调用函数文件; D. 函数文件中的第一行必须以function开始; 3. 在Command Window窗口中分别输入下列命令，对应输出结果错误的是 ( C ) A. x=[-3:2] x=[-3 -2 1 0 1 2] B. x=zeros(1,2);x>0 ans=[0 0] C. y=diag(eye(3),2)’ y=[0 0] D. 3-2*rand(1,2) ans=[1.0997 2.5377] 4. 对于矩阵B，统计其中大于A的元素个数，可以使用的语句是( B ) A. length(B) - length(find(B<=A)) B. sum(sum(B>A)) C. length(sum(B>A)) D. sum(length(B>A)) 5. 已知str1=’find’, str2=’fund’, str3=’I like you’,有：k1=sum(str1==str2), k2=sum(strrep(str1,’i’,’u’)==str2), k3=findstr(str3,’ke’), 则k1，k2，k3的结果分别为：( B ) A. 3, 3, 5 B.3, 4, 5 C. 4, 3, 5 D. 3, 4, 6 6. 已知a=2:2:8, b=2:5，下面的运算表达式中，出错的为（C） (A) a'*b (B) a .*b (C) a*b (D) a-b 7. 角度x=[30 45 60]，计算其正弦函数的运算为（D） (A) SIN（deg2rad(x)） (B) SIN(x) (C) sin(x) (D) sin(deg2rad(x)) 8. 下面的程序执行后array的值为（A） for k=1:10 if k>6 break; else array(k) = k; end end (A) array = [1, 2, 3, 4, 5, 6] (B) array = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] (C) array =6 (D) array =10. 9．i=2; a=2i; b=2*i; c=2*sqrt(-1); 程序执行后；a, b, c的值分别是多少？（ C） (A)a=4, b=4, c=2.0000i (B)a=4, b=2.0000i, c=2.0000i (C)a=2.0000i, b=4, c=2.0000i (D) a=2.0000i, b=2.0000i, c=2.0000i 10. 求解方程x4-4x3+12x-9 = 0 的所有解（A） (A)1.0000, 3.0000, 1.7321, -1.7321 (B)1.0000, 3.0000, 1.7321i, -1.7321i (C)1.0000i, 3.0000i, 1.7321, -1.7321 (D)-3.0000i, 3.0000i, 1.7321, -1.7321 1.下列哪个变量的定义是不合法的（A） (A) abcd-3 (B) xyz_3 (C) abcdef (D) x3yz 2. 下列哪条指令是求矩阵的行列式的值（C） (A) inv (B) diag (C) det (D) eig 3. 在循环结构中跳出循环，执行循环后面代码的命令为（B） (A) return (B) break (C) continue (D) keyboard 4. 清空Matlab工作空间内所有变量的指令是（C） (A) clc (B) cls (C) clear (D) clf 5. 用round函数四舍五入对数组[2.48 6.39 3.93 8.52]取整，结果为（C） (A) [2 6 3 8] (B) [2 6 4 8] (C) [2 6 4 9] (D) [3 7 4 9]

matlab基本函数的用法

一. Matlab中常见函数基本用法 1.sum (1 )sum(A)A为矩阵得出A矩阵每列的和组成的一个矢量; A为矢量得出A的各元 素之和 (2)sum(diag(A))得矩阵A的对角元素之和 (3)sum(A,dim) A为矩阵，sum(A,1)按列求和;sum(A,2)按行求和 2.max(min) (1)max(A) 若A为矩阵则得出A矩阵每列的最大元素组成的一个矢量 若A为矢量则得出A中最大的元 （2）max(A,B) A与B为同维矩阵得出取A 与B中相同位置元素中较大者组成的新矩阵 （3）max(A,[],dim) max(a,[ ],1),求每列的最大值；max(a,[ ],2)求每行的最大值 3.find （1）find（X）若X为行向量则得出X中所有非零元素所在的位置（按行）若X为列向量或矩阵则得出X中所有非零元素的位置（按列）（2）ind = find(X, k)/ind = find(X,k,'first') 返回前k个非零元的指标ind = find(X,k,'last') 返回后k个非零元的指标 （3）[row,col] = find(X) row代表行指标，col代表列指标 [row,col,val] = find(X) val表示查找到对应位置非零元的值 [row,col] = find(A>100 & A<1000) 找出满足一定要求的元素 4.reshape （1）B = reshape(A,m,n) 把A变成m*n的矩阵 5.sort （1）B = sort(A) 把A的元素按每列从小到大的顺序排列组成新矩阵

（2）B = sort(A,dim) dim=1同（1）； dim=2 把A按每行从小到大的顺序排列组成新矩阵 6.cat （1）C = cat(dim, A, B) dim=1相当于[A;B]；dim=2相当于[A,B] （2）C = cat(dim, A1, A2, A3, A4, ...) 类推（1） 7.meshgrid （1）[X,Y] = meshgrid(x,y) 将向量x和y定义的区域转换成矩阵X和Y，矩阵X的行向量是向量x的简单复制，而矩阵Y的列向量是向量y的简单复制。（2）[X,Y] = meshgrid(x) （1）y=x中情形 8.diag （1）X = diag(v,k) 向量v作为X的第k对角线上的元素X的其他元素为零（2）X = diag(v) （1）中k=0的情况 （2）v = diag(X,k) v为矩阵X的第k对角线的元素组成的列向量 （4）v = diag(X) （3）中k等于零的情况

matlab基本运算与函数

1-1、基本运算与函数 在MATLAB下进行基本数学运算，只需将运算式直接打入提示号（>>）之後，并按入Enter 键即可。例如： >> (5*2+1.3-0.8)*10/25 ans =4.2000 MATLAB会将运算结果直接存入一变数ans，代表MATLAB运算後的答案（Answer）并显示其数值於萤幕上。 小提示： ">>"是MATLAB的提示符号（Prompt），但在PC中文视窗系统下，由於编码方式不同，此提示符号常会消失不见，但这并不会影响到MATLAB的运算结果。 我们也可将上述运算式的结果设定给另一个变数x： x = (5*2+1.3-0.8)*10^2/25 x = 42 此时MATLAB会直接显示x的值。由上例可知，MATLAB认识所有一般常用到的加（+）、减（-）、乘（*）、除（/）的数学运算符号，以及幂次运算（^）。 小提示： MATLAB将所有变数均存成double的形式，所以不需经过变数宣告（Variable declaration）。MATLAB同时也会自动进行记忆体的使用和回收，而不必像C语言,必须由使用者一一指定.这些功能使的MATLAB易学易用，使用者可专心致力於撰写程式，而不必被软体枝节问题所干扰。 若不想让MATLAB每次都显示运算结果，只需在运算式最後加上分号（；）即可，如下例：y = sin(10)*exp(-0.3*4^2); 若要显示变数y的值，直接键入y即可： >>y y =-0.0045 在上例中，sin是正弦函数，exp是指数函数，这些都是MATLAB常用到的数学函数。 下表即为MATLAB常用的基本数学函数及三角函数： 小整理：MATLAB常用的基本数学函数 abs(x)：纯量的绝对值或向量的长度 angle(z)：复数z的相角(Phase angle) sqrt(x)：开平方 real(z)：复数z的实部 imag(z)：复数z的虚部 conj(z)：复数z的共轭复数 round(x)：四舍五入至最近整数 fix(x)：无论正负，舍去小数至最近整数 floor(x)：地板函数，即舍去正小数至最近整数 ceil(x)：天花板函数，即加入正小数至最近整数 rat(x)：将实数x化为分数表示 rats(x)：将实数x化为多项分数展开 sign(x)：符号函数 (Signum function)。 当x<0时，sign(x)=-1； 当x=0时，sign(x)=0; 当x>0时，sign(x)=1。 > 小整理：MATLAB常用的三角函数 sin(x)：正弦函数

Matlab中函数文件prime.m的用法

1.函数文件prime.m function prime(a) if size(a)~=1|round(a)~=a %判断输入的变量是否为一个整数标量 error('The input argument must be a interger number') end for i=2:sqrt(a) %判断a是否能被2到（a-1）之中的任一个数除尽 b=rem(a,i); if b==0 disp(sprintf('%i is not a prime',a)) break; end end if b~=0 %若循环完后，余数最终还不为零，则此数为素数 disp(sprintf('%i is a prime',a)) end 2.函数文件dele.m function [remstr,charnum]=dele(a) %只有ASCII码在32到126之间的整数所对应的符号才能被显示或打印出来,而且该程序只对标量或行、列向量进行操作 for i=1:length(a) if a(i)<32 | a(i)>126| round(a(i))~=a(i) | any(ismember(size(a),1))==0 error('The input must be an interger array,whose element value is between 32 and 126'); end end b=[]; for i=1:length(a) if a(i)~=32 b=[b,a(i)]; end end remstr=char(b); charnum=length(b); 3.函数文件zeronum.m统计一个数值中的0的个数 function c=zeronum(a) if size(a)~=1 error('The input argument must be a scalar') end b=num2str(a); %提问：为什么要这一步？ c=0; for i=1:length(b) if b(i)=='0' %先给不加引号，保存程序后，在命令行用zeronum（700）测试， %结果为何错误

MATLAB中常用的函数

[转]MATLAB 主要函数（一） (2008-05-11 17:09:43) 转载 标签： 分类：ＩＴ matlab 函数 杂谈 MATLAB主要函数指令表（按功能分类）原贴地址:https://www.wendangku.net/doc/351466763.html,/casularm/archive/2007/04/20/1572638.aspx 1常用指令(General Purpose Commands) 1.1通用信息查询(General information) demo 演示程序 help 在线帮助指令 helpbrowser 超文本文档帮助信息 helpdesk 超文本文档帮助信息 helpwin 打开在线帮助窗 info MATLAB 和MathWorks 公司的信息 subscribe MATLAB 用户注册 ver MATLAB 和TOOLBOX 的版本信息 version MATLAB 版本 whatsnew 显示版本新特征 1.2工作空间管理(Managing the workspace) clear 从内存中清除变量和函数 exit 关闭MATLAB load 从磁盘中调入数据变量 pack 合并工作内存中的碎块 quit 退出MATLAB save 把内存变量存入磁盘 who 列出工作内存中的变量名

whos 列出工作内存中的变量细节 workspace 工作内存浏览器 1.3管理指令和函数(Managing commands and functions) edit 矩阵编辑器 edit 打开M 文件 inmem 查看内存中的P 码文件 mex 创建MEX 文件 open 打开文件 pcode 生成P 码文件 type 显示文件内容 what 列出当前目录上的M、MAT、MEX 文件 which 确定指定函数和文件的位置 1.4搜索路径的管理(Managing the seach patli) addpath 添加搜索路径 rmpath 从搜索路径中删除目录 path 控制MATLAB 的搜索路径 pathtool 修改搜索路径 1.5指令窗控制(Controlling the command window) beep 产生beep 声 echo 显示命令文件指令的切换开关 diary 储存MATLAB 指令窗操作内容 format 设置数据输出格式 more 命令窗口分页输出的控制开关 1.6操作系统指令(Operating system commands) cd 改变当前工作目录 computer 计算机类型 copyfile 文件拷贝 delete 删除文件 dir 列出的文件 dos 执行dos 指令并返还结果

Matlab中常用的函数集

sort (排序) xlsread ( exl文件导入) load （txt 文件，mat文件等导入） 附录Ⅰ工具箱函数汇总 Ⅰ.1 统计工具箱函数 表Ⅰ-1 概率密度函数 函数名对应分布的概率密度函数 betapdf 贝塔分布的概率密度函数 binopdf 二项分布的概率密度函数 chi2pdf 卡方分布的概率密度函数 exppdf 指数分布的概率密度函数 fpdf f分布的概率密度函数 gampdf 伽玛分布的概率密度函数 geopdf 几何分布的概率密度函数 hygepdf 超几何分布的概率密度函数normpdf 正态（高斯）分布的概率密度函数lognpdf 对数正态分布的概率密度函数nbinpdf 负二项分布的概率密度函数 ncfpdf 非中心f分布的概率密度函数nctpdf 非中心t分布的概率密度函数 ncx2pdf 非中心卡方分布的概率密度函数poisspdf 泊松分布的概率密度函数 raylpdf 雷利分布的概率密度函数 tpdf 学生氏t分布的概率密度函数unidpdf 离散均匀分布的概率密度函数unifpdf 连续均匀分布的概率密度函数weibpdf 威布尔分布的概率密度函数 表Ⅰ-2 累加分布函数 函数名对应分布的累加函数 betacdf 贝塔分布的累加函数 binocdf 二项分布的累加函数 chi2cdf 卡方分布的累加函数 expcdf 指数分布的累加函数 fcdf f分布的累加函数 gamcdf 伽玛分布的累加函数 geocdf 几何分布的累加函数 hygecdf 超几何分布的累加函数

logncdf 对数正态分布的累加函数 nbincdf 负二项分布的累加函数 ncfcdf 非中心f分布的累加函数 nctcdf 非中心t分布的累加函数 ncx2cdf 非中心卡方分布的累加函数 normcdf 正态（高斯）分布的累加函数 poisscdf 泊松分布的累加函数 raylcdf 雷利分布的累加函数 tcdf 学生氏t分布的累加函数 unidcdf 离散均匀分布的累加函数 unifcdf 连续均匀分布的累加函数 weibcdf 威布尔分布的累加函数 表Ⅰ-3 累加分布函数的逆函数 函数名对应分布的累加分布函数逆函数 betainv 贝塔分布的累加分布函数逆函数 binoinv 二项分布的累加分布函数逆函数 chi2inv 卡方分布的累加分布函数逆函数 expinv 指数分布的累加分布函数逆函数 finv f分布的累加分布函数逆函数 gaminv 伽玛分布的累加分布函数逆函数 geoinv 几何分布的累加分布函数逆函数hygeinv 超几何分布的累加分布函数逆函数logninv 对数正态分布的累加分布函数逆函数nbininv 负二项分布的累加分布函数逆函数ncfinv 非中心f分布的累加分布函数逆函数nctinv 非中心t分布的累加分布函数逆函数 ncx2inv 非中心卡方分布的累加分布函数逆函数icdf norminv 正态（高斯）分布的累加分布函数逆函数poissinv 泊松分布的累加分布函数逆函数 raylinv 雷利分布的累加分布函数逆函数 tinv 学生氏t分布的累加分布函数逆函数unidinv 离散均匀分布的累加分布函数逆函数unifinv 连续均匀分布的累加分布函数逆函数weibinv 威布尔分布的累加分布函数逆函数 表Ⅰ-4 随机数生成器函数

MatLab常用函数大全

1、求组合数 C，则输入： 求k n nchoosek(n,k) 例：nchoosek(4,2) = 6. 2、求阶乘 求n!.则输入： Factorial(n). 例：factorial(5) = 120. 3、求全排列 perms(x). 例：求x = [1,2,3]; Perms(x)，输出结果为： ans = 3 2 1 3 1 2 2 3 1 2 1 3 1 2 3 1 3 2 4、求指数 求a^b：Power(a,b) ; 例：求2^3 ; Ans = pow(2,3) ; 5、求行列式 求矩阵A的行列式：det(A); 例：A=[1 2;3 4] ; 则det(A) = -2 ; 6、求矩阵的转置 求矩阵A的转置矩阵：A’ 转置符号为单引号. 7、求向量的指数 求向量p=[1 2 3 4]'的三次方：p.^3 例： p=[1 2 3 4]' A=[p,p.^2,p.^3,p.^4] 结果为：

注意：在p 与符号”^”之间的”.”不可少. 8、求自然对数 求ln(x)：Log(x) 例：log(2) = 0.6931 9、求矩阵的逆矩阵 求矩阵A 的逆矩阵：inv(A) 例：a= [1 2;3 4]; 则 10、多项式的乘法运算 函数conv(p1,p2)用于求多项式p1和p2的乘积。这里，p1、p2是两个多项式系数向量。 例2-2 求多项式43810x x +-和223x x -+的乘积。 命令如下： p1=[1,8,0,0,-10]; p2=[2,-1,3]; c=conv(p1,p2) 11、多项式除法 函数[q ，r]=deconv(p1，p2)用于多项式p1和p2作除法运算，其中q 返回多项式p1除以p2的商式，r 返回p1除以p2的余式。这里，q 和r 仍是多项式系数向量。 例2-3 求多项式43810x x +-除以多项式223x x -+的结果。 命令如下： p1=[1,8,0,0,-10]; p2=[2,-1,3]; [q,r]=deconv(p1,p2) 12、求一个向量的最大值 求一个向量x 的最大值的函数有两种调用格式，分别是：

matlab常用的几个适应度评价函数

https://www.wendangku.net/doc/351466763.html,/niuyongjie/article/details/1619496 粒子群算法(6)-----几个适应度评价函数 下面给出几个适应度评价函数，并给出图形表示 头几天机子种了病毒，重新安装了系统，不小心把程序全部格式化了，痛哭！！！没办法，好多程序不见了，现在把这几个典型的函数重新编写了，把他们给出来，就算粒子群算法的一个结束吧！痛恨病毒！！！！ 第一个函数：Griewank函数，图形如下所示： 适应度函数如下：（为了求最大值，我去了所有函数值的相反数） function y = Griewank(x) % Griewan函数 % 输入x,给出相应的y值,在x = ( 0 , 0 ,…, 0 )处有全局极小点0. % 编制人： % 编制日期： [row,col] = size(x); if row > 1 error( ' 输入的参数错误 ' ); end y1 = 1 / 4000 * sum(x. ^ 2 );

y2 = 1 ; for h = 1 :col y2 = y2 * cos(x(h) / sqrt(h)); end y = y1 - y2 + 1 ; y =- y; 绘制函数图像的代码如下： function DrawGriewank() % 绘制Griewank函数图形 x = [ - 8 : 0.1 : 8 ]; y = x; [X,Y] = meshgrid(x,y); [row,col] = size(X); for l = 1 :col for h = 1 :row z(h,l) = Griewank([X(h,l),Y(h,l)]); end end surf(X,Y,z); shading interp 第二个函数：Rastrigin函数，图形如下所示：

MATLAB常用的基本数学函数解读

基本运算与函数 下表即为 MATLAB 常用的基本数学函数及三角函数:小整理:MATLAB 常用的基本数学函数 abs(x:纯量的绝对值或向量的长度 angle(z:复数 z 的相角 (Phase angle sqrt(x:开平方 real(z:复数 z 的实部 imag(z:复数 z 的虚部 conj(z:复数 z 的共轭复数 round(x:四舍五入至最近整数 fix(x:无论正负,舍去小数至最近整数 floor(x:地板函数,即舍去正小数至最近整数 ceil(x:天花板函数,即加入正小数至最近整数 rat(x:将实数 x 化为分数表示 rats(x:将实数 x 化为多项分数展开 sign(x:符号函数 (Signum function。 当 x<0时, sign(x=-1; 当 x=0时, sign(x=0; 当 x>0时, sign(x=1。 > 小整理 :MATLAB 常用的三角函数

sin(x:正弦函数 cos(x:馀弦函数 tan(x:正切函数 asin(x:反正弦函数 acos(x:反馀弦函数 atan(x:反正切函数 atan2(x,y:四象限的反正切函数 sinh(x:超越正弦函数 cosh(x:超越馀弦函数 tanh(x:超越正切函数 asinh(x:反超越正弦函数 acosh(x:反超越馀弦函数 atanh(x:反超越正切函数 其他函数: sy msum(f(x , n,a, b 求级数 sum(x : sum([1:10],运行结果一定是 55 sum(A 的用法,是对矩阵 A ,按列计算,得到每一列的和工具箱函数汇总Ⅰ .1统计工具箱函数 表Ⅰ -1概率密度函数

MATLAB部分函数使用方法

读取图像：用imread函数读取图像文件，文件格式可以是TIFF、JPEG、GIF、BMP、PNG 等。比如 >> f = imread('chestxray.jpg'); 读进来的图像数据被保存在变量f中。尾部的分号用来抑制输出。如果图片是彩色的，可以用rgb2gray转换成灰度图： >> f = rgb2gray(f); 然后可以用size函数看图像的大小 >> size(f) 如果f是灰度图像，则可以用下面的命令把这个图像的大小赋给变量M和N >> [M, N] = size(f); 用whos命令查看变量的属性 >> whos f 显示图像：用imshow显示图像 imshow(f, G) 其中f是图像矩阵，G是像素的灰度级，G可以省略。比如 >> imshow(f, [100 200]) 图像上所有小于等于100的数值都会显示成黑色，所有大于等于200的数值都会显示成白色。pixval命令可以用来查看图像上光标所指位置的像素值。 pixval 例如 >> f = imread('rose_512.tif'); >> whos f >> imshow(f) 如果要同时显示两幅图像，可以用figure命令，比如 >> figure, imshow(g) 用逗号可以分割一行中的多个命令。imshow的第二个参数用一个空的中括号： >> imshow(h, []) 可以使动态范围比较窄的图像显示更清楚。 写图像。用imwrite写图像 imwrite(f, 'filename') 文件名必须包括指明格式的扩展名。也可以增加第三个参数，显式指明文件的格式。比如

>> imwrite(f, 'patient10_run1.tif', 'tif') 也可以写成 >> imwrite(f, 'patient10_run1.tif') 还可以有其他参数，比如jepg图像还有质量参数： >> imwrite(f, 'filename.jpg', 'quality', q) q是0到100之间的一个整数。对比不同质量的图像效果。用imfinfo命令可以查看一个图像的格式信息，比如 >> imfinfo bubbles25.jpg 可以把图像信息保存到变量中 >> K = imfinfo('bubbles25.jpg'); >> image_bytes = K.Width * K.Height * K.BitDepth / 8; >> compressed_btyes = K.FileSize; >> compression_ratio = image_bytes / compressed_bytes 数据类型。MA TLAB的数据类型包括： double 双精度浮点 uint8 无符号8位整数 uint16 无符号16位整数 uint32 无符号32位整数 int8 有符号8位整数 int16 有符号16位整数 int32 有符号32位整数 single 单精度 char 字符 logical 逻辑型（二值） 数据类型转换 B = data_class_name(A) 比如 >> C = [1.4 1.5] >> D = uint8(C) 图像类型分为： Intensity image 灰度图 Binary image 二值图 Indexed image 索引图 RGB image 彩色图 在灰度图中每个像素可以是整型、浮点型或者逻辑型。图像类型的像素类型可以转换

matlab的常用函数及函数库

表2.1基本矩阵和矩阵运算（elmat）（d） 基本矩阵zeros全零矩阵（m×n）logspace对数均分向量1×n维数组ones全一矩阵（m×n）Freqspace频率特性的频率区间 rand随机数矩阵（m×n）meshgrid画三维曲面时的X，Y网格randn正态随机数矩阵（m×n）Linspace均分向量（1×n维数组）Eye（n）单位矩阵（方阵）…（竖的）将元素按列取出排成一列 特殊变量和函数ans最近的答案inf Infinity（无穷大）eps浮点数相对精度NaN Not-a-Number（非数）realmax最大浮点实数flops浮点运算次数realmin最小浮点实数computer计算机类型 pi 3.14159235358579inputname输入变量名 i，j虚数单位size多维矩阵的各维长度length一维矩阵的长度 矩阵结构提取和变换cat*链接数组diag提取或建立对角阵 fliplr矩阵左右翻转ind2sub把元素序号变为矩阵下标flipud矩阵上下翻转sub2ind把矩阵下标变为元素序号repmat复制和排成矩阵tril取矩阵的左下三角部分reshape维数重组triu去矩阵的右上三角部分rot90矩阵整体逆时针旋转90° 特殊矩阵company Companion矩阵magic魔方矩阵 gallery Higham测试矩阵pascal Pascal矩阵 hadamard Hadamard矩阵rosser经典的对称特征值测试问题hankle Hankle矩阵Toeplitz Toeplitz矩阵 hilb Hilbert矩阵vander vandermonde矩阵 invhilb Hilbert逆矩阵wilkinson Wilkinson’s特征值测试矩阵表2.5简单的元素群运算 运算式输出结果z=x.*y z=41018 z=x.\y z=4.0000 2.5000 2.0000 z=x.^y z=132729 z=x.^2z=149 z=2.^[x y]z=248163264 注：x=[1,2,3]y=[4,5,6]

常见的matlab的运算函数

三角函数： （）里如果是角度必须是弧度，如果是矩阵的话则为对每个元素执行。cos(),tan()也是一样。 以2为底对数函数：log2（4）=2 以10为底对数函数：log10() 自然对数：log（） 绝对值函数：abs（-2）=2 平方根函数：sqrt（2）=1.41 符号函数：sign（正数）=1 sign(负数)=-1 sign(0)=0 天花板函数ceil（）向大的方向 地板函数floor（）向小的方向 fix()向0的方向 圆整函数round()对数进行4舍5入，负数的话也对对应的正数4舍5入

取模函数 mod（5，3）=2 rem（5，3）=2 区别rem（-5，3）=-2 mod（-5，3）=1 多项式相乘函数：

conv（）deconv（）是相除 取最大和最小函数： max() min() 图中b为行向量或者是列向量 如果（）里为矩阵，则输出每列的最大值（以行向量的形式）如果要求矩阵的最大值max（max（A）） mean（A）输出对应每列的平均值（以行向量的形式）

向量的求和和求积：

整个矩阵的总和sum（sum（A）），求积函数prod同理

多项式乘多项式展开的表达式： [1,1]表示x+1，1 2 1的意思是x^2+2*x+1 复数的函数 real（1+2i）=1(取实部) imag(1+2i)=2(取虚部) abs（1+2i）=2.23 angle（1+2i）=1.107 （在坐标系中对应的角度，即arctan 2=1.107 ）取共轭复数： （1+2i）’=1-2i conj（1+2i）=1-2i dot（a，b）向量的内积 det（a）求行列式的值

MATLAB常用函数说明

MATLAB常用函数 2008-04-23 09:47 matlab常用函数- - 1、特殊变量与常数 ans 计算结果的变量名 computer 确定运行的计算机 eps 浮点相对精度 Inf 无穷大 I 虚数单位 inputname 输入参数名 NaN 非数 nargin 输入参数个数 nargout 输出参数的数目 pi 圆周率 nargoutchk 有效的输出参数数目 realmax 最大正浮点数 realmin 最小正浮点数 varargin 实际输入的参量 varargout 实际返回的参量 操作符与特殊字符 + 加- 减 * 矩阵乘法.* 数组乘（对应元素相乘） ^ 矩阵幂.^ 数组幂（各个元素求幂） \ 左除或反斜杠/ 右除或斜面杠 ./ 数组除（对应元素除） kron Kronecker张量积 : 冒号() 圆括 [] 方括. 小数点 .. 父目录... 继续 , 逗号（分割多条命令）; 分号（禁止结果显示）% 注释! 感叹号 ' 转置或引用= 赋值 == 相等<> 不等于 & 逻辑与| 逻辑或 ~ 逻辑非xor 逻辑异或 2、基本数学函数 abs 绝对值和复数模长 acos,acodh 反余弦，反双曲余弦 acot,acoth 反余切，反双曲余切 acsc,acsch 反余割，反双曲余割 angle 相角 asec,asech 反正割，反双曲正割

secant 正切 asin,asinh 反正弦，反双曲正弦 atan,atanh 反正切，双曲正切 tangent 正切 atan2 四象限反正切 ceil 向着无穷大舍入 complex 建立一个复数 conj 复数配对 cos,cosh 余弦，双曲余弦 csc,csch 余切，双曲余切 cot,coth 余切，双曲余切 exp 指数 fix 朝0方向取整 floor 朝负无穷取整 *** 最大公因数 imag 复数值的虚部 lcm 最小公倍数 log 自然对数 log2 以2为底的对数 log10 常用对数 mod 有符号的求余 nchoosek 二项式系数和全部组合数 real 复数的实部 rem 相除后求余 round 取整为最近的整数 sec,sech 正割，双曲正割 sign 符号数 sin,sinh 正弦，双曲正弦 sqrt 平方根 tan,tanh 正切，双曲正切 3、基本矩阵和矩阵操作 blkding 从输入参量建立块对角矩阵 eye 单位矩阵 linespace 产生线性间隔的向量 logspace 产生对数间隔的向量 numel 元素个数 ones 产生全为1的数组 rand 均匀颁随机数和数组 randn 正态分布随机数和数组 zeros 建立一个全0矩阵colon) 等间隔向量cat 连接数组 diag 对角矩阵和矩阵对角线 fliplr 从左自右翻转矩阵

matlab中函数解析

A a abs 绝对值, 模 acos 反余弦 acosh 反双曲余弦 acot 反余切 acoth 反双曲余切 acsc 反余割 acsch 反双曲余割 all 所有元素均非零则为真 alpha 透明控制 angle 相角 ans 最新表达式的运算结果 any 有非零元则为真 area 面域图 asec 反正割 asech 反双曲正割 asin 反正弦 asinh 反双曲正弦 atan 反正切 atan2 四象限反正切 atanh 反双曲正切 autumn 红、黄浓淡色 axis 轴的刻度和表现 B b bar 直方图 binocdf 二项分布概率 binopdf 二项分布累积概率 binornd 产生二项分布随机数组blanks 空格符号 bode 给出系统的对数频率曲线 bone 蓝色调浓淡色阵 box 坐标封闭开关 break 终止最内循环 brighten 控制色彩的明暗 butter ButterWorth低通滤波器 C c caxis （伪）颜色轴刻度 cd 设置当前工作目录 cdf2rdf 复数对角型转换到实块对角型ceil 朝正无穷大方向取整 cell 创建单元数组 char 创建字符串数组或者将其他类型 变量转化为字符串数组 charfcn Maple函数 Children 图形对象的子对象 clabel 等高线标注 class 判别数据类别 clc 清除指令窗中显示内容 clear 从内存中清除变量和函数 clf 清除当前图形窗图形 close 关闭图形窗 collect 合并同类项 Color 图形对象色彩属性 colorbar 显示色条 colorcube 三浓淡多彩交错色 colordef 定义图形窗色彩 colormap 设置色图 comet 彗星状轨迹图 comet3 三维彗星动态轨迹线图compass 射线图;主用于方向和速度 cond 矩阵条件数 conj 复数共轭 continue 将控制转交给外层的for或while 循环 contour 等高线图 contourf 填色等高线图 conv 卷积和多项式相乘 cool 青和品红浓淡色图 copper 线性变化纯铜色调图 corrcoef 相关系数 cos 余弦 cosh 双曲余弦 cot 余切 coth 双曲余切 cov 协方差矩阵 csc 余割

matlab实验四

本科实验报告 课程名称：Matlab电子信息应用实验项目：M文件的编写 实验地点：电机馆跨越机房专业班级：学号： 学生姓名： 指导教师： 2014年 5 月21 日

一、实验目的 1．学习MATLAB中的关系运算和逻辑运算，掌握它们的表达形式和用法。 2．掌握MATLAB中的选择结构和循环结构。 3．学会用MATLAB进行M文件的编写和调用。 二、预备知识 1．关系和逻辑运算 关系运算符用来完成关系运算，在控制程序流程方面有着极为重要的作用。MATLAB常用的关系符有：<、>、<=小于或等于、>=大于或等于、==等于、～=不等于。 关系运算符可以用来比较两个数值，若所描述的关系成立，则结果为1，表示逻辑真，反之，若所描述的关系不成立，结果为0，表示逻辑假。 MATLAB中的逻辑运算符有＆与、∣或、～非。 逻辑运算法则 A B A ＆B A ∣B xor(A ,B) ～ A 000 0 0 1 010 1 1 1 100 1 1 0 11 1 1 0 0 2．选择结构 if语句和switch语句 if语句的一般形式如下： if A1 %表达式1 B1 %命令1 else if A2 %表达式2 B2 %命令2 else B3 %命令3 end switch语句的一般结构如下：

switch a %读入一个语句 case A1 %情况1 B1 %命令1 case A2 B2 case .… … other case %其余情况 Bn %最后一个命令 3．循环结构 for 语句一般用于循环次数已知的情况，而while 语句一般用于循环次数未知的情况。 for 语句的格式为：for 变量=表达式 命令1 命令2 … end while 语句的格式为：while 表达式 命令 end 1． 实验内容与步骤 1． 创建一个矩阵，用函数all 和any 作用于该矩阵，比较结果。 2． 编写一个switch 语句，判断输入数的奇偶性。 3． 编写一个程序画出下列分段函数所表示的曲面，并用M 文件存储。 ()??? ? ???-≤+≤+<->+=+-------1 e 5457.011e 7575.01e 5457.0,215.175.375.0216215.175.375.0211 212221 2212 122x x x x x x x x f x x x x x x x x 步骤1 打开MATLAB 的M 文件编辑器file/new/M-file,编写以下内容： %first.m This is my first example a=2;b=2;